{{ post.content | xml_escape }}

Up to now only linear interpolation was performed and this significantly lowered the accuracy if a higher order solver was used.

I then implemented a series of interpolation function:

**linear_interpolation**:

Given the time span \(t=[t_0, t_1]\) and the function values \(x=[x_0, x_1]\), it returns the linear interpolation value \(x_{out}\) at the point \(t_{out}\).`x_out = linear_interpolation (t, x, t_out)`

**quadratic_interpolation**:

Given the time span \(t=[t_0, t_1]\), the function values \(x=[x_0, x_1]\) and the derivative of the function at the point \(x_0\), it returns the quadratic interpolation value \(x_{out}\) at the point \(t_{out}\).`x_out = quadratic_interpolation (t, x, der, t_out)`

**hermite_cubic_interpolation**:

Given the time span \(t=[t_0, t_1]\), the function values \(x=[x_0, x_1]\) and the derivatives of the function at both points \(x_0\) and \(x_1\), it returns the 3rd order approximation \(x_{out}\) at the point \(t_{out}\) by performing Hermite interpolation.`x_out = hermite_cubic_interpolation (t, x, der, t_out)`

**hermite_quartic_interpolation**:

Given the time span \(t=[t_0, t_1]\), the function values \(x=[x_0, x_{1/2}, x_1]\) (where \(x_{1/2}\) is the value of the function at the time \(t_0+dt/2\)) and the derivatives of the function at the extremes \(x0\) and \(x1\), it returns the 4th order approximation \(x_{out}\) at the point \(t_{out}\) by performing Hermite interpolation.`x_out = hermite_quartic_interpolation (t, x, der, t_out)`

**dorpri_interpolation**:

This interpolation method is specific for the Dormand-Prince Runge-Kutta scheme. Given the time span \(t=[t_0, t_1]\), the function value \(x=x_0\) and the vector \(k\) with the function evaluations required in the Dormand-Prince method, it returns the 4th order approximation \(x_{out}\) at the point \(t_{out}\). For more information on the method have a look at`x_out = dorpri_interpolation (t, x, k, t_out)`

*Hairer, Noersett, Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (pag. 191-193)*.**hermite_quintic_interpolation**:

Given the time span \(t=[t_0, t_1]\), the function values \(x=[x_0, x_{1/2}, x_1]\) and the derivatives of the function at each point, it returns the 5th order approximation \(x_{out}\) at the point \(t_{out}\) by performing Hermite interpolation.`x_out = hermite_quintic_interpolation (t, x, der, t_out)`

% if next tspan value is caught, update counter

if( (z(end) == tspan(counter)) || (abs (z(end) - tspan(counter)) / ...

(max (abs (z(end)), abs (tspan(counter)))) < 8*eps) )

counter++;

% if there is an element in time vector at which the solution is required

% the program must compute this solution before going on with next steps

elseif( vdirection*z(end) > vdirection*tspan(counter) )

% initializing counter for the following cycle

i = 2;

while ( i <= length (z) )

% if next tspan value is caught, update counter

if( (counter <= k) && ...

( (z(i) == tspan(counter)) || (abs (z(i) - tspan(counter)) / ...

(max (abs (z(i)), abs (tspan(counter)))) < 8*eps)) )

counter++;

endif

% else, loop until there are requested values inside this subinterval

while((counter <= k) && (vdirection*z(i) > vdirection*tspan(counter)))

% choose interpolation scheme according to order of the solver

switch order

case 1

u_interp = linear_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], tspan(counter));

case 2

if (~isempty (k_vals))

der = k_vals(1);

else

der = feval (func, z(i-1) , u(:,i-1), options.vfunarguments{:});

endif

u_interp = quadratic_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], der, tspan(counter));

case 3

% only ode23 - use k_vals

u_interp = hermite_cubic_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], [k_vals(:,1) k_vals(:,end)], tspan(counter));

case 4

% if ode45 used without local extrapolation this function doesn't require a new function evaluation.

u_interp = dorpri_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], k_vals, tspan(counter));

case 5

% ode45 with Dormand-Prince scheme:

% 4th order approximation of y in t+dt/2 as proposed by Shampine in Lawrence, Shampine, "Some Practical Runge-Kutta Formulas", 1986.

u_half = u(:,i-1) + 1/2*dt*((6025192743/30085553152)*k_vals(:,1) + (51252292925/65400821598)*k_vals(:,3) - (2691868925/45128329728)*k_vals(:,4) + (187940372067/1594534317056)*k_vals(:,5) - (1776094331/19743644256)*k_vals(:,6) + (11237099/235043384)*k_vals(:,7));

u_interp = hermite_quartic_interpolation ([z(i-1) z(i)], [u(:,i-1) u_half u(:,i)], [k_vals(:,1) k_vals(:,end)], tspan(counter));

% it is also possible to do a new function evaluation and the quintic hermite interpolator

%f_half = feval (func, t+1/2*dt, u_half, options.vfunarguments{:});

%u_interp = hermite_quintic_interpolation ([z(i-1) z(i)], [u(:,i-1) u_half u(:,i)], [k_vals(:,1) f_half k_vals(:,end)], tspan(counter));

otherwise

warning ('high order interpolation not yet implemented: using cubic iterpolation instead');

der(:,1) = feval (func, z(i-1) , u(:,i-1), options.vfunarguments{:});

der(:,2) = feval (func, z(i) , u(:,i), options.vfunarguments{:});

u_interp = hermite_cubic_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], der, tspan(counter));

end

% add the interpolated value of the solution

u = [u(:,1:i-1), u_interp, u(:,i:end)];

% add the time requested

z = [z(1:i-1);tspan(counter);z(i:end)];

% update counters

counter++;

i++;

endwhile

% if new time requested is not out of this interval

if ((counter <= k) && (vdirection*z(end) > vdirection*tspan(counter)))

% update the counter

i++;

else

% else, stop the cycle and go on with the next iteration

i = length (z) + 1;

endif

endwhile

endif

It is important to notice that:

- The 1st order approximation doesn't require any additional function evaluation.
- The 2nd order approximation may require the evaluation of the function at the current time. This can be avoided if the stepper already returns that value.
- The only 3rd order solver implemented is
**ode23**. The 3rd order approximation exploits the Runge-Kutta \(k\) values to avoid further function evaluations. - There are no 4th order schemes as yet implemented. However if ones were to use
**ode45**without local extrapolation then the**dorpri_interpolation**function can be used to obtain a 4th order approximation without any additional function evaluation. For any other 4th order scheme the**hermite_quartic_interpolation**function can be used. - For the 5th order method
**ode45,**Shampine proposes to obtain a 4th order approximation at the middle point and to use quartic interpolation. It is however possible to directly do quintic interpolation but this require an additional function evaluation without (according to Shampine) a significant improvement. - For the higher order solvers (
**ode78**), a suitable interpolator has not yet been implemented.

by Jacopo Corno (noreply@blogger.com) at August 27, 2014 03:00 AM

During this week I have been reorganizing all the code, docs and tests in a better way for integrating into Octave. As Rik kindly suggested, I decided to organize things this way:

- Inside libinterp/dldfcn directory I have created two files, __ichol__.cc and __ilu__.cc

- Within those files there are the dld functions that implements the each of the algorithms. They are ment to be built-in functions and follows the __foo__.cc naming convention.

* **__ilu__.cc:** contains __ilu0__() , __iluc__() and __ilutp__()

* **__ichol__.cc:** contains __ichol0__() and __icholt__().

- I have moved all the tests from .cc files to .m scripts so no tests are performed for built-in functions.

The code is ready to be pulled from my repo to be reviewed :

https://edu159@bitbucket.org/edu159/octave-edu159

It is interesting to show how preconditioning techniques can improve the convergency of some iterative solvers. In this case I am running a Matlab example using the Poisson matrix (that is positive definite) obtained with gallery() function. The scritp:

A = gallery ('Poisson', N);

b = ones (size (A, 1), 1);

tol = 1e-6; maxit = 100;

[x0, fl0, rr0, it0, rv0] = pcg (A, b, tol, maxit);

L1 = ichol (A);

[x1, fl1, rr1, it1, rv1] = pcg(A, b, tol, maxit, L1, L1');

opts.type = 'nofill'; opts.michol = 'on';

L2 = ichol (A, opts);

e = ones (size (A, 2), 1);

norm(A * e - L2 * (L2' * e))

[x2, fl2, rr2, it2, rv2] = pcg (A, b, tol, maxit, L2, L2');

semilogy (0:maxit, rv0 ./ norm (b), 'b.');

hold on;

semilogy (0:it1, rv1 ./ norm(b), 'r.');

semilogy (0:it2, rv2 ./ norm(b), 'k.');

xlabel ('iterations');

ylabel ('error');

legend ('No Preconditioner', 'IC(0)', 'MIC(0)');

Octave |

Matlab |

L3 = ichol(A, struct('type', 'ict', 'droptol', 1e-1));

[x3, fl3, rr3, it3, rv3] = pcg (A, b, tol, maxit, L3, L3');

L4 = ichol (A, struct ('type', 'ict', 'droptol', 1e-2));

[x4, fl4, rr4, it4, rv4] = pcg (A, b, tol, maxit, L4, L4');

L5 = ichol (A, struct ('type', 'ict', 'droptol', 1e-3));

[x5, fl5, rr5, it5, rv5] = pcg (A, b, tol, maxit, L5, L5');

figure; semilogy (0:maxit, rv0 ./ norm (b), 'b-', 'linewidth', 2);

hold on;

semilogy (0:it3, rv3 ./ norm(b), 'b-.', 'linewidth', 2);

semilogy (0:it4, rv4 ./ norm(b), 'b--', 'linewidth', 2);

semilogy (0:it5, rv5 ./ norm(b), 'b:', 'linewidth', 2);

ylabel ('error');

xlabel ('iterations');

legend ('No Preconditioner', 'ICT(1e-1)', 'ICT(1e-2)', ...

'ICT(1e-3)', 'Location', 'SouthEast');

Octave |

Matlab |

As it can be seen Octave plots are the same as Matlab's ones. Both lead to a decrease in the number of steps upt to convergence of the pcg method. ILU algorithms could also have been used here, but due to the simetry of the problem matrix ICHOL is faster.

Regards,

Eduardo

by Eduardo (noreply@blogger.com) at August 18, 2014 08:19 PM

The input and output arguments of the steppers have then be modified. As an example here is the

`function varargout = runge_kutta_23 (f, t, x, dt, varargin)`

options = varargin{1};

k = zeros (size (x, 1), 4);

if (nargin == 5) % only the options are passed

k(:,1) = feval (f, t , x, options.vfunarguments{:});

elseif (nargin == 6) % both the options and the k values are passed

k(:,1) = varargin{2}(:,end); % FSAL property

endif

k(:,2) = feval (f, t + (1/2)*dt, x + dt*(1/2)*k(:,1), options.vfunarguments{:});

k(:,3) = feval (f, t + (3/4)*dt, x + dt*(3/4)*k(:,2), options.vfunarguments{:});

%# computing new time and new values for the unkwnowns

varargout{1} = t + dt; %t_next

varargout{2} = x + dt.*((2/9)*k(:,1) + (1/3)*k(:,2) + (4/9)*k(:,3)); % return the 3rd order approximation x_next

%# if the estimation of the error is required

if (nargout >= 3)

%# new solution to be compared with the previous one

k(:,4) = feval (f, t + dt, varargout{2}, options.vfunarguments{:});

varargout{3} = x + dt.*((7/24)*k(:,1) + (1/4)*k(:,2) + (1/3)*k(:,3) + (1/8)*k(:,4)); %x_est

varargout{4} = k;

endif

endfunction

And the call within the solver becomes:

`[s, y, y_est, k_vals] = stepper (func, z(end), u(:,end), dt, options, k_vals);`

where

This implementation will reduce the number of function evaluation for each step.

Furthermore, after some tests in MATLAB, the return values for the solution and the estimate in the runge_kutta_23 and runge_kutta_45 steppers have been swapped to automatically perform local extrapolation. The MATLAB functions are in fact of order 3 and 5 respectively.

by Jacopo Corno (noreply@blogger.com) at August 18, 2014 07:52 AM

**odeset**and**odeget**functions have been slightly modified to be compliant with MATLAB. Each MATLAB option is present and all the options are tested. The coding style has been adapted to the GNU-Octave standard.**ode_struct_value_check**: this function has been introduced by Roberto in addition to**odepkg_structue_check**. The relation between the two functions has to be clarified: in particular it is necessary to understand if it is really necessary to have two different functions or one is sufficient.

- The
**runge_kutta_78**stepper has been implemented. - Two 4th order steppers have been implemented:
**runge_kutta_45_dopri**(*Dormand-Prince*coefficients) and**runge_kutta_45_fehlberg**(*Fehlberg*coefficients).

**ode78**solver has been updated to the new structure. It now exploits the**runge_kutta_78**stepper.- A series of tests has been added to each solver to check all the functionalities and the all options. This has made me possible to detect some bugs that have been corrected. In particular the adaptive timestep evaluation had some issues that lead to the use of too short timesteps. This has been corrected and now the algorithm proposed in [1] is used.
- Furthermore the current implementation uses linear interpolation to evaluate the solution at the user specified times. This leads to a considerable loss in accuracy and is not consistent with MATLAB (which guarantees the same order of accuracy of the scheme used). In [1] different methods are proposed for the
*dense output*: these will be used as a reference for the implementation of a better solution. - In the released version of
**odepkg**some of the solvers perform*local extrapolation*, that is the higher-order estimate is chosen as the solution. With the new stepper structure, as it is now, this choice effects all the solvers. It have to be decided whether to perform it or not (MATLAB doesn't seem to do it, thus I suggest to avoid it). - MATLAB implementation of
**ode45**uses the*Dormand-Prince*(DP) coefficients. In the released version of**odepkg**there exits two solvers:**ode45**that uses the*Fehlberg*coefficients and**ode54**that uses the DP coefficients. To be consistent with MATLAB,**ode45**now uses the DP method. This makes the**runge_kutta_45_fehlberg**stepper and the**ode54**solver, as it is now, redundant. Either their elimination or a change of the solver might be considered. However one of the advantages of DP coefficients is the possibility to reuse the last function evaluation at a given step as the first evaluation of the subsequent one. This is not easily done with the stepper structure introduced by Roberto.

**InitialStep**option has been modified to be MATLAB compatible (it must be a__positive__scalar).**RelTol**defalut value has been changed to**1e-3**instead of**1e-6**to be MATLAB compatible.**MaxStep**option has been implemented.**NormControl**option has been implemented.

- Clarify the relation between
**ode_struct_value_check**and**odepkg_structue_check**. - Decide if
*local extrapolation*has to be used or not. My opinion (and the current implementation) is to avoid it to be compliant to what MATLAB seems to be doing. - Solve the
**dense output**problem in a way that guarantees the consistency with MATLAB. - Consider if it's possible to reduce the number of function evaluation for the Dormand-Prince stepper (
**ode45**) and the Bogacki-Shampine stepper (**ode23**) exploiting the FSAL property (first same as last). - Decide if in the future releases of
**odepkg****ode54**has to be removed or maybe changed to become a*Fehlberg*solver.

[1] E. Hairer, S.P. N{\o}rsett, G. Wanner, Solving Ordinary Differential Equations, 1993, Springer.

by Jacopo Corno (noreply@blogger.com) at August 13, 2014 02:28 AM

As said in my previous post, a missing feature in **fem-fenics** was the marking of subdomains. Indeed, I proposed an example that needed a file generated with a run of the corresponding Python code, which is not, honestly, the best approach. In order to address this issue, these days I have implemented a new class, subdomain, which can be used to mark mesh entities. In the following I will describe how to use this new functionality. Here is the code:

pkg load fem-fenics msh

ufl start Subdomains

ufl fe = FiniteElement "(""CG"", triangle, 2)"

ufl u = TrialFunction (fe)

ufl v = TestFunction (fe)

ufl

ufl a0 = Coefficient (fe)

ufl a1 = Coefficient (fe)

ufl g_L = Coefficient (fe)

ufl g_R = Coefficient (fe)

ufl f = Coefficient (fe)

ufl

ufl a = "inner(a0*grad(u), grad(v))*dx(0) + inner(a1*grad(u), grad(v))*dx(1)"

ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1)

ufl end

# Create mesh and define function space

x = y = linspace (0, 1, 65);

[msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1));

V = FunctionSpace ("Subdomains", msh);

# Define boundary conditions

bc1 = DirichletBC (V, @(x, y) 5.0, facets, 2);

bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4);

# Define problem coefficients

a0 = Constant ("a0", 1.0);

a1 = Constant ("a1", 0.01);

g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2));

g_R = Constant ("g_R", 1.0);

f = Constant ("f", 1.0);

# Define subdomains - Here are the edits #

domains = MeshFunction ("dx", msh, 2, 0);

obstacle = SubDomain (@(x,y) (y >= 0.5) && (y <= 0.7) && ...

(x >= 0.2) && (x <= 1.0), false);

domains = mark (obstacle, domains, 1);

# Define variational form

a = BilinearForm ("Subdomains", V, V, a0, a1, domains);

L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains);

# Assemble system

[A, b] = assemble_system (a, L, bc1, bc2);

sol = A \ b;

u = Function ("u", V, sol);

# Save solution in VTK format

save (u, "subdomains");

# Plot solution

[X, Y] = meshgrid (x, y);

U = u (X, Y);

surf (X, Y, U);

As you can see, it is basically the same as in the previous post, except the line used to import the meshfunction. I wrote in the corresponding comment where the edits are to be found. Now the workflow comprises these steps: first of all, a meshfunction needs to be created, then a subdomain, in the end we should mark cells.

The call to MeshFunction is something new, since it is now possible to instantiate a meshfunction given a mesh, the required topological dimension and the value to initialise it with. Moreover, the optional label "dx" means that it can be used in calls to BilinearForm or LinearForm to supply markers for subsets of the integration domains. In the example, this instruction returns a meshfunction of dimension 2, which means it holds values associated with each triangle in the mesh, initialised to be 0 in every entry.

The subsequent instruction, instead, defines a subdomain, passing as arguments a function handle and a logical flag. The former will be the override of the dolfin::SubDomain::inside method, so it must return true for entities contained in the subset and false otherwise. In facts it checks whether the coordinates are inside the 2-interval defining the obstacle. The latter, instead, can be used to ask for a boundary subdomain, when set to true.

At last, mark is called to set the entries corresponding to cells inside the subdomain to 1, so that the returned meshfunction now represents the obstacle: after these lines, the variable named domains assumes value 1 on cells inside the obstacle region and 0 outside. Thus, it is now possible to solve a problem whose formulation entails subdomains entirely using fem-fenics.

by Eugenio Gianniti (noreply@blogger.com) at August 12, 2014 10:54 PM

It's been quite long since I posted here due to some personal situations. Anyway to sum up: I have finished **ilu **and **ichol** functions as I have planned in the beginning with great results.

Things done after mid-term evaluation:

Before going into the details of the algorithms' implementation, I want to point out some details about how ichol behave in MATLAB.

**Details of implementation**

** -->src/ichol0.cc **

In this file is located the implementation of ICHOL(0) algorithm. The zero-pattern of the output matrix is the same as the input one so it is known from the beginning how much memory is needed to be allocated. The milu = ['on'|'off'] parameter indicates whether the dropped elements are added to the pivot or not (that keeps the colum sumation).

I will show two examples, one that corresponds to a big matrix with a very low density and the one that used Kai last year in his blog.

**Example 1:**

A = gallery ('poisson', 500);

size (A)

ans =

250000 250000

tic; L = ichol (A); toc;

Elapsed time is 0.031718 seconds.

density = nnz (A) / (size (A)(1))^2

density = 1.9968e-05

norm (A - L*L', 'fro') / norm (A, 'fro')

ans = 0.0924207846384523

norm(A-(L*L').*spones(A),'fro')./norm(A,'fro')

ans = 2.28617974245061e-17

It can be seen that the product L*L' is quite different from A, but the product L*L' will match A on its pattern (that is expected for the ICHOL(0) algorithm. The execution time is just given to give an idea of how fast the code is. It is executed in a i7 2.4GHz.

**Example 2:**

This example is taken from that post, written by Kai the past year. He faced problems with the michol option, obtaining different results from Matlab.

input:

A = sparse (A);

opts.michol = 'on';

L = ichol (A, opts);

Octave:

ans =

0.60828 0.00000 0.00000 0.00000

-0.08220 0.32014 0.00000 0.00000

-0.08220 0.00000 0.32014 0.00000

-0.11508 -0.18573 -0.18573 0.34607

Matlab:

ans =

0.6083 0 0 0

-0.0822 0.3201 0 0

-0.0822 0 0.3201 0

-0.1151 -0.1857 -0.1857 0.3461

Works fine.

**-->src/icholt.cc **

This file contains the implementation of ICHOLT algorithm. In this case the final structure of the output matrix is unknown. Therefore, a policy should be adopted for allocating memory. After trying different ways of doing that I end up using that one:** **

** ** // max_len is the maximun length of ridx and data arrays for the output sparse matrix.

max_len = sm.nnz ();

max_len += (0.1 * max_len) > n ? 0.1 * max_len : n;

What is done here is just to increment 10% of the actual size of the ridx and data internal arrays of the output sparse matrix. But only if that amount is larger than the dimension of the input matrix (n). In other case the increment in size is just n. That policy seems to work very well in every case I tested and do not slow down the process at all due to reallocations.

**Example 3:**

icholt accepts a parameter for controling the sparsity of the ouput matrix called**droptol**. If droptol = 0 then the complete factorization takes place. If we increase that value the output matrix will become more sparse as more elements will be dropped. Taking the same matrix than in example 1:

A = gallery ('poisson', 500);

opts.type= 'ict'

% Complete factorization

opts.droptol = 0;

tic;L = ichol(A, opts);toc;

Elapsed time is** 46.0734** seconds.

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =**7.8595e-16**

% droptol = 1e-2

opts.droptol=1e-2

tic;L = ichol(A, opts);toc;

Elapsed time is**0.0650802** seconds.

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =**0.016734**

% droptol = 1e-3

opts.droptol=1e-3

tic;L = ichol(A, opts);toc;

Elapsed time is**0.183416** seconds.

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =**0.0021773**

% droptol = 1e-4

opts.droptol=1e-4

tic;L = ichol(A, opts);toc;

Elapsed time is**0.589693** seconds.

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =**2.4820e-04**

As it can be seen, the higher the droptol parameter is, the sparser the matrix become. That lead to less execution times but on the other hand a higher error is obtained in the factorization. The complete factorization obviously have practically no error. Cool.

**Location of source files inside Octave core**

Now I've finished with the development of the algorithms, the final step is to integrate them into Octave core. For doing so I will create a subrepo of the default Octave repository and add the files. I have chosen the location for the functions looking at the last year repository Kai set.

**Location:**

*libinterp/dldfcn:* ilutp.cc ilu0.cc iluc.cc ichol0.cc icholt.cc

**scripts/sparse: **ilu.m ichol.m

That is just a sugestion and should be revised and accepted by the maintainers.

**Future contributions**

There is a week left that I want to use it to start (and hopefully finish) the development of**sprandsym** extra parameters that Matlab have but Octave does not. As I submitted in the past a changeset for a similar functionality in sprand and sprandn, it will be much easier to implement for me.

Also I am interested in developing some sparse linear solvers like**minres** and **lsqr** that Octave lacks. They are tightly related to the preconditioners I have been working on, and would be nice if they could be assigned to me for developing them.

Regards,

Eduardo

Things done after mid-term evaluation:

- Implementing
**ICHOLT**and**ICHOL0**algorithms. - Fixing several bugs in ILU algorithms and introducing some enhancements for big sparse matrices with verly low densities.

- src/icholt.cc
- src/ichol0.cc
- ichol.m

- https://edu159@bitbucket.org/edu159/gsoc2014-edu15

Before going into the details of the algorithms' implementation, I want to point out some details about how ichol behave in MATLAB.

- In the real case the matrix must be symetric positive definite. In the complex case the input matrix must be hermitian. That means: diagonal elements of the input and output matrix have to be
**non-zero**,**positive**and**real**values. So that, at each iteration those conditions have to be fullfilled. - If ichol is called just as L = ichol (A), Matlab ignores complex numbers and only work with their real part. Using L = ichol (A, setup) call, complex numbers are considered. Seriusly I do not understand why they do that and I have not followed that behaviour. Anyway if to be 100% compatible I must change that, it would be only a line of code extra.

In this file is located the implementation of ICHOL(0) algorithm. The zero-pattern of the output matrix is the same as the input one so it is known from the beginning how much memory is needed to be allocated. The milu = ['on'|'off'] parameter indicates whether the dropped elements are added to the pivot or not (that keeps the colum sumation).

I will show two examples, one that corresponds to a big matrix with a very low density and the one that used Kai last year in his blog.

A = gallery ('poisson', 500);

size (A)

ans =

250000 250000

tic; L = ichol (A); toc;

Elapsed time is 0.031718 seconds.

density = nnz (A) / (size (A)(1))^2

density = 1.9968e-05

norm (A - L*L', 'fro') / norm (A, 'fro')

ans = 0.0924207846384523

norm(A-(L*L').*spones(A),'fro')./norm(A,'fro')

ans = 2.28617974245061e-17

It can be seen that the product L*L' is quite different from A, but the product L*L' will match A on its pattern (that is expected for the ICHOL(0) algorithm. The execution time is just given to give an idea of how fast the code is. It is executed in a i7 2.4GHz.

This example is taken from that post, written by Kai the past year. He faced problems with the michol option, obtaining different results from Matlab.

input:

A = [ 0.37, -0.05, -0.05, -0.07;

-0.05, 0.116, 0.0, -0.05;

-0.05, 0.0, 0.116, -0.05;

-0.07, -0.05, -0.05, 0.202];

A = sparse (A);

opts.michol = 'on';

L = ichol (A, opts);

Octave:

ans =

0.60828 0.00000 0.00000 0.00000

-0.08220 0.32014 0.00000 0.00000

-0.08220 0.00000 0.32014 0.00000

-0.11508 -0.18573 -0.18573 0.34607

Matlab:

ans =

0.6083 0 0 0

-0.0822 0.3201 0 0

-0.0822 0 0.3201 0

-0.1151 -0.1857 -0.1857 0.3461

Works fine.

This file contains the implementation of ICHOLT algorithm. In this case the final structure of the output matrix is unknown. Therefore, a policy should be adopted for allocating memory. After trying different ways of doing that I end up using that one:

max_len = sm.nnz ();

max_len += (0.1 * max_len) > n ? 0.1 * max_len : n;

What is done here is just to increment 10% of the actual size of the ridx and data internal arrays of the output sparse matrix. But only if that amount is larger than the dimension of the input matrix (n). In other case the increment in size is just n. That policy seems to work very well in every case I tested and do not slow down the process at all due to reallocations.

icholt accepts a parameter for controling the sparsity of the ouput matrix called

A = gallery ('poisson', 500);

opts.type= 'ict'

% Complete factorization

opts.droptol = 0;

tic;L = ichol(A, opts);toc;

Elapsed time is

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =

% droptol = 1e-2

opts.droptol=1e-2

tic;L = ichol(A, opts);toc;

Elapsed time is

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =

% droptol = 1e-3

opts.droptol=1e-3

tic;L = ichol(A, opts);toc;

Elapsed time is

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =

% droptol = 1e-4

opts.droptol=1e-4

tic;L = ichol(A, opts);toc;

Elapsed time is

norm (A - L*L', 'fro') / norm (A, 'fro')

ans =

As it can be seen, the higher the droptol parameter is, the sparser the matrix become. That lead to less execution times but on the other hand a higher error is obtained in the factorization. The complete factorization obviously have practically no error. Cool.

Now I've finished with the development of the algorithms, the final step is to integrate them into Octave core. For doing so I will create a subrepo of the default Octave repository and add the files. I have chosen the location for the functions looking at the last year repository Kai set.

That is just a sugestion and should be revised and accepted by the maintainers.

There is a week left that I want to use it to start (and hopefully finish) the development of

Also I am interested in developing some sparse linear solvers like

Regards,

Eduardo

by Eduardo (noreply@blogger.com) at August 11, 2014 06:47 AM

As you may recall from my last post, for DirichletBC to work in parallel runs I had to implement a new class, meshfunction. However it was still quite unfinished, with no way for the user to create one, except extracting it from a mesh produced by the **msh** package, no description to display, no way to save it. These days I have been tackling this issue: while at it I wondered what one could do with meshfunction and found out that they can come in handy when you are dealing with obstacles.

At this link you can find a detailed explanation of the problem. It is a Poisson equation with variable diffusion coefficient on the unit square. Precisely, on [0.2, 1]x[0.5, 0.7] its value is 0.01, otherwise it is 1. The mentioned subset is the obstacle to diffusion, so we study its effect applying*u* = 0 on the *y* = 0 edge and *u* = 5 on *y* = 1. Here is the **fem-fenics** code:

In the beginning there is the now familiar ufl block. As you might have noticed, subscripted measures appear in the definition of the bilinear form*a* and of the linear functional *L*. This is UFL notation for the integration on specific subsets of the computational domain. For instance, dx(1) is an integral over the subdomain marked with label 1, while ds(3) is an integral over the exterior edges marked with label 3. A third possibility, even if not used in this example, is to use dS for integrals on interior facets, which could be of use for interior penalty methods. Going back to the example, you can see that markers are used to enforce non-homogeneous Neumann conditions on the side edges and to assign the proper coefficient on the two subdomains.

After defining the problem in UFL language, there are instructions to define the mesh, the function space, the essential boundary conditions and all the coefficients involved. All such lines come from fem-fenics before this summer or have been described in my previous posts, so I will not cover them in detail. The same applies for the assembly, solve and all the output in the end of the script. The only note is that the very last lines will error out in parallel runs: point-wise evaluations in**DOLFIN** can be performed only on local cells, but with meshgrid we are providing to every process the whole domain.

In between there are my latest efforts. At first, the brand new MeshFunction. With this, providing a mesh and a file name you can import a dolfin::MeshFunction. In this case it was saved in the XDMF format, here you can find the files needed to execute the script. DOLFIN uses this format for parallel input/output. It comprises a .h5 file storing data and a .xdmf with metadata useful to read the other one. The optional first argument is a string identifying the role of the returned meshfunction in the variational problem. In this case, with "dx" it will be searched for markers of the integrals on cells. All the previously mentioned measures are available, and "ds" is automatically attached to the meshfunction returned by Mesh. In the example this behaviour is exploited for the measure on edges.

Afterwards, the mesh functions are passed as arguments to BilinearForm and LinearForm, so that the markers are available to assemble the system. In addition to the usual parameters, such as the name of the imported UFL problem, the function spaces and the coefficients, it is now possible to provide mesh functions properly labeled and they will be used.

Currently fem-fenics allows for easily marking subdomains and exterior edges copying markers from the PDEtool representation returned by the functions of the**msh** package, which makes it quite tricky to properly identify the obstacle in the example. The approach used in the python interface to DOLFIN entails subclassing dolfin::Subdomain with the proper implementation of the inside method, then use an object of the derived class to mark a dolfin::MeshFunction. This could be an interesting feature to implement in the future also in fem-fenics.

At this link you can find a detailed explanation of the problem. It is a Poisson equation with variable diffusion coefficient on the unit square. Precisely, on [0.2, 1]x[0.5, 0.7] its value is 0.01, otherwise it is 1. The mentioned subset is the obstacle to diffusion, so we study its effect applying

`pkg load fem-fenics msh`

ufl start Subdomains

ufl fe = FiniteElement "(""CG"", triangle, 2)"

ufl u = TrialFunction (fe)

ufl v = TestFunction (fe)

ufl

ufl a0 = Coefficient (fe)

ufl a1 = Coefficient (fe)

ufl g_L = Coefficient (fe)

ufl g_R = Coefficient (fe)

ufl f = Coefficient (fe)

ufl

ufl a = "inner(a0*grad (u), grad (v))*dx(0) + inner(a1*grad (u), grad (v))*dx(1)"

ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1)

ufl end

# Create mesh and define function space

x = y = linspace (0, 1, 65);

[msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1));

V = FunctionSpace ("Subdomains", msh);

# Define boundary conditions

bc1 = DirichletBC (V, @(x, y) 5.0, facets, 2);

bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4);

# Define problem coefficients

a0 = Constant ("a0", 1.0);

a1 = Constant ("a1", 0.01);

g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2));

g_R = Constant ("g_R", 1.0);

f = Constant ("f", 1.0);

# Define subdomains

domains = MeshFunction ("dx", msh, "cells.xdmf");

# Define variational form

a = BilinearForm ("Subdomains", V, V, a0, a1, domains);

L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains);

# Assemble system

[A, b] = assemble_system (a, L, bc1, bc2);

sol = A \ b;

u = Function ("u", V, sol);

# Save solution in VTK format

save (u, "subdomains");

# Plot solution

[X, Y] = meshgrid (x, y);

U = u (X, Y);

surf (X, Y, U);

In the beginning there is the now familiar ufl block. As you might have noticed, subscripted measures appear in the definition of the bilinear form

After defining the problem in UFL language, there are instructions to define the mesh, the function space, the essential boundary conditions and all the coefficients involved. All such lines come from fem-fenics before this summer or have been described in my previous posts, so I will not cover them in detail. The same applies for the assembly, solve and all the output in the end of the script. The only note is that the very last lines will error out in parallel runs: point-wise evaluations in

The computed solution |

Afterwards, the mesh functions are passed as arguments to BilinearForm and LinearForm, so that the markers are available to assemble the system. In addition to the usual parameters, such as the name of the imported UFL problem, the function spaces and the coefficients, it is now possible to provide mesh functions properly labeled and they will be used.

Currently fem-fenics allows for easily marking subdomains and exterior edges copying markers from the PDEtool representation returned by the functions of the

by Eugenio Gianniti (noreply@blogger.com) at August 09, 2014 01:30 PM

After quite a struggle, I have been able to obtain a working implementation of **fem-fenics** supporting MPI parallelism. Let's go through an example and highlight what has changed lately.

ufl start Poisson

ufl element = FiniteElement '("Lagrange", triangle, 1)'

ufl u = TrialFunction (element)

ufl v = TestFunction (element)

ufl f = Coefficient (element)

ufl g = Coefficient (element)

ufl a = "inner (grad (u), grad (v))*dx"

ufl L = f*v*dx + g*v*ds

ufl end

# Create mesh and define function space

x = y = linspace (0, 1, 33);

[mesh, facets] = Mesh (msh2m_structured_mesh (x, y, 1, 1:4));

V = FunctionSpace ('Poisson', mesh);

# Define boundary condition

bc = DirichletBC (V, @(x, y) 0.0, facets, [2;4]);

f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));

g = Expression ('g', @(x,y) sin (5.0 * x));

a = BilinearForm ('Poisson', V, V);

L = LinearForm ('Poisson', V, f, g);

# Compute solution

[A, b] = assemble_system (a, L, bc);

sol = A \ b;

u = Function ('u', V, sol);

# Save solution in VTK format

save (u, 'poisson');

The basic structure has remained the same. **DOLFIN** boasts the capability to be run both in serial and in parallel execution without intervening on the code, so I did my best to have the same behaviour from fem-fenics. The Poisson.m m-file above can be run either as you usually would do with any other m-file, or from the command line with an invocation such as:

mpiexec -np 4 octave --eval Poisson

Now, how is this possible? In the beginning, with the ufl block, the variational problem is defined in UFL language, written to an .ufl file and compiled via **FFC**. Since IO is performed, ufl.m ensures that only process zero will open and write to the file. Moreover, a MPI barrier makes sure that no process will proceed before the .ufl file is imported.

As soon as the just-in-time compilation is over, there are two instructions to build the mesh, in this case on the unit square. For this, we rely on the **msh** package, which returns a PDE-tool-like representation of it. Mesh.oct must, then, convert it to DOLFIN internal representation and distribute it among processes. Here comes an issue: fem-fenics relies on markers present in the PDE-tool format to impose essential boundary conditions, and in serial runs dolfin::Mesh can store them, so that DirichletBC.oct needs just to know the boundary subset label. Unfortunately, this feature is not supported yet in parallel by the DOLFIN library, then Mesh.oct has been edited to return, if requested, also a meshfunction holding this information, in the example above facets. This way markers can be conveyed to DirichletBC.oct and boundary conditions can be applied on the correct edges.

Further intervention was needed for the assembly and solve phase. In assemble_system.oct both the matrix and the vector are assembled locally on each portion of the mesh and, afterwards, gathered on process zero and joined, so that the system can be solved with the backslash instruction of Octave. In order to allow output in VTK format, in Function.oct the solution is split up and properly distributed among processes, so that each one holds the portion of degrees of freedom related to its subdomain and to the neighbouring vertices. After save.oct has written the solution to poisson.pvd and its auxiliary files, it can be visualised with **ParaView**.

by Eugenio Gianniti (noreply@blogger.com) at August 04, 2014 01:31 PM

Lately I have not been very active on the blog since I am encountering some difficulties in the attempt to introduce MPI parallelism. Meanwhile, I have extended support to the latest version of the DOLFIN library.

Among other changes, one that strongly affects **fem-fenics** is the shift from the shared pointer implementation by the Boost libraries to the one included in the Standard Template Library with the new C++11 standard. This change alone calls for edits in almost all the codebase of the package, as basically all DOLFIN data structures are stored via smart pointers in the corresponding fem-fenics classes. However, currently version 1.3.0 is still present in the official repositories of the main Linux distributions, thus switching abruptly to the latest version would have prevented further releases of the package for a while.

In order to tackle the above mentioned issue, I resorted to the preprocessor capabilities, so as to infer from the DOLFIN version available on the compiling computer the right kind of pointers to use. Among other options, the preprocessor flags obtained using pkg-config define also a macro reporting the DOLFIN version. It is, then, possible to check it and choose the correct pointer implementation right before compilation. Currently in fem-fenics every occurrence of boost::shared_ptr has been replaced by a SHARED_PTR macro, which in turn is defined in a new header that takes care of setting it to the right value. There is just a catch: preprocessor conditionals cannot compare strings, but the DOLFIN_VERSION macro is indeed defined as a string. In order for this approach to work, the package Makefile, for the initial compilation, and the get_vars.m function, for the just-in-time ones, perform the actual check and define an auxiliary macro if the latest version is found on the system.

by Eugenio Gianniti (noreply@blogger.com) at July 30, 2014 11:58 AM

On the occasion of SoCiS 2014 I will take over the work that has been done last year on **odepkg** and continue it. The final goal is to release a new stable version of **odepkg** and to insert the most common solver in **core-Octave**in such a way that everything is MATLAB-compatible.

The following list explains the main points of the timeline for my SOCIS project:

- Check of the current status of the code, in particular with respect to the current release on SourceForge. The two repository will be merged so that every test present in the old version will be added to the new code. Verify if there are missing features and add them if necessary.
- Comparison of the performance between the old and the new structure. In particular we expect that the introduction of the Levenshtein algorithm for the string comparisons will be a critical issue. If necessary implement
*levenshtein.m*and fuzzy_compare.m in C++. - Verify that the functions
**odeset**and**odeget**are MATLAB-compatible and compliant to Octave core. Add the two functions to the core. - Move
**ode45**,**ode23**and**ode23s**to Octave core. - Implement
**ode15s**solver. This solver is still missing in odepkg but is highly suitable for stiff problems. - Move
**ode15s**to Octave core. - New release of
**odepkg**.

by Jacopo Corno (noreply@blogger.com) at July 30, 2014 05:26 AM

The translation from DOLFIN's to Octave's data structures is logically a well defined task, whilst its implementation needs to vary according to its serial or parallel execution. Furthermore, it strictly depends on the linear algebra back-end used, for each of them stores a different representation and exposes a different interface to access it. To address these difficulties, I wrote a hierarchy of factories to provide the adequate methods, based on the run-time necessities. Moreover, this way the code is easily expandable to allow for more back-ends to be used in fem-fenics (currently only uBLAS is available). There is an abstract class, to declare the interface of its derived ones, and a concrete factory implementing the uBLAS-specific methods.

Since in a future fem-fenics there will be several algebraic back-ends available for use, the hierarchy will expand. This means that the checks of the run-time configuration will eventually become more complex. Another issue comes from the need to use different types depending on information available only at run-time. Both to encapsulate those checks, avoiding code duplication, and to solve the problem of choosing the right class, I added to the hierarchy a class implementing the Pimpl idiom. With this design, the "user code" in the C++ implementation of assemble.oct and assemble_system.oct needs just to create a femfenics_factory object and use it to extract the data structures of interest, while every other hassle is dealt with behind the scenes by this class.

UML diagram of the new hierarchy |

In the diagram above you can see the already implemented classes and an example class to point out where others will collocate amongst them. femfenics_factory has private methods to check which is the right concrete class to use each time, and implements the public methods of the abstract class dispatching the call through a reference. uBLAS_factory, as other concrete classes are expected to do, holds the real code for creating Octave matrices and vectors and exposes a static method, instance, which allows for access to the singleton object of this type. femfenics_factory, in turn, obtains with it the reference needed for dispatching.

by Eugenio Gianniti (noreply@blogger.com) at July 12, 2014 06:40 PM

These days I have started my investigations on the actual implementation of the MPI parallelisation in **fem-fenics**. I found out some information that I will point out here, together with my goals for the next weeks.

First of all, apparently MPI can be used without user intervention on the serial code. This is a feature that DOLFIN boasts, but I would expect it not to pass on to fem-fenics, at least not without some effort on the implementation side. Furthermore, DOLFIN offers also a wrapper for MPI functionalities, thus probably it can be helpful in managing data transfers among threads in C++ code.

An issue that will need to be addressed is making ufl.m robust to parallel execution, since its serial implementation leads to all workers trying to open the same file, thus leading to an error that stops computation. Anyway, even if they could all open the file and write to it, this would entail that lines are copied in random order or more than once, so it must be fixed.

In the end, it seems that the partitioning procedure produces matrices that are not slices of the one assembled in serial execution. Due to this fact, I must go deep in the algorithm to find out the proper way to merge the pieces and obtain the complete matrix, which will be stored as octave_value to allow for further computation using Octave's features.

by Eugenio Gianniti (noreply@blogger.com) at July 12, 2014 04:37 PM

Hi all,

The purpose of this post is to explain the details behind the implementation of the ilu function, my work during this first period of GSOC program. The files involved are:

**:**

**--> src/ilu0.cc**

This file contains the implementation of ILU0 algorithm, the easiest one. In this version the zero-pattern of the input matrix is not modified so it is known the final structure of the output matrix. That simplifies things. For the milu=['col'|'row'] option, it is needed to implement both the IKJ and JKI versions of the algorithm to efficiently compute the compensation of the diagonal element with dropped elements. I managed to do both in the same function, just changing a few lines of code. Lets use Matlab's documentation example:

**Example 1:**

A = gallery('neumann', 1600) + speye(1600);

setup.type = 'nofill';

setup.milu = 'row';

[L,U] = ilu(A,setup);

e = ones(size(A,2),1);

norm(A*e-L*U*e)

ans = 1.4660e-14 (Very low, good)

The following little function can be used, when milu = ['row'|'col'] to check that all the columns/rows preserve its sumation (not only with ilu0 but with iluc and ilutp). Just run it after calling ilu in any form.

**benchmark/check_sums.m **(It can be found here in the repo)

function check_sums (A, L, U, milu)

b = L * U;

dim = 1;

if (milu == 'row')

dim = 2;

endif

c = sum (b, dim);

d = sum (A, dim);

v = abs (c - d);

num_zeros = length (find (v > sqrt (eps)));

printf('Number of rows-columns not well compensated: %d\n', num_zeros);

if (num_zeros > 0)

v (find (v > sqrt (eps)))

endif

endfunction

**NOTE:** I have found in Matlab 2013a that the row and col sumation does not work well always, and the row and column compensation fails for ilutp and iluc. I will show an example later.

**--> src/ilutp.cc**

** **This algorithm is the trickiest one due to pivoting, and has caused me more than one headache during its coding because it is not well described in Saad's book, just a few indications. I have found here several bugs in Matlab's 2013a implementation that make me a bit reticent about trusting results correctness.

**Error 1**

A = sparse ([3 4 4 3 -1.1; 2 0 8 9 2.2; 2 1 9 9 1; 3.2 10 2.3 2 4.5; 9 2 6 2 1]);

setup =

type: 'ilutp'

milu: 'col'

droptol: 0.2000

thresh: 0

udiag: 0

>> [L, U, P] = ilu(a,setup);

sum(A(:, 2)) => 17

sum(L*U(:, 2) => 14.4857

Clearly the sum of the second column is not preserved :/.

**Error 2**

A = sparse([3 1.5 0 0 1.44; 0 1 0 0 -2;0 0 8 0 0; 0 2 0 2 -4.5; 0 -1 0 0 1]);

setup =

type: 'ilutp'

milu: 'col'

droptol: 0.5000

thresh: 0.2000

udiag: 0

>> [L, U, P] = ilu(a,setup);

The output is:

U =

3.0000 1.5000 0 0 0

0 0 0 0 0

0 0 8.0000 0 0

0 0 0 2.0000 Inf

0 0 0 0 -Inf

L =

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 Inf 0 1 0

0 0 0 0 1

What are those Inf doing there? Clearly the are not detecting correctly 0 pivots.

**Error 3**

A= sparse([3 1 0 0 4; 3 1 0 0 -2;0 0 8 0 0; 0 4 0 4 -4.5; 0 -1 0 0 1]);

setup =

type: 'ilutp'

milu: 'row'

droptol: 0

thresh: 0

udiag: 0

>> [L, U, P] = ilu(a,setup);

Output:

L =

1.0000 0 0 0 0

1.0000 0 0 0 0

0 0 1.0000 0 0

0 1.0000 0 1.0000 0

0 -0.2500 0 0 1.0000

That 0 cannot be there. By construction L has to be a lower unit triangular matrix and that zero element spoils the L*U product. Again WRONG.

I have encountered more issues when testing Matlab using some testing matrices with 2000x2000 and 5000x5000 dimensions. With them my output is not the same as Matlab's (nnz of L and U are different from Matlab's), but taking into account the errors I found, I trust the most my version and not theirs. BTW in my case the rows and columns sums were preserved, theirs not. Obviously I have checked that those examples behave correctly in my code detecting 0 pivots

A similar example** **can be run as with ilu0:

**Example 2:**

A = gallery('neumann', 1600) + speye(1600);

setup.droptol = 1e-2;

setup.type = 'ilutp';

setup.thresh = 0.5;

setup.milu = 'row';

[L,U] = ilu(A,setup);

e = ones(size(A,2),1);

norm(A*e-L*U*e)

ans = 2.5170e-14 (Nice)** **

**Pivoting:** It worths to mention how pivoting is performed in that algorithm. When milu = 'row' the U matrix is column permuted (IKJ version used) but when milu=['off',|'col'] L is the permuted one and it is row permuted (JKI version used). Both algorithms share a lot of similarities and the code is designed to work in one version or another depending on milu option. That way code duplication is avoided. That was one of my primary fears when I realized that both versions were needed to attain Matlab compatibility.** **

**--> src/iluc.cc**

This is the file containing the crout version of ILU. This version is an enhancement of pure IKJ and JKI variants of gaussian eliminations. At iteration k the k:n section of k column and k row is computed. The enhancement is noticed in the execution time for the same input matrix. The following example is a comparison between my versions of ilutp and iluc:

For a 2000x2000 matrix ( I have not included this matrix in the repository due to it size):

With setup.droptol = 0.01 and setup.milu = 'off'.

Octave:

ilutp --> 12.3458 seconds

iluc --> 6.31089 seconds

Matlab:

ilutp --> 12.868686 seconds

iluc --> 7.498106 seconds

That is just to illustrate the performance of different versions.

**NOTE:** In iluc the dropping strategy for elements in U (stored as CRS) is to drop the element aij if (abs(aij) < droptol * norm(A(i, :))). For the L part (stored as CCS) aij is dropped if (abs(aij) < droptol * norm(A(:, j))).

Finally the numeric example:

**Example 3:**

A = gallery('neumann', 1600) + speye(1600);

setup.droptol = 1e-2;

setup.type = 'crout';

setup.milu = 'row';

[L,U] = ilu(A,setup);

e = ones(size(A,2),1);

norm(A*e-L*U*e)

ans = 2.5212e-14 (Nice)** **

That is all I wanted to show till now. I have written tests for the functions and adapted several ones from Kai last year work. However I want to add some more function-specific ones for validating results. The last thing pending is to place the source files inside the Octave source tree. I am not totally sure where they should go. On the other hand I have already started to work on ichol function and next week I'll report about my progress.

I know the post is a bit long but I think it is needed due to the poor verbosity I had through the blog during this period. I am aware of that (Jordi pointed me out a few days ago) and I will take into account for the following weeks.

Regards,

Eduardo

The purpose of this post is to explain the details behind the implementation of the ilu function, my work during this first period of GSOC program. The files involved are:

- src/ilu0.cc
- src/iluc.cc
- src/ilutp.cc
- ilu.m

- https://edu159@bitbucket.org/edu159/gsoc2014-edu159

This file contains the implementation of ILU0 algorithm, the easiest one. In this version the zero-pattern of the input matrix is not modified so it is known the final structure of the output matrix. That simplifies things. For the milu=['col'|'row'] option, it is needed to implement both the IKJ and JKI versions of the algorithm to efficiently compute the compensation of the diagonal element with dropped elements. I managed to do both in the same function, just changing a few lines of code. Lets use Matlab's documentation example:

A = gallery('neumann', 1600) + speye(1600);

setup.type = 'nofill';

setup.milu = 'row';

[L,U] = ilu(A,setup);

e = ones(size(A,2),1);

norm(A*e-L*U*e)

ans = 1.4660e-14 (Very low, good)

The following little function can be used, when milu = ['row'|'col'] to check that all the columns/rows preserve its sumation (not only with ilu0 but with iluc and ilutp). Just run it after calling ilu in any form.

function check_sums (A, L, U, milu)

b = L * U;

dim = 1;

if (milu == 'row')

dim = 2;

endif

c = sum (b, dim);

d = sum (A, dim);

v = abs (c - d);

num_zeros = length (find (v > sqrt (eps)));

printf('Number of rows-columns not well compensated: %d\n', num_zeros);

if (num_zeros > 0)

v (find (v > sqrt (eps)))

endif

endfunction

A = sparse ([3 4 4 3 -1.1; 2 0 8 9 2.2; 2 1 9 9 1; 3.2 10 2.3 2 4.5; 9 2 6 2 1]);

setup =

type: 'ilutp'

milu: 'col'

droptol: 0.2000

thresh: 0

udiag: 0

>> [L, U, P] = ilu(a,setup);

sum(A(:, 2)) => 17

sum(L*U(:, 2) => 14.4857

Clearly the sum of the second column is not preserved :/.

A = sparse([3 1.5 0 0 1.44; 0 1 0 0 -2;0 0 8 0 0; 0 2 0 2 -4.5; 0 -1 0 0 1]);

setup =

type: 'ilutp'

milu: 'col'

droptol: 0.5000

thresh: 0.2000

udiag: 0

>> [L, U, P] = ilu(a,setup);

The output is:

U =

3.0000 1.5000 0 0 0

0 0 0 0 0

0 0 8.0000 0 0

0 0 0 2.0000 Inf

0 0 0 0 -Inf

L =

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 Inf 0 1 0

0 0 0 0 1

What are those Inf doing there? Clearly the are not detecting correctly 0 pivots.

A= sparse([3 1 0 0 4; 3 1 0 0 -2;0 0 8 0 0; 0 4 0 4 -4.5; 0 -1 0 0 1]);

setup =

type: 'ilutp'

milu: 'row'

droptol: 0

thresh: 0

udiag: 0

>> [L, U, P] = ilu(a,setup);

Output:

L =

1.0000 0 0 0 0

1.0000 0 0 0 0

0 0 1.0000 0 0

0 1.0000 0 1.0000 0

0 -0.2500 0 0 1.0000

That 0 cannot be there. By construction L has to be a lower unit triangular matrix and that zero element spoils the L*U product. Again WRONG.

I have encountered more issues when testing Matlab using some testing matrices with 2000x2000 and 5000x5000 dimensions. With them my output is not the same as Matlab's (nnz of L and U are different from Matlab's), but taking into account the errors I found, I trust the most my version and not theirs. BTW in my case the rows and columns sums were preserved, theirs not. Obviously I have checked that those examples behave correctly in my code detecting 0 pivots

A similar example

A = gallery('neumann', 1600) + speye(1600);

setup.droptol = 1e-2;

setup.type = 'ilutp';

setup.thresh = 0.5;

setup.milu = 'row';

[L,U] = ilu(A,setup);

e = ones(size(A,2),1);

norm(A*e-L*U*e)

ans = 2.5170e-14 (Nice)

This is the file containing the crout version of ILU. This version is an enhancement of pure IKJ and JKI variants of gaussian eliminations. At iteration k the k:n section of k column and k row is computed. The enhancement is noticed in the execution time for the same input matrix. The following example is a comparison between my versions of ilutp and iluc:

For a 2000x2000 matrix ( I have not included this matrix in the repository due to it size):

With setup.droptol = 0.01 and setup.milu = 'off'.

Octave:

ilutp --> 12.3458 seconds

iluc --> 6.31089 seconds

Matlab:

ilutp --> 12.868686 seconds

iluc --> 7.498106 seconds

That is just to illustrate the performance of different versions.

Finally the numeric example:

A = gallery('neumann', 1600) + speye(1600);

setup.droptol = 1e-2;

setup.type = 'crout';

setup.milu = 'row';

[L,U] = ilu(A,setup);

e = ones(size(A,2),1);

norm(A*e-L*U*e)

ans = 2.5212e-14 (Nice)

That is all I wanted to show till now. I have written tests for the functions and adapted several ones from Kai last year work. However I want to add some more function-specific ones for validating results. The last thing pending is to place the source files inside the Octave source tree. I am not totally sure where they should go. On the other hand I have already started to work on ichol function and next week I'll report about my progress.

I know the post is a bit long but I think it is needed due to the poor verbosity I had through the blog during this period. I am aware of that (Jordi pointed me out a few days ago) and I will take into account for the following weeks.

Regards,

Eduardo

I’ve neglected this blog. Sorry

**What I’ve done so far:**

Changeset 1: Removed second internal representation from PermMatrix (all PermMatrix are strictly column-major now): Accepted!

Changeset 2: Created a generic templated “find” method which consolidates the many different implementations of “find”. This change includes adding nz_iterators, ie iterators over the nonzero elements of any of the four array types (Array, Sparse, DiagArray2, PermMatrix): Pending jwe’s review

Changeset 3: Added a “dispatch” function which dispatches interpreter function calls to the proper matrix template-type. jwe warned that the community might be uncomfortable with such a major change, but so far the community’s been almost entirely unresponsive except for one enthusiastic and supportive email from Carne. (Thanks!)

Note, these first three changesets have not changed significantly since I implemented them in the FIRST WEEK of GSoC, yet only one of them has actually been fully reviewed and accepted so far.

Changeset 4 WIP: Started working on an “accumulate” function to use as the back-end implementation of sum, product, any, all, min, and max. In particular I was writing a version that will properly accumulate over the rows of tall sparse matrices. I stopped this in the middle and figured I’d come back to it. I still plan to

Changeset 5: Added bounds-checking and reference-checking to Array::xelem (when debugging options are turned on). I’ve been advised I should profile this change, but there aren’t any recommendations on what benchmark to use. Experience suggests whatever benchmark I use, the community won’t accept it.

Changeset 6 WIP: Logical indexing into sparse matrices. This is the primary change my GSoC project is meant to be centered around. I have implemented a form of it, but it doesn’t yet handle any of the edge cases. Also my implementation relies heavily on the iterators and dispatch function from changesets 2 and 3 respectively so if jwe ultimately doesn’t accept those, then I don’t know what I’ll do.

Experimenting with Matlab shows that Matlab will reshape the index matrix to match the height of the result matrix and Matlab’s reshape apparently use >64-bit arithmetic to determine the size of the resulting matrix. This leaves me in somewhat of a bind because if I want to do the same, I need to add gmp as a dependency, but no one wants to add new dependencies.

Considering that in this particular instance, I’m guaranteed to know the column height (and nobody asked me to re-implement reshape so I won’t), I think I’ll just use repeated addition/subtraction to do the multiply and and divide. It will still run in only O(#columns) running time which is how long a reshape should take.

On one hand, I’m nearly done with the primary thing my project is supposed to do. That’s good. But on the other hand I should have finished it and have moved on to other stuff by now. I understand that community/mentor approval is an important thing, but if the community is really so strict, then why is Octave such a mess of buggy copy-pasted ill-considered code? Who approved all of that?

Hi all,

This is a short post just to clarify my state at midterm. As I have arranged with Kai, at the beginning of the next week I will write a verbose post to explain all the details related to the development of*"ilu"* function. I need a couple of days to tidy up all the code and make it presentable.

The state of the code now is functional. It lacks of tests and documentation and need a bit of re-factorization I will do this weekend. I will list several relevant points about the implementation of the function.

__Future work__: The second part of the GSOC I will implement *"ichol" *function. There are several points to discuss about its development with Kai because he implemented the code last year but there were some kind of issues with the licensing of it. This period is a bit longer and I will have no classes nor exams. Because of that, if I have time remaining at the end, I can start implementing *"minres" *or *"lsqr"* algorithms that Octave lacks of too. So there will be no time wasted.

See you,

Eduardo

This is a short post just to clarify my state at midterm. As I have arranged with Kai, at the beginning of the next week I will write a verbose post to explain all the details related to the development of

The state of the code now is functional. It lacks of tests and documentation and need a bit of re-factorization I will do this weekend. I will list several relevant points about the implementation of the function.

- It is programmed to output the same as Matlab.

- My version is at least as fast as Matlab's, outperforming by small amounts of time in large output cases.

- I found a bug. At least on Matlab2013a regarding "ilutp" option when using milu="col". The col sum is not preserved in at least one case so I found in my testing cases that my function does not output the same. I will explain with more detail that issue next week.

See you,

Eduardo

I will try to give a comprehensive feel of what I achieved in this first part of the Google Summer of Code, since it is time for the mid term evaluation. Let's start with an example: as usual, it is the Poisson equation, but today, as a twist, we consider a fully Neumann problem. In order for such a problem to be well posed there is the need of an additional constraint, otherwise the solution would not be unique, so in the Octave code there is the Lagrange multiplier *c*. Here you can find more details and the C++ and Python code, I will just write down the differential problem for convenience:

- Δ*u* = *f* in Ω

∇*u* ⋅ *n* = *g* on ∂Ω

Here is the Octave code that solves the above mentioned problem:

pkg load fem-fenics msh

ufl start NeumannPoisson

ufl CG = FiniteElement '("CG", triangle, 1)'

ufl R = FiniteElement '("R", triangle, 0)'

ufl W = CG * R

ufl

ufl "(u, c)" = TrialFunctions (W)

ufl "(v, d)" = TestFunctions (W)

ufl f = Coefficient (CG)

ufl g = Coefficient (CG)

ufl

ufl a = "(inner (grad (u), grad (v)) + c*v + u*d)*dx"

ufl L = f*v*dx + g*v*ds

ufl end

# Create mesh and function space

x = y = linspace (0, 1, 33);

mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));

W = FunctionSpace ("NeumannPoisson", mesh);

# Define variational problem

f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));

g = Expression ('g', @(x,y) - sin (5.0 * x));

a = BilinearForm ("NeumannPoisson", W, W);

L = LinearForm ("NeumannPoisson", W, f, g);

# Compute solution

[A, b] = assemble_system (a, L);

sol = A \ b;

solution = Function ('solution', W, sol);

u = Function ('u', solution, 1);

# Plot solution

[X, Y] = meshgrid (x, y);

U = u (X, Y);

surf (X, Y, U);

At the very beginning you can see a block with every line starting with ufl. That is what you would have to put in a separate UFL file before this summer. In a sense it is not plain UFL, but there are extra quotes and apices. They are needed because, using the current version of Octave, those brackets with commas inside would otherwise be interpreted as function calls. After this blocks closes with the ufl end line, the resulting UFL file is compiled to obtain a FunctionSpace, a BilinearForm and a LinearForm. These are oct-files that fem-fenics will use later on to define the corresponding variables in Octave. A robust implementation of ufl.m, the function that provides this binding to the UFL language, is one of the results of the first term.

In the end of the snippet you can see that the solution*u* is evaluated in its domain exactly as you expect to do with a regular function taking two arguments and returning one value. This is due to the new subsref method of the function class, which is used to represent the elements of a function space. Aside from surface plots, this feature can be of interest to generalise methods that rely on analytical solutions to differential problems, or to apply basically any algorithm to such functions. Here is the plot you will obtain with this script:

I wrote in an earlier post of the interpolate function: with this you can get the representation of a Function or Expression on a given FunctionSpace. It is useful, for instance, to compare your numerical solution with an exact one you happen to know. Or, in the example above, you might want to view what is the forcing term like:

f_cg = interpolate ("f_cg", f, u);

F = f_cg (X, Y);

surf (X, Y, F);

There is one last achievement to highlight for the mid term evaluation: currently both the initial compilation of the package and all the ones performed just-in-time when importing UFL instructions proceed smoothly without user intervention. To this end, now the build system relies on pkg-config to get at once all the flags needed for proper compilation and linking, since some dependencies of dolfin, the FEniCS interface, are not to be found in standard directories. In order to exploit the extracted information also for the subsequent run time builds, the autoconf substitution is performed also in the get_vars.m auxiliary function, which in turn provides it to generate_makefile.m. An implementation detail that proved quite tricky is how to pass all the preprocessor flags to mkoctfile: only a subset of the options of g++ are hard-coded in it, so I needed to resort to a workaround. Indeed, CPPFLAGS are always passed as environment variables and not as command line flags, so that mkoctfile will just copy and deliver them to the real compiler.

To further enhance the build system, I implemented other internal functions that hash the UFL file that was compiled and, later, check it to understand if it changed between the previous and the freshly requested build. In the example above, you will find in your working directory four new files after a run: the three already mentioned oct-files and a text file storing the md5 sum of the UFL that has been imported. Until one of these files gets somehow deleted or the problem in the ufl block changes, you will not need to take on a time consuming compilation any more.

pkg load fem-fenics msh

ufl start NeumannPoisson

ufl CG = FiniteElement '("CG", triangle, 1)'

ufl R = FiniteElement '("R", triangle, 0)'

ufl W = CG * R

ufl

ufl "(u, c)" = TrialFunctions (W)

ufl "(v, d)" = TestFunctions (W)

ufl f = Coefficient (CG)

ufl g = Coefficient (CG)

ufl

ufl a = "(inner (grad (u), grad (v)) + c*v + u*d)*dx"

ufl L = f*v*dx + g*v*ds

ufl end

# Create mesh and function space

x = y = linspace (0, 1, 33);

mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));

W = FunctionSpace ("NeumannPoisson", mesh);

# Define variational problem

f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));

g = Expression ('g', @(x,y) - sin (5.0 * x));

a = BilinearForm ("NeumannPoisson", W, W);

L = LinearForm ("NeumannPoisson", W, f, g);

# Compute solution

[A, b] = assemble_system (a, L);

sol = A \ b;

solution = Function ('solution', W, sol);

u = Function ('u', solution, 1);

# Plot solution

[X, Y] = meshgrid (x, y);

U = u (X, Y);

surf (X, Y, U);

At the very beginning you can see a block with every line starting with ufl. That is what you would have to put in a separate UFL file before this summer. In a sense it is not plain UFL, but there are extra quotes and apices. They are needed because, using the current version of Octave, those brackets with commas inside would otherwise be interpreted as function calls. After this blocks closes with the ufl end line, the resulting UFL file is compiled to obtain a FunctionSpace, a BilinearForm and a LinearForm. These are oct-files that fem-fenics will use later on to define the corresponding variables in Octave. A robust implementation of ufl.m, the function that provides this binding to the UFL language, is one of the results of the first term.

In the end of the snippet you can see that the solution

I wrote in an earlier post of the interpolate function: with this you can get the representation of a Function or Expression on a given FunctionSpace. It is useful, for instance, to compare your numerical solution with an exact one you happen to know. Or, in the example above, you might want to view what is the forcing term like:

f_cg = interpolate ("f_cg", f, u);

F = f_cg (X, Y);

surf (X, Y, F);

There is one last achievement to highlight for the mid term evaluation: currently both the initial compilation of the package and all the ones performed just-in-time when importing UFL instructions proceed smoothly without user intervention. To this end, now the build system relies on pkg-config to get at once all the flags needed for proper compilation and linking, since some dependencies of dolfin, the FEniCS interface, are not to be found in standard directories. In order to exploit the extracted information also for the subsequent run time builds, the autoconf substitution is performed also in the get_vars.m auxiliary function, which in turn provides it to generate_makefile.m. An implementation detail that proved quite tricky is how to pass all the preprocessor flags to mkoctfile: only a subset of the options of g++ are hard-coded in it, so I needed to resort to a workaround. Indeed, CPPFLAGS are always passed as environment variables and not as command line flags, so that mkoctfile will just copy and deliver them to the real compiler.

To further enhance the build system, I implemented other internal functions that hash the UFL file that was compiled and, later, check it to understand if it changed between the previous and the freshly requested build. In the example above, you will find in your working directory four new files after a run: the three already mentioned oct-files and a text file storing the md5 sum of the UFL that has been imported. Until one of these files gets somehow deleted or the problem in the ufl block changes, you will not need to take on a time consuming compilation any more.

by Eugenio Gianniti (noreply@blogger.com) at June 25, 2014 09:13 PM

As said in the previous post, my latest result is the possibility to evaluate a fem-fenics function on a point of its domain. This way it is possible to generalise methods that, otherwise, would rely on analytical solutions. In [1] we have an example of such a method.

The paper deals with the deposition of nanoparticles in tissues, for the treatment of cancer. The phenomenon is described with a Monte Carlo simulation of these particles' trajectories, assuming that the velocity field of the carrying fluid is known. In this study, some simplifying hypotheses about the geometry of the cells and the fluid layer nearby allow for an analytical solution of the Stokes equation. Unfortunately, these assumptions do not hold generally in human tissues: for instance, in the liver cells have cubic shape, contrasting to the spherical one used in this paper. Now, if this method is implemented in Octave, we can solve numerically the Stokes equation on a realistic domain and obtain right away a more general approach to this significant application.

The evaluation of a fem-fenics function was already possible via the feval method, but it had some glitches. One aspect is that the solution of a differential problem could not be used *as if* it was a regular Octave function, then a user should have adapted his/her algorithms to take advantage of it. One more critical issue is that the previous implementation did not handle the exception raised by the underlying FEniCS method when it gets as argument the coordinates of a point outside of the domain, thus leading to a crash of Octave.

In order to address these problems, I added the subsref method to the **function** class and implemented the proper exception handling in feval. To avoid code duplication, the former relies on the latter for the real computation, so it basically just forwards the parameters after checking that the right type of indexing was used. As a result, it is now possible to solve the equations:

- *ν* Δ**u** + ∇*p* = 0

∇ ⋅ **u** = 0

with relevant border conditions, on a proper mesh and finite element space, and then evaluate the solution with the Octave expression values = u (points), where points is a matrix holding the coordinates of every point where to do so, one per column. Moreover, a careless evaluation will not result in your Octave session crashing any more.

Even if this feature of the package underwent some improvement, there is still room for more. Two issues I have not addressed yet are the somehow weird interface and the possibility to create a function handle to perform evaluations with. Regarding the former, we might observe that the above mentioned expression remains exactly the same no matter what the geometrical dimension of the domain is. I should modify the implementation so that a vectorial function on a 3D space is evaluated with [ux, uy, uz] = velocity (x, y, z). Moving to the latter, in my understanding the class design should be modified to allow the exploitation of the Octave internals managing functions, so this would require a careful reflection on all the possible collateral effects of such a change.

Even if this feature of the package underwent some improvement, there is still room for more. Two issues I have not addressed yet are the somehow weird interface and the possibility to create a function handle to perform evaluations with. Regarding the former, we might observe that the above mentioned expression remains exactly the same no matter what the geometrical dimension of the domain is. I should modify the implementation so that a vectorial function on a 3D space is evaluated with [ux, uy, uz] = velocity (x, y, z). Moving to the latter, in my understanding the class design should be modified to allow the exploitation of the Octave internals managing functions, so this would require a careful reflection on all the possible collateral effects of such a change.

by Eugenio Gianniti (noreply@blogger.com) at June 23, 2014 01:11 AM

The mid-term review is approaching, so it is time to highlight what is done, what is underway and what are the future goals. In this post I will try to do so as clearly as possible.

The main effort during the first part of the Google Summer of Code was the implementation of the bindings for the UFL language in Octave. Now UFL code can be written directly in m-files, without the need of a separate file to define the problem. To this end, ufl has been implemented for opening a file, writing to it and importing the variational problem when it is complete.

Further, I implemented interpolate, which allows the interpolation of a Function or an Expression on a given FunctionSpace. This can be of interest to test the validity of a discretisation method, for instance if an analytical solution is available in closed form, so that it is possible to compare it with the numerically obtained one.

Lately, I focused on the build system, both for the package compilation and for the just-in-time ones needed to import variational problems in Octave. The former is now backed by pkg-config, so that all the proper compiling and linking options required by the dependencies are obtained at once. Thinking about the latter, this information is used to accordingly configure the get_vars function, which provides it to the one that generates the Makefiles used to compile oct-files just-in-time. In the end, currently these oct-files are compiled again only when necessity arises, for example if one of them has been deleted or if the UFL file has been changed.

In the upcoming week I will add another feature: it will be possible to get a function handle for the evaluation of a Function. This way the solution of a variational problem can be used exactly as any other function in Octave, for instance allowing the generalisation of algorithms relying on exact solutions of differential problems, which are thus limited to simple cases. I will provide some details on an application in my post about this feature.

In the second part of the project I will be mainly committed to the parallelisation of the package execution via MPI. As noted in an earlier post, the parallelisation through the OpenMP paradigm has been quickly abandoned because it does not provide a significant performance gain, while opening the way to bugs and errors. Parallelism is, anyway, an interesting feature for the package's use cases, so it will be the main goal of the final hand in.

by Eugenio Gianniti (noreply@blogger.com) at June 16, 2014 01:13 AM

One of the known issues of the fem-fenics package was related to the errors during the just-in-time compilation, due to missing include directories. Among my preliminary contributions there is a changeset addressing the problem in the initial build of the package, when installing it into Octave. Now I went on and solved it also during the usage of fem-fenics.

At the moment of the first build, **autoconf** is in charge of finding out the relevant compiler and linker flags through **pkg-config**. They are, then, substituted in the Makefile, which compiles the package making use of them. This piece of information is needed also when an UFL file is imported and transformed into the oct-files used to transform the weak formulation at hand into an algebraic system, but until now the user had to supply it by the means of an environment variable.

Currently, I added a new utility function that provides those flags. In the configuration process they are substituted also in get_vars.m, which is called by generate_makefile.m when a differential problem is imported. The latter replaces two placeholders and writes the *ad hoc* Makefile with all the necessary compile and link options. This way users will not need to provide compilation flags anymore, instead the package will manage this aspect on its own.

As noted in a previous post, however, this just-in-time build is relatively time consuming, taking around half a minute each time. Nonetheless, a common usage pattern could entail the resolution of the same weak formulation on varying meshes or with different boundary conditions, forcing terms or physical parameters. Every mentioned situation does not need the recompilation of the problem's oct-files, since they carry information only about the function space and the formal expressions of the bilinear form and the linear operator. It is useful, then, to take on the build process only when needed.

To add this feature, I created three function to perform appropriate actions. After every successful just-in-time compilation, save_hash.m takes care of hashing the imported UFL file and writing the result to <ufl-filename>.md5sum. On the other hand, at the beginning of every import_ufl_*.m function, a check like this is performed:

if (check_hash (var_prob) ||

! check_oct_files (var_prob, "Problem"))

You can see in it the remaining two functions implemented lately. The first one, check_hash.m, receives as argument the name of the variational problem, reconstructs the UFL file name, looks for a saved hash sum and compares it with the current file's. It returns true if the proper .md5sum file is not found or if the new and old hashes are not the same. Clearly, the oct-files should be rebuilt if one of them is missing: check_oct_files.m looks for the relevant files, with its second option stating which import is underway (thus, which files are expected as output), and returns true if they are all available.

by Eugenio Gianniti (noreply@blogger.com) at June 15, 2014 04:18 PM

These days I have worked on the implementation of the interface to the OpenMP-powered assembly offered by FEniCS. Despite being potentially a one-line intervention, it proved quite tricky: indeed, with the needed addition, the fem-fenics function for system assembly broke with a huge number of run time errors, probably due to a change in the underlying data structure that is transparent to the library users, but does not go unnoticed if you need to access it directly, as fem-fenics does. This led me to leave this functionality behind.

My choice is backed by some computational experiments. They show that the approach enacted by the FEniCS library is quite effective, with times required for assembly reduced by half using four threads instead of just one. However, they are negligible compared to the linear system solve phase, even when the OpenMP parallelisation is disabled. I used a great number of mesh nodes in order to have meaningful timings: even if linear systems took some minutes for resolution, the assembly phase lasted as much as a couple of hundredth of a second in serial code. If we add to these findings that the fem-fenics package requires a just-in-time compilation lasting around half a minute, we understand that there is no point in devoting effort for the implementation of this feature.

by Eugenio Gianniti (noreply@blogger.com) at June 15, 2014 12:49 PM

It has been a bit more than two week since my last posting. I just wanted something solid enough to show before doing it again :). Because one image is better than a 1000 words. This is the state of my project till now:

In green color is what it is finished and working (obvious...) and in pink what it is partially finished. Red stuff is not working at all.

** ILUTP implementation:**

As I did with ilu0 function, I started the implementation of ilutp using the IKJ variant of the Gaussian elimination as prof. Saad does in his book. For working efficiently with CCS(column compressed storage) structure of sparse matrices it is only needed a transposition before and after the process. So I came up with a working version without pivoting using this strategy a week before this post (src/ilutp_crs.cc file in the repository). All OK till that point. Well ... it was not all OK. When pivoting comes into play, all get messy. It is not feasible to do row-pivoting efficiently after transposing the matrix and using the CCS structure with the IKJ algorithm. What I realized is that Matlab, by default, implements for milu="col" and milu="off" options a JKI variant of the algorithm. This way row- pivoting can be used and no transposition is needed using the CCS structure. So for the whole last week I had to almost rewrite entirely the function to implement it in the JKI way. That was a serious delay because I was not familiar with that variant. On the other hand I also got to the conclusion that milu="row" option demands a IKJ implementation with column pivoting. It can be infer from the documentation:

* "....When SETUP.milu == 'row', U is a column permuted upper triangular factor. Otherwise, L is a row-permuted unit lower triangular factor.*"

Column pivoting means that if CCS is used as storage structure (Octave does), the strategy must be to [transpose - perform IKJ algorithm with column pivoting - transpose again]. So it is needed another implementation. That is the reason milu="row" is not working with ilutp. I had no time to implement that variant with pivoting. However, I have half way traversed because of my early IKJ implementation. So I am working on it.

I am taking special care to output exactly the same as Matlab, that means figuring out some nuances of their implementation that can only be understood after trial and error experimentation with their ilu version. I tried to test intensively the function and for my test cases my version outputs the same as Matlab's.

I have integrated the ilu0 and ilutp function inside a m-file wrapper called ilu.m located in the root directory of the repository. The file was written last year by Kai and need to be changed a bit. But for now it is OK to provide a user-friendly interface to try my functions. Use it the same way as you were in Matlab.

A quick script to test it could be:

A = sprand(100, 0.5);

setup.thresh = 0.4;

setup.droptol = 0.005;

setup.type = 'ilutp';

[L, U, P] = ilu(a, setup);

To get the code pull from here:

https://edu159@bitbucket.org/edu159/gsoc2014-edu159

Just execute*make* in the root directory and then open the Octave interpreter inside it too.

For the next week I am planning to finish the implementation for the milu option in both ilu0 and ilutp. (You can find the files as src/ilutp.cc and src/ilu0.cc in the project directory)

P.D: For who cares about performance ( I do), my version is a bit faster than Matlab's. You can try it for big matrices. I did, and for low values of droptol (means few terms of the matrix will be dropped), using pivoting and relatively big matrices (5000x5000) my version lasted around 200 secs and Matlab 220 secs. For a 2000x2000 one, the times were 19secs Matlab's, 13 secs mine. The numbers are just for you to get an idea. But they are good news.

See you!

In green color is what it is finished and working (obvious...) and in pink what it is partially finished. Red stuff is not working at all.

As I did with ilu0 function, I started the implementation of ilutp using the IKJ variant of the Gaussian elimination as prof. Saad does in his book. For working efficiently with CCS(column compressed storage) structure of sparse matrices it is only needed a transposition before and after the process. So I came up with a working version without pivoting using this strategy a week before this post (src/ilutp_crs.cc file in the repository). All OK till that point. Well ... it was not all OK. When pivoting comes into play, all get messy. It is not feasible to do row-pivoting efficiently after transposing the matrix and using the CCS structure with the IKJ algorithm. What I realized is that Matlab, by default, implements for milu="col" and milu="off" options a JKI variant of the algorithm. This way row- pivoting can be used and no transposition is needed using the CCS structure. So for the whole last week I had to almost rewrite entirely the function to implement it in the JKI way. That was a serious delay because I was not familiar with that variant. On the other hand I also got to the conclusion that milu="row" option demands a IKJ implementation with column pivoting. It can be infer from the documentation:

Column pivoting means that if CCS is used as storage structure (Octave does), the strategy must be to [transpose - perform IKJ algorithm with column pivoting - transpose again]. So it is needed another implementation. That is the reason milu="row" is not working with ilutp. I had no time to implement that variant with pivoting. However, I have half way traversed because of my early IKJ implementation. So I am working on it.

I am taking special care to output exactly the same as Matlab, that means figuring out some nuances of their implementation that can only be understood after trial and error experimentation with their ilu version. I tried to test intensively the function and for my test cases my version outputs the same as Matlab's.

I have integrated the ilu0 and ilutp function inside a m-file wrapper called ilu.m located in the root directory of the repository. The file was written last year by Kai and need to be changed a bit. But for now it is OK to provide a user-friendly interface to try my functions. Use it the same way as you were in Matlab.

A quick script to test it could be:

A = sprand(100, 0.5);

setup.thresh = 0.4;

setup.droptol = 0.005;

setup.type = 'ilutp';

[L, U, P] = ilu(a, setup);

To get the code pull from here:

https://edu159@bitbucket.org/edu159/gsoc2014-edu159

Just execute

For the next week I am planning to finish the implementation for the milu option in both ilu0 and ilutp. (You can find the files as src/ilutp.cc and src/ilu0.cc in the project directory)

P.D: For who cares about performance ( I do), my version is a bit faster than Matlab's. You can try it for big matrices. I did, and for low values of droptol (means few terms of the matrix will be dropped), using pivoting and relatively big matrices (5000x5000) my version lasted around 200 secs and Matlab 220 secs. For a 2000x2000 one, the times were 19secs Matlab's, 13 secs mine. The numbers are just for you to get an idea. But they are good news.

See you!

Note: Yech, the code formatting on this post came out terrible. I’ll come back and fix it up later, I promise.

Almost immediately, when writing any function in Octave I run into a basic problem. I’m not sure how many other people have recognized this problem, but it seems like no one’s yet bothered to deal with it. So this seems like an unavoidable first task:

For efficiency purposes, everyone avoids OO polymorphism in favor of template-polymorphism (all the indirection inevitably will slow down processing). But if I want to write a templated function for dealing with matrices

template<typename Mat>

octave_value_list myAwesomeFunction(Mat m)

{

// Does something awesome

}

It turns out to be hard to call this function when all I have is an object of type “octave_value”.

For every type I want my function to support I need to handle this explicitly:

octave_value_list

callMyFunction(octave_value m)

{

if(m.isType1)

myAwesomeFunction(m.type1Value ());

else if(m.isType2)

myAwesomeFunction(m.type2Value ());

else if...

}

You get the picture. But it’s far worse than this, because with matrices, the types themselves are parameterized (by the element type), so now it looks more like:

octave_value_list callMyFunction(octave_value m)

{

if(m.isType1)

{

if(m.innerTypeIsA)

myAwesomeFunction(m.type1AValue());

else if(m.innerTypeIsB)

myAwesomeFunction(m.type1BValue());

...

}

else if(m.isType2)

{

if(m.innerTypeIsA)

myAwesomeFunction(m.type2AValue());

else if(m.innerTypeIsB)

myAwesomeFunction(m.type2BValue());

...

}

else if...

}

And then this is re-written for every single core function in Octave. Needless to say this process is buggy and there are many missing cases. This is probably the primary reason why so many Sparse operations are broken. The authors simply forgot to include isSparse in their list of checks. Or if they remembered, then they forgot to include isDiag or isPerm. And even if they managed to include all the matrix types, they might not have handled isComplex etc.

So I’m creating a utility file called “dispatch.h”.

The idea of dispatch.h is that it takes a template template parameter which is a functor to call back with the proper template argument.

Ideally, dispatch.h should contain the one-and-only implementation of that long if-else-if shown above and then other functions can call the dispatch function to get the right template type. From there, they can use overloading, specialization typedefs etc. whatever they need to generalize it (or not). I’m first implementing it by copying the if-else-if clause from the “find” method and then I’ll do the accumulate/reduce methods (any, all, sum, product, min, and max)

If it can handle all those different functions, I’ll be satisfied that it does the right thing. In this way, we can have code that doesn’t work sporadically depending on the type passed into it. ex.

octave:2> [a,b,c,d,e,f] = find(speye(10))

panic: impossible state reached in file 'corefcn/find.cc' at line 221

panic: Aborted -- stopping myself...

attempting to save variables to 'octave-workspace'...

save to 'octave-workspace' complete

octave exited with signal 6

(nargout wasn’t properly checked for sparse matrices, only full ones)

or

octave:1> [i,j] = find(eye(1000000))

error: out of memory or dimension too large for Octave's index type

(Someone forgot to handle diagonal matrices)If there’s one source of truth for what qualifies as “all-encompassing”, then the compiler will tell you when you forgot to handle this type or that type. No more ad-hoc dispatching.

In my mind there’s a big difference between **duplication** and **boilerplate**.

Code **duplication** occurs when two functions (or two blocks of code more generally) do approximately the same thing with minor variations. If those two blocks of code occur in different places in the code-base, then likely when one gets changed (ie to fix a bug) the other will be ignored. Such is the case with the old implementation of “find” in which there were multiple places where the dimensions of the return-matrix are computed (among other things). When you eliminate code duplication, you generally have to make the replacement code more generic, but then you also need ways to construct specific instances to be passed to the generic code.

In an intelligent modern language like D, or a language which relies on higher-level functions like haskell, Ocaml, lisp (any functional language really), the story ends there. You make your code more generic and pass the proper instantiations to it.

In an old cluttered imperative language like C++, constructing the proper instances of a generic function tends to require a lot of **boilerplate** code which is code that consists of many short blocks with the same structure. Boilerplate occurs all in one place, and unlike with code duplication, there’s no danger of an unsuspecting future developer bugfixing your code by changing one block while forgetting to change the others. But it does indeed include “duplicated code”. The difference is, the code in question tends to be a single function call or a method header. It’s short, and difficult to abstract any further (without changing languages).

If you see a line of code that looks like:

switch (n)

{

case 1: return call_func<1> (args);

case 2: return call_func<2> (args);

case 3: return call_func<3> (args);

default:

error(“call_func should be called with a number from 1 to 3″);

}

please recognize this for what it is. This is boilerplate. It’s necessarily ugly because it’s written in C++, but it’s not “code duplication” in the same way two implementations of binary search in different files is “code duplication”. You can be sure whoever wrote this code struggled with it and thought about it and determined that this was the most generic way to do whatever they were trying to do.

*Note: in order to satisfy the exquisite tastes of today’s discerning internet readers, the following blog post is written in Cracked.com style.*

We have been using open source for so long that we have forgotten, culturally, where it came from. It seems so natural and ubiquitous that we can no longer remember how things were before it. Some of us are young enough to have never even lived through times were open source wasn’t everywhere.

I am here to set the record straight on a few things, because I have noticed that even people who *have* lived through ye olden times have forgotten where things came from. Open source wasn’t spawned single-handedly by the sheer might of Linus Torvalds’s virtual gonads. Open source doesn’t mean that money is forbidden. Open source doesn’t mean that Richard Stallman is a twit.

First things first, and the #1 thing most people have forgotten about open source: the term did not arise naturally. It was invented in early 1998 during the release of Netscape Navigator as the free Mozilla suite. The Open Source Initiative, composed of trailblazers such as Eric Raymond and Bruce Perens, decided that we needed a new name for what was about to happen. They got together with other people and Christine Petersen suggested the term, to much rejoicing. She then vanished back into the shadows and went back to being a nanotechnologist or something.

OSI is an organisation that got together for a single purpose: to keep saying “open source, open source, open source” so much until everyone else was saying it too. This was all in February 1998, remember. That means open source is barely a year older than The Matrix. Neo had probably not even heard about it, because…

The greatest testament to how good OSI’s marketing campaign was is that we have come to believe that the term is so natural that we always just called it that. They have convinced us all that “open source” was our idea, without needing to get into our dreams to do so.

Needless to say, it was not our idea. By far, the most common way to refer to “open source” before 1998 was “free software”.

Now, I know what you’re thinking. “Oh god, not this stupid flamewar again. Jordi, we know you’re a FSF-spouting ~~propaganda~~ drivel machine, why do you keep pushing the stupid term for open source that Richard Stallman keeps talking about?”

Wait, wait, hear me out. It wasn’t just Richard Stallman who called it “free software”. You know FreeBSD? The “free” in there doesn’t just mean “without a fee”. They really do mean free as in freedom. Or look at what OpenBSD calls itself a few times while rocking out to sweet, sweet, pufferfish freedom:

[...] we instead celebrate the 10 years that we have been given (so far) to write free software, express our themes in art, and the 5 years that we have made music with a group of talented musicians.

That’s right, even the biggest haters of the FSF and the GPL, and the most ardent opponents of His Exalted Bearded Gnuliness Richard the Stallman call themselves “free software”.

Amusingly enough, you probably never really noticed this, but the very same Mozilla for whom “open source” was initially coined, tried to call itself “organic software” for a while.

Now, here’s the thing: OSI didn’t just say, “here is open source, go wild and free, call anything you want open source!” Nope, in what might appear at first blush to be a cruel ironic twist, OSI did not make the definition of “open source” itself open source. In fact, they even trademarked “open source”, and ask that you only use the phrase according to their trademark guidelines!

Alright, so what does “open source” mean?

Well, in the beginning, Bruce Perens wrote the Debian Free Software Guidelines (there’s that pesky “free” term again). Then, he decided he was just going to grab those very same guidelines, run `sed -i s/Debian/Open Source/g`

, and make that the official definition of open source.

This means that “open source” means a lot more than just “show me the code”. In particular it means that,

- If you don’t let people sell it, it’s not open source.
- If you don’t let people give it to their friends, it’s not open source.
- If you don’t treat all receipients of your software equally, it’s not open source.

So why did OSI insist so much on a precise definition of open source? Well, because…

Okay, this is one that really gets people riled and the one where the flamewars arise. I am here to tell everyone that if you’re flaming over whether stuff is open source or if it’s free software, you guys need to chill the fuck out: everything that is open source is also free software, and vice versa.

I bet that declaration alone is gonna rile everyone up even more, eh?

Okay, let’s look at this from a different angle with an analogy. The issue here is with something that philosophers like to call intensionality vs extensionality.

You know how Canada is a constitutional monarchy, right? And you know how there is a Queen of Canada who is the head of government? The Constitution Act of 1867 establishes that Canada has a monarch. She has fun duties such as for example being assigned the copyright of anything an employee of Her Majesty’s Government does. Great fun, I once had someone send us Octave patches under the name of Her Majesty the Queen in Right of Canada.

Now, you might recognise that lady above, and you probably also know that England also has a queen, and by now my astute readers and you have doubtlessly put together that the Queen of Canada also happens to be the Queen of England. Two names for the same person!

However, Canada’s Constitution Act doesn’t actually specify “The Queen of Canada will be whoever occupies the position of Queen of England”. It just says that Canada has a queen and goes on to list the duties of said queen. This is called the *intensionality*, the words by which we describe what something is. The *extensionality* refers to the actual objects in the world that are described by these words. In this case, “Queen of Canada” and “Queen of England” could, perhaps, under some weird political shenanigans end up being two different people, but in practice they end up referring to the same person. So the extensionalities of “Queen of Canada” and “Queen of England” are the same.

It is the same with free software and open source. The definitions look different, but in practice the software that they refer to ends up being the same. Oh, sure, there are some very minor disagreements over whether this or that license is OSI-approved but not FSF-approved or vice versa, but the whole point of coining “OSI” was to have another word to refer to “free software”.

In other words, it was always OSI’s intention for “open source” to be a synonym for “free software”. Hell, even Bruce Perens said so. Why did OSI want a synonym?

The whole point of coining the phrase “open source” was to push a certain point of view. The biggest proponent for the “open source” phrase was Eric Raymond. He and OSI have always described open source as marketing for free software.

So this marketing campaign came with certain promises, promises that we have forgotten were ever part of a marketing campaign by OSI, because they’re so ingrained into open source itself. Stop me if you’ve heard any of these before…

- Open source is a cheaper model to develop software
- Open source ensures that software has fewer bugs, because more eyes can look at the source code
- Release early, release often.
- The best software is created by scratching an itch.

And so on… the whole point was to make free software attractive to business by de-emphasising the whole “freedom” part of it. Instead, OSI promised that by making your software open source, you would have *better* software, that open source was a better development model, leading to cheaper, less buggy software.

The “cheaper model” thing is also still a fairly popular meme nowadays. When you look at free projects in Ohloh.com, one of the lines is how much money it would have cost to build this or that under some model called COCOMO.

I’m not trying to say that OSI is right or wrong about its promises. Some free software really is less buggy than non-free variants. It probably is way cheaper to develop Linux when all of the big companies chip in a few developers here and there to maintain it. All I’m saying is that we have forgotten that with the word “open source”, certain promises came attached to it. Some of these promises might even appear to be broken in some cases.

So next time you hear someone tell you that there will be fewer bugs and everyone will come sending you patches the moment you reveal your source code, remember that they’re repeating campaign slogans. And remember that even if those slogans might not always be true, there might be other reasons why you should give everyone else freedom to enjoy and distribute and hack your software.

So this post made the rounds a couple of days ago, and it got me thinking… can Mercurial (hg) do any better? I think it can, especially with Evolve. Here is me describing how Evolve works:

As to the movie, if you have not seen it yet, you might want to wait until after you do, but the basic gist is a time-travel plot where they go back and fix timelines.

History is terribly wrong, an awful, crippling bug has been discovered way back in history, and it’s so terrible that a big chunk of current history has to be thrown out. Someone created evil sentinels, so evil that they decided to exterminate all mutants and most humans.

Everyone digs back through the logs to find the cause of the problem. They know everything is bad now,

$ hg bisect --bad

but remember that some time in the past it was ok

$ hg bisect --good xmen-release-1.0

After some discussion,

$ hg bisect --good

$ hg bisect --bad

$ hg bisect --good

$ hg bisect --bad

$ hg bisect --bad

$ hg bisect --good

$ hg bisect --bad

the problem is revealed:

The first bad revision is:

changeset: 1024:0adf0c6e2698

user: Raven Darkhölme <mystique@x-men.org>

date: Fri May 18 12:24:50 1973 -0500

summary: Kill Trask, get DNA stolen

changeset: 1024:0adf0c6e2698

user: Raven Darkhölme <mystique@x-men.org>

date: Fri May 18 12:24:50 1973 -0500

summary: Kill Trask, get DNA stolen

A bookmark is placed here for future reference

$ hg bookmark mystiques-first-kill -r 1024

Professor X and Magneto brief Wolverine on his impending task. The history has been made public, but the situation is so hopeless that hg admin Kitty Pryde decides to operate on Wolverine’s repo, the only one that could withstand the changes:

$ cd /home/wolverine/xmen

$ hg phases --draft --force -r 'descendants("mystiques-first-kill")'

$ hg phases --draft --force -r 'descendants("mystiques-first-kill")'

Now Wolverine’s repo can endure any change. It’s a desperate move, but these are desperate times. Kitty sends Logan back:

$ hg update -r mystiques-first-kill

Wolverine dispatches some minor thugs and squashes a few bugs, but the first change needs to alter the timeline,

$ hg amend -m "Attempt some wisecracks with some thugs"

137 new unstable changesets

Now all of the history that was based on top of this commit is unstable. It’s still there, for now, but things are rocky. Sentinels are approaching in the bad future and might kill everyone. Shit will get real there.

That’s ok, though, Wolverine is badass, doesn’t give a fuck, and goes about his business,

$ hg ci -m "Psychoanalyse Charles Xavier" #Acceptable spelling for a Canadian

$ hg ci -m "New recruit: Peter Maximoff <quicksilver@x-men.org>"

$ hg ci -m "Use Quicksilver to rescue Magneto"

$ hg ci -m "Stop Mystique from killing Trask (WIP)"

$ hg ci -m "Stop Mystique again from killing Trask"

$ hg fold -r .^ -m "Stop Mystique from killing Trask"

$ hg ci -m "Get metal painfully inserted into body. Then get drowned for good measure"

$ hg ci -m "New recruit: Peter Maximoff <quicksilver@x-men.org>"

$ hg ci -m "Use Quicksilver to rescue Magneto"

$ hg ci -m "Stop Mystique from killing Trask (WIP)"

$ hg ci -m "Stop Mystique again from killing Trask"

$ hg fold -r .^ -m "Stop Mystique from killing Trask"

$ hg ci -m "Get metal painfully inserted into body. Then get drowned for good measure"

He decided that he didn’t want two separate commits for the same effect of stopping Mystique, so he folded those two commits into one. This is ok, because he’s still in draft mode.

Now Wolverine can’t do much about his current situation, and it’s up to others. So he decides to put his memory away for a while,

$ hg shelve

and now it’s up Mystique’s less buggy version, disguised as Stryker, to revive Wolverine,

$ hg ci -m "Rescue Wolverine from only thing that *might* kill him"

and a whole lot of other merry developments happen offscreen:

$ hg ci -m "Rebuild the school"

$ hg ci -m "Get new recruits"

$ hg ci -m "Everyone's happy"

$ hg ci -m "Etc, etc"

$ hg ci -m "Get new recruits"

$ hg ci -m "Everyone's happy"

$ hg ci -m "Etc, etc"

At this point, the unstable history with the bad timeline is no longer needed. If the X-Men had wanted to keep any part of it, they might have used the `hg evolve`

command, but they just want to forget the whole mess

$ hg bookmark --delete mystiques-first-kill

$ hg prune -r "unstable()"

$ hg prune -r "unstable()"

and the whole thing just fades away. Wolverine reawakens in the future, along with his memories,

$ hg unshelve

and it’s up to him and future Professor X in the good timeline to fix all the merge conflicts that will ensue from this unshelving.

These days I have been studying the documentation of the **FEniCS** project, mainly the FEniCS book, in order to understand the features related to parallel execution that it boasts. This preliminary study is aimed at adding them to the fem-fenics package. First of all I will summarise my findings, then I will comment the problems I need to address to implement this functionality.

FEniCS implements parallelism in such a way to be transparent to the user of the library. Moreover, it scales on different architectures, ranging from multi-core personal computers to distributed clusters. To this end, FEniCS makes use of two paradigms, which can be exploited both separately and together.

The first approach is tailored for shared memory architectures, such as the vast majority of the PCs nowadays, but also in many cases each node of a computational cluster. The implementation is based on **OpenMP**, and adding a simple instruction one can enable parallelisation to speed up the matrix assembly phase. It should be noted that this paradigm has little support in the underlying linear algebra libraries, so the resolution phase can take advantage of multi-threading only with the **PaStiX** solver. Since in a shared memory model parallel programs might suffer *race conditions*, the mesh is coloured to identify subsets, so that no two neighbouring elements belong to the same set. Obviously, the notion of proximity depends on the particular function space, then this is considered in the colouring algorithm. The assembly proceeds iterating over colours and splitting their nodes among threads: with this technique race conditions are avoided and the user can enjoy the benefits of parallelisation without incurring in unpredictable behaviour.

Contrasting to the first approach, the second paradigm is based on **MPI** and addresses the needs of distributed memory architectures. Unfortunately, the latter is less immediate than the former, requiring a DOLFIN program to be launched with the MPI execution utility, but in this case the code need not be modified. In this implementation, the mesh is split so that each process gets its part of it, with an algorithm striving to minimise inter-process communication. With scalability in mind, no single process holds the full matrix and, moreover, everything happens behind the scenes: this way the user has no need of taking care of low level issues. The distributed memory paradigm is diffusely supported in the algebraic back-ends, so it allows the usage of several solvers, both indirect and direct. As already noted, this and the previous approach can be combined, for instance distributing the computation on a cluster and further speeding up the assembly process enabling multi-threading within each node, provided they are multi-core machines.

The shared memory paradigm should be quite straightforward to implement in **fem-fenics**. I expect to operate on a couple of functions: the private generate_makefile and the two assemble and assemble_system. The former should have the proper compilation flag (-fopenmp) added. The latter should have a new line reading like:

dolfin::parameters["num_threads"] = femfenicsthreads;

The number of threads could be passed to those functions as an argument, but this would ruin the interface compatibility with FEniCS, so this is a poor approach. Another way of addressing the issue is to define a global Octave variable in PKG_ADD and store in it the desired number of concurrent threads to use for the assembly.

The implementation of the distributed memory paradigm, instead, seems quite tricky. Basically, **Octave** does not use MPI, at least not Octave core. Nonetheless, there are two Forge packages with this goal, **mpi** and **parallel**. I will go through the documentation of these packages to understand if and, in case, how they address the problem of launching the oct-file with mpirun or mpiexec. Even leaving this aspect aside, I still do not know how easily the distributed objects storing matrices and vectors can be accessed to obtain the whole data.

In conclusion, I will initially work to add shared memory parallelism, at the same time looking deeper into the issues related to the distributed memory paradigm, which I suspect of being more than the ones highlighted.

by Eugenio Gianniti (noreply@blogger.com) at June 03, 2014 02:19 AM

Here I am again.

This week has been more about researching than coding. I have finally been able to reproduce the output from the*ilutp(ilu with threshold and pivoting)* Matlab's algorithm with an m-script (named ILU_pc.m in my project's directory). The fact is that Matlab does not implement the algorithm as is described in Yousef Saad's book in a few ways. Because of that I had to do reverse engineering, testing many cases and matrices. That is the function, **ugly** as hell, but is just for testing purposes.

function [A, P] = ILU_pc(A, tau, thresh)

B = A;

n = length(A);

P = speye(n);

for i = 1:n

for k = i:n

A(k:n,k) *= thresh;

A(k,k) /= thresh;

[m,mi] = max(abs(A(k:n,k)))

A(k,k) *= thresh;

A(k:n,k) /= thresh;

mi = mi + k -1;

tmp = A(mi,:);

A(mi,:) = A(k,:);

A(k,:) = tmp;

e = speye(n);

e(mi,mi) = 0; e(k,mi) = 1;

e(k,k) = 0; e(mi,k) = 1;

P = e*P;

endfor

for k = 1:i-1

if ( (A(i,k) == 0) || (abs(A(i,k)) < (tau*norm(B(:,k)))))

A(i,k) = 0;

continue

endif

A(i,k) = A(i,k) / A(k,k);

A(i,k+1:n) = A(i,k+1:n) - A(i,k) * A(k,k+1:n);

endfor

endfor

for i = 1:n

for j = i+1:n

if (abs(A(i,j)) < (tau*norm(B(:,j))))

A(i,j) = 0;

end

end

end

end

This week has been more about researching than coding. I have finally been able to reproduce the output from the

function [A, P] = ILU_pc(A, tau, thresh)

B = A;

n = length(A);

P = speye(n);

for i = 1:n

for k = i:n

A(k:n,k) *= thresh;

A(k,k) /= thresh;

[m,mi] = max(abs(A(k:n,k)))

A(k,k) *= thresh;

A(k:n,k) /= thresh;

mi = mi + k -1;

tmp = A(mi,:);

A(mi,:) = A(k,:);

A(k,:) = tmp;

e = speye(n);

e(mi,mi) = 0; e(k,mi) = 1;

e(k,k) = 0; e(mi,k) = 1;

P = e*P;

endfor

for k = 1:i-1

if ( (A(i,k) == 0) || (abs(A(i,k)) < (tau*norm(B(:,k)))))

A(i,k) = 0;

continue

endif

A(i,k) = A(i,k) / A(k,k);

A(i,k+1:n) = A(i,k+1:n) - A(i,k) * A(k,k+1:n);

endfor

endfor

for i = 1:n

for j = i+1:n

if (abs(A(i,j)) < (tau*norm(B(:,j))))

A(i,j) = 0;

end

end

end

end

- The next goal to achieve is obviously to implement the function as .oct file translating this algorithm into a sparse one using Octave's internal data types.

- All the testing I did was at college using their Matlab license. That delayed me because I couldn't do almost nothing in the weekend. Now I have a function that reproduce the behavior of Matlab's version I can test against it my c++ code.

I’ve been working on some stuff. I’ll get to that in my next post. For the sake of having something written, I’ll pontificate on what “has” means as it caused some confusion for me while working with Octave.

After a brief discussion with John on the Freenode channel, I discovered that my idea of the word “has” is different from in Octave terminology.

If every full carton *has* a dozen (12) eggs and a fridge drawer *has *three full egg-cartons, then the drawer *has 3×12 = *36 eggs.

We can also work backwards. If a fridge shelf contains only full cartons of eggs and we know that it has 36 eggs, then it must be the case that this shelf *has* 36/12 = 3 cartons

But for 3-dimensional matrices, this is not so. How many columns does this matrix have?

octave:1> zeros(2,2,2)

ans =

ans(:,:,1) =

0 0

0 0

ans(:,:,2) =

0 0

0 0

Well, every column *has* 2 elements and the matrix *has* 8 elements total so I would say it *has *8 / 2 = 4 columns (we could also say it *has* 2 pages and each page has 2 columns so there are 2 * 2 = 4 columns).

But although everyone agrees on the definition of a column (even in the three-dimensional setting; a single column is of the form A(:,i,j) for some i and j), apparently the definition of the word “*has”* is overloaded when talking about matrix dimensions. In Octave terminology, the above matrix *has* only 2 columns, which means “the length of the row-dimension is 2″. Equivalently, A has x rows translates to “the length of the column-dimension is x”

I don’t know if this terminology generalizes.

This week I started my work on the ufl function: it is now possible to write ufl code on-the-go, directly in your m-files. You can see below how the Poisson.ufl file of the homonymous example provided with **fem-fenics** (on the left) can be translated to a snippet of Octave code:

# Copyright (C) 2005-2009 Anders Logg element = FiniteElement("Lagrange", triangle, 1) u = TrialFunction(element) v = TestFunction(element) f = Coefficient(element) g = Coefficient(element) a = inner(grad(u), grad(v))*dx L = f*v*dx + g*v*ds | # Copyright (C) 2005-2009 Anders Logg ufl start Poisson ufl element = FiniteElement("Lagrange", triangle, 1) ufl ufl u = TrialFunction(element) ufl v = TestFunction(element) ufl f = Coefficient(element) ufl g = Coefficient(element) ufl ufl a = inner(grad(u), grad(v))*dx ufl L = f*v*dx + g*v*ds ufl end |

Basically, you just need to prepend what you would have written in your .ufl file with ufl. As you can see, anyway, there are also two new instructions. fem-fenics still needs to store your code in a separate file, which is then compiled using ffc, the FEniCS form compiler, but now ufl takes care of the process.

Your code should begin with the start command, and optionally with the name you want to assign to the file: in this example, we choose to open a new Poisson.ufl file. Be aware that ufl will not overwrite an existing file so, if you plan to use your script for several runs, my suggestion is to keep your working directory clean and tidy with a delete ('Poisson.ufl') after the snippet above.

When you are fine with your ufl code, the end command will tell ufl that it can compile and provide you with your freshly built problem. You can also specify options like BilinearForm (it is not the only one available, find a comprehensive list in the help message, in Octave), in case you wrote just part of the problem in your last lines.

A lot of commitment was devoted to this function. This is not due to intrinsic difficulties: a sketch of the function's code has been around for a while and the current implementation has not consistently slid away from it. The goal was to obtain a robust piece of code, since it will be the cornerstone of a new paradigm in fem-fenics usage. At least each and every example provided with the package needs to be modified to take advantage of this change, and this will be my next task.

by Eugenio Gianniti (noreply@blogger.com) at May 23, 2014 02:05 PM

As said in my previous post, I have been working on extending the implementation of interpolate to allow for an Expression as input. Currently it can also be used as in the Python dolfin interface, see here. Let's see how to use this new function in **fem-fenics**.

This example can be found in the FEniCS Book, it is the very first. The problem at hand is the Poisson equation with Dirichlet boundary conditions:

*- *Δ*u = f *in* Ω*

*u = u*_{0} on* ∂Ω*

We will solve this problem on the unit square, with *f* constant and equal to -6 and u_{0} = 1 + x^{2} + 2y^{2}. It can be verified that the exact solution is u_{ex} = 1 + x^{2} + 2y^{2}. With the following ufl file:

element = FiniteElement("Lagrange", triangle, 1)

u = TrialFunction(element)

v = TestFunction(element)

f = Coefficient(element)

a = inner(grad(u), grad(v))*dx

L = f*v*dx

and Octave script:

pkg load fem-fenics msh

import_ufl_Problem ('Poisson')

# Create mesh and define function space

x = y = linspace (0, 1, 20);

mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));

V = FunctionSpace('Poisson', mesh);

func = @(x,y) 1.0 + x^2 + 2*y^2;

# Define boundary condition

bc = DirichletBC(V, func, 1:4);

f = Constant ('f', -6.0);

# Define exact solution

u_e = Expression ('u_ex', func);

a = BilinearForm ('Poisson', V, V);

L = LinearForm ('Poisson', V, f);

# Compute solution

[A, b] = assemble_system (a, L, bc);

sol = A \ b;

u = Function ('u', V, sol);

# Save solution

save (u, 'poisson');

# Interpolate and save the exact solution

u_e_int = interpolate (u_e, V);

save (u_e_int, 'exact');

it is possible to compute the numerical solution, interpolate the analytical one on the same function space and then compare them. Using a visualisation tool like Paraview, one can verify that the computed solution and the interpolation of the exact one are practically the same. This is due to the fact that the Finite Elements Method with triangle elements on a rectangular domain can exactly represent a second order polynomial, as the solution of the problem at hand.

Here you can see a good solution poorly post-processed in Paraview to the Poisson problem solved in the example.

by Eugenio Gianniti (noreply@blogger.com) at May 19, 2014 07:27 PM

As code period is starting today, I want to write a brief timeline for the first period of the GSOC here:

**19 May-20 June:**Implement ilu related functions (ilu0.cc, iluc.cc, ilutp.cc) and merge them together with ilu.m script

**20-25 June:**Automated test writing and documentation. Integration to mainstream octave code should be achieved here.

**27 June:****(Millstone 1)**ilu.m is fully functional and integrated with Octave core.

- Taking the idea from Kai's last year blog, I will keep track of what is already done with the following figure.

Regarding repository setup, Kai helped me to configure a subrepository using bitbucket service. At present, it only contains an outdated Octave development version just to make sure things work. For cloning:

hg clone https://edu159@bitbucket.org/edu159/octave-subrepo

However, I would not need to use this subrepo until the final integration of my code into Octave. For development purposes I have set another repository for daily work, as I am working with .oct files that compile standalone. Here is the repo you should check for my updated work.

hg clone https://edu159@bitbucket.org/edu159/gsoc2014-edu159

See you next week!

Lately I have coded my first function for **fem-fenics**: it is interpolate, which wraps the homonymous method of the dolfin::Function class. This allows to interpolate a Function, G, on the FunctionSpace of Function F, even if they are not defined on the same mesh, with a call like this:

res = interpolate (F, G)

I am working on extending it to allow for an Expression as input. With this function it is possible to make a quantitative comparison between the results of different discretisation approaches or to check the accuracy of a method, comparing the computed solution and an analytically obtained one.

#### The implementation

I provide here a comprehensive overview of the code for interpolate. First of all, the number of input arguments is obtained and an octave_value is declared, in order to hold the output. Then there is a check ensuring that exactly two arguments are provided and no more than one output value is asked for.

After verifying these preliminary conditions, there are some instructions checking that the function type is loaded and, if necessary, registering it. This way Octave is aware of it and can store it in an octave_value.

Eventually, the real computation is performed. After checking that the inputs are of the function type, with a static_cast the actual objects are extracted from the arguments:

const function & u0 = static_cast<const function&> (args(0).get_rep ());

const function & u1 = static_cast<const function&> (args(1).get_rep ());

Here comes the tricky part. The classes in fem-fenics are designed to hold constant dolfin objects, but dolfin::Function::interpolate is not a constant method. In order to be able to call it, a local dolfin::Function is constructed, used to perform the interpolation, then fed to the function constructor and assigned to the return value:

boost::shared_ptr<dolfin::Function> output (new dolfin::Function (u0.get_fun ()));

const dolfin::Function & input = u1.get_fun ();

output->interpolate (input);

std::string name = u1.get_str ();

retval = new function (name, output);

by Eugenio Gianniti (noreply@blogger.com) at May 17, 2014 05:22 PM

Octave has been selected as a mentor organization for the European Space Agency‘s Summer of Code in Space!

This is our third year in SOCIS. See Roberto Porcù’s blog for the work from last year.

If you are an eligible student and interested in applying, check out our Ideas page. May 15 is the student application deadline.

In this post I identify my project's goals, as already published on Melange.

### list of tasks

### details

### tentative agenda

- reduce copies when passing matrices from Octave to dolfin;
- avoid useless compilations of .oct files when they are not needed;
- avoid using separate .ufl files, introducing the possibility to write UFL code in .m files;
- implement in fem-fenics further FEniCS functionalities, preferably according to the FEniCS developers' directions;
- improve the build and distribution system, so that end users can enjoy full functionality right away after installing from Forge.

I will address point 3 implementing an .m function which accepts strings as arguments and writes them to a file. There should be two keywords, such as start and end, to identify where the UFL input begins and finishes. After writing this code, it will be compiled when needed. This way UFL instructions could be written directly in .m files in the following manner:

ufl start filename

ufl <first line>

ufl <second line>

ufl <...>

ufl <last line>

ufl end

ufl <first line>

ufl <second line>

ufl <...>

ufl <last line>

ufl end

To address point 5, instead, I will add instructions to PKG_ADD to automatically find out, through pkg-config, the proper flags to allow for the just in time compilation of .oct files. I will also add instructions to PKG_DEL to restore the environment at its previous state when the package is unloaded. This would allow end users to use the package without taking care of the problems reported here.

- 22 April - 18 May: Study of the documentation and interaction with my mentor and the Octave community to understand thoroughly the code base and the contribution expected from my project
- 19 May - 22 June: First phase of the project. I will implement bindings for the UFL language in Octave and adapt accordingly the provided examples. I will also work on the build and distribution system to allow for feedback on it from the community. In the end, I will commit a first set of new functions
- 23 June - 27 June: Period for the submission of the mid-term review, I will double check the functionalities already implemented and improve their documentation
- 28 June - 10 August: Second phase of the project. I will improve the package performance, both reducing copies of matrices between Octave and dolfin and implementing checks to avoid useless just in time compilations. Furthermore, I will add a second, probably larger, set of new functions, as suggested by the community and FEniCS developers. I expect to code some new examples which make use of the freshly introduced capabilities
- 11 August - 22 August: Week devoted to final minor fixes and to the improvement of the documentation

by Eugenio Gianniti (noreply@blogger.com) at April 27, 2014 08:45 PM

My name is Eugenio, a student in Mathematical Engineering at Politecnico di Milano. This summer I will be working with GNU Octave to continue the implementation of fem-fenics, an Octave Forge package started in last year Google Summer of Code. It is intended as a wrapper of FEniCS, a general purpose finite elements library, and its goal is to provide such numerical methods in the familiar interface offered by Octave.

In this blog you will find up-to-date information about the state of my contribution.

In this blog you will find up-to-date information about the state of my contribution.

by Eugenio Gianniti (noreply@blogger.com) at April 27, 2014 08:14 PM

My name is Eduardo (edu159), and that blog has the purpose of tracking the state of my project with Octave during the GSOC2014 program ~~(if I become selected)~~ .

You can visit my public profile at the Octave wiki here:

Feedback would be welcome. Feel free to comment :).

Eduardo

You can visit my public profile at the Octave wiki here:

Feedback would be welcome. Feel free to comment :).

Eduardo

I have to announce that I have been selected for the 2014 GSOC program and I am very happy with that. In a few days I will continue from where I left my project.

Thanks to all!

Thanks to all!

If you've ever been in a situation where you wanted to see *only* the additions,
or *only* the deletions, in a git diff, then you probably know that it's not as
simple as running `git diff --diff-filter=A`

or `git diff --diff-filter=D`

. At
least, not if you're looking for added or deleted *lines*, as opposed to files.

From the man page for `git diff`

:

```
--diff-filter=[ACDMRTUXB*]
Select only files that are
* A Added
* C Copied
* D Deleted
* M Modified
* [omitted]
```

That's pretty cool, but this only works on **files**. So this would let us see
only new files, or only deleted files. But what about lines? What if you only
want to see the lines that have been added (or deleted) in one file?

Unfortunately, git does not appear to have an option for this built-in. On the bright side, if you run a *NIX system, you can do this:

`git diff | grep ^+`

This gets you all the additions. To see all the deletions:

`git diff | grep ^-`

Note that the output of the first command will also give you something like

`+++ b/filename`

for each affected file. If this is a problem, you can alter the regex to account for this (or just delete it manually).

You can of course pass other parameters to `git diff`

, such as the specific
commit or commit range you want to see the diff for, or the filename(s).

So why would anyone ever need to do something like this?

If you've ever been to a Model United Nations conference, you probably know that in exchange for paying a nominal registration fee and giving up your weekend to try to convince someone (who is not actually a representative of the United States) that you (someone who is not actually a representative of Iran) have absolutely no intention of producing nuclear weapons, definitely not, let's go out for a drink and I'm sure we can clear up this whole misunderstanding, you get a nice shiny badge that says your name, your school, and the country that you are supposed to pretend to represent, which looks something like this:

For the last year or so, I was part of the organising committee for a Model United Nations conference that is sadly not named BADPUNS. As the USG-IT, one of my responsibilities was to take the registration information that was provided by conference attendees through the website and send it to the person in charge of printing badges. Since the website was completely custom-built and I didn't feel like writing a ton of code to make this into an actual feature (since, theoretically, it would only have to be used once), I just wrote a simple Django management command for exporting the information for the badges as a CSV file.

It worked fine, except for the fact that it was a week before the conference and some of the attendees still had not filled in their names. But we needed to get at least some of the ~1400 badges printed now, so I sent off what I had, and those badges got printed.

Two days later, when I exported the new badge list, I used git to figure out the difference between the current badges.csv and the previous one. The difficulty arose from the fact that not only did we have new badges to print (because some attendees didn't fill in their information before), we had badges to get rid of as well (because some attendees had last-minute cancellations). So I needed to use the exported badges.csv file to generate a new CSV containing just the new badges to be printed, as well as one listing the badges to be discarded.

That's where git and grep came to the rescue. To get rid of the + and - at the beginning of each line, I just used Vim's visual block mode. Much easier than manually checking each name, which is probably what I would have done if I didn't have git in my life.

There may actually be a way of doing this with just git, but I haven't found it. This might be a pretty hacky solution, but hey, it works.

Know of a better solution? Do tell! I'm @dellsystem on Twitter.

I just spent 5 days at PyCon 2014 here in Montréal (3 days for the actual conference, 2 days sprinting), and wow, what a great conference that was.

There are many things I want to praise about the whole experience. The venue was great, the organisation was superb, the talks were interesting, the infrastructure was amazing, the atmosphere was friendly… but most of all, I think I want to praise the entire culture of inclusiveness that the Python community is trying to promote.

It is interesting that the only true common thread at the conference was a programming language (and not even that, sometimes, some of the talks were hardly about the Python programming language at all). Python was originally conceived as a programming language that was meant to be as easy as possible to understand. Whether it has succeeded from a purely language-design point of view is hard to say, and not *everything* about Python is great. The language has its gotchas here and there, just like any other language. And yet, despite not being a perfect language programming language, it’s able to bring together such a diverse group of individuals together to accomplish common goals.

Python is an excellent programming *lingua franca* for everyone, not just for Unix nerds (witness: Windows support is taken seriously) and not just for programming geeks (witness: Software Carpentry). Just take a look at the wide range of topics covered in the talks. General software development, the benefits of software freedom, cryptography and security (lol, heartbleed)…

Of particular note is that 1/3 of all attendees *and* speakers were female. Can any other tech conference boast such inclusiveness of the usually tech-neglected half of humankind? Look at all the talks related to gender issues: sexism in rap lyrics via machine learning, being a transgender Python hacker, or how to bring more Python to girls in school.

Now, to be clear, I don’t think that PyCon has eliminated sexism or that we have “won” this battle. As I overheard someone say, PyCon will not be inclusive enough for women unless the lines for the women’s bathroom are as long as the lines for the men’s. And there are still many issues, such as women still being “invisible” and ignored, or as I overheard someone else say, she actually had to say to some guy to look up from her breasts while she was talking to him. It’s not there all the way yet.

This just seems like a good start. I hope next year at PyCon 2015, we’ll be able to get 50% women attendees and speakers!

My intention is upgrading some functions related with sparse matrices so they become compliant with Matlab and implement others that are not present in Octave right now. This is the list plus some comments about how I expect to do things.

**some modifications are needed **to improve performance. Anyway I have mailed Rik (the last one that modified the source code of that functions) and has told me that he will give it a look in a couple of weeks.

2: I have not yet investigated enough to give a road map for implementing that function.

3 & 4: For implementing that functions I found those links:

In this website there are codes written in several programming languages that implement the algorithms (the authors are the ones that written the paper that Matlab gives as reference in their documentation for both functions). I have emailed professor Michael Saunders about adapting them into Octave versions and he answered me that I am welcome to do while I respect the license (CPL or BSD licenses). They are meant to be very unrestrictive but I need to get informed about the compatibility with GPL3.

5: Here comes the big one. That function has a big chunk of options and the last year was almost implemented by Kai Torben as his GSOC project. He interfaced Octave with ITSOL/ZITSOL libraries but in the end there were some issues with that approach:

__My approach:__ I will write from scratch all the functions needed as oct-files (ILUTP, ILU0, ILUC and ILUT) for real and complex numbers implementing the algorithms described by Yousef Saad in his book *"Iterative methods for sparse linear systems Ed. 2". *I can use some of the code that Kai wrote, mostly the tests and the m-file "ilu.m" that glue together all the functions.

6:~~ That function need less work since it is almost all implemented. Just some complex implementations are missing and the modified version of the algorithms too. Since Kai implemented those functions from scratch there are no dependency problems. That is nice.~~

Update 1: Kai has commented me that there are some license issues on the FORTRAN code he used (see here). So ichol related functions need to be implemented from scratch.

Update 2: Following Kai's recommendations I will focus on ilu/ichol functions for the GSoC period and just in case there is some time left go ahead with the other functions.

~~Implement sprand/sprandn with the 4rth parameter.~~(link) (Patch accepted)- sprandsym (implement arguments needed to be compliant with Matlab)
- lsqr
- minres
- ilu (complete last year GSOC project)
- ichol (complete last year GSOC project)

2: I have not yet investigated enough to give a road map for implementing that function.

3 & 4: For implementing that functions I found those links:

In this website there are codes written in several programming languages that implement the algorithms (the authors are the ones that written the paper that Matlab gives as reference in their documentation for both functions). I have emailed professor Michael Saunders about adapting them into Octave versions and he answered me that I am welcome to do while I respect the license (CPL or BSD licenses). They are meant to be very unrestrictive but I need to get informed about the compatibility with GPL3.

5: Here comes the big one. That function has a big chunk of options and the last year was almost implemented by Kai Torben as his GSOC project. He interfaced Octave with ITSOL/ZITSOL libraries but in the end there were some issues with that approach:

- ILUTP algorithm did not work
- He had to patch the library to get things work!
- modified versions of algorithms ("milu" option) were not implemented in the libraries
- That "ugly" scenario lead to finally not being able to include ITSOL as a dependency with Octave. Bottom line, the integration with the development repository could not be achieved.

6:

Update 1: Kai has commented me that there are some license issues on the FORTRAN code he used (see here). So ichol related functions need to be implemented from scratch.

Update 2: Following Kai's recommendations I will focus on ilu/ichol functions for the GSoC period and just in case there is some time left go ahead with the other functions.

.nobrtable br { display: none } tr {text-align: center;} tr.alt td {background-color: #eeeecc; color: black;} tr {text-align: center;} caption {caption-side:bottom;}

I have implemented the most basic function ilu0 that do the incomplete LU decomposition with 0-fill. I have drawn a table with the execution times of that algorithm using Matlab, my version and using the code that Kai implemented last year using ITSOL. The function right now can be used with real and complex numbers and has the milu='row' option implemented.

The table shows for a NxN sparse matrix with a number of non-zero elements NNZ, the time of execution (tic - toc was used).

It can be seen that the implementation using ITSOL is the slowest. Maybe just because the overhead of copying and translating data back and forth between Octave and ITSOL. Between my version and Matlab's there is almost no difference.

Here you can download the code (ilu0.cc). It does not have any test nor documentation written yet.

I have implemented the most basic function ilu0 that do the incomplete LU decomposition with 0-fill. I have drawn a table with the execution times of that algorithm using Matlab, my version and using the code that Kai implemented last year using ITSOL. The function right now can be used with real and complex numbers and has the milu='row' option implemented.

The table shows for a NxN sparse matrix with a number of non-zero elements NNZ, the time of execution (tic - toc was used).

N-NNZ | ILU0-mine | ILU0-Matlab | ILU0-ITSOL |
---|---|---|---|

50 - 683 | 0.000055 s | 0.000065 s | 0.00085 s |

400 - 72435 | 0.027 s | 0.024 s | 0.04 s |

2000 - 1805571 | 3.35 s | 3.25 s | 4.88 s |

5000 - 6482839 | 14.2 s | 14.5 s | 22.75 s |

It can be seen that the implementation using ITSOL is the slowest. Maybe just because the overhead of copying and translating data back and forth between Octave and ITSOL. Between my version and Matlab's there is almost no difference.

Here you can download the code (ilu0.cc). It does not have any test nor documentation written yet.

Octave has been accepted into Google Summer of Code, for our first time as an independent organization! Student applications are due March 21, and decisions about accepting students will be made in early April.

To make GSoC a success, we need not only strong student programmers but also committed mentors to supervise the students and work with them to have contributions be useful for Octave. This can be tough to achieve because mentors are unpaid (though Google may send you a nifty T-shirt).

Each project should have a primary mentor and a backup mentor. In my experience, primary mentors should plan to devote 5-10 hours per week to the project, including speaking with the students and reviewing their code. Backup mentors have a smaller time commitment but still should keep up with project progress and be available to step in when the primary mentor is otherwise occupied.

If you’d like to be a project mentor:

1) Go to the wiki projects page and put your name next to projects you’d be interested in mentoring. Feel free to improve the project description at the same time.

2) Sign up as a mentor for Octave on the GSoC Melange site.

3) Start interacting with student applicants who have similar topics of interest.

I am cautiously hopeful for bitcoin. I just ate pizza and poutine with a group of friends, paying at the restaurant with bitcoins!

I am not a speculator. I am not an investor. I am not a miner. I am not trying to get rich nor make any money whatsoever by manipulating bitcoins. I just want to be able to earn and spend money on the internet without needing the permission of a bank, or the permission of Paypal, or making it anyone’s business but mine and the person I’m actually doing business with.

I acquired some bitcoins about a year ago, at the time it was about 1.8 bitcoins worth around 30 USD. I did not buy them. I did not invest on bitcoins. I did not mine them. I received them as a payment for tutoring mathematics in an IRC channel for an hour and a half. I earned these bitcoins through my labour, just like I would earn any other currency.

At the time there was not much I could do with bitcoins. Although I freely encouraged and accepted the payment in bitcoins, I did it more with amusement than conviction. I am not a criminal, but since at the time Silk Road still had a sort of forbidden underground allure, my client suggested that I could use the bitcoins to buy illegal drugs on the black market. I had no intention to do so, but I decided to keep the bitcoins around, as a cautious hope I could use them for something else.

I kept the bitcoins for quite some time, since I could not find anything to spend them on. The first thing I could find that seemed interesting was a 4chan “account” (in reality, just an exemption from its captcha). Despite the website’s bad reputation as being the cesspool of the internet, I think that there is value in its pro-anonymity ethos, something that is slowly being eroded away in today’s online world. This was also my first test case for the possibility of anonymous online currency. In keeping with this ethos, I made the payment, and when there was a slight hiccup with the transaction, used a throw-away email address to resolve the issue. No chargebacks, but people are still people and are still mostly nice. I thus obtained my 4chan captcha exemption. I have not really used it since then, but I am satisfied knowing that I have made a small contribution towards promoting anonymity.

I kept my eye out for news stories of more opportunities to spend bitcoins on. The currency seemed to be slowly gaining more adoption, especially for non-material goods and services. My next purchase was one month of reddit gold as a gift for a particularly witty commentator. These two purchases together had already given a significant blow to my bitcoin balance, but I was not duly concerned. After all, this was just pocket change I acquired for 90 minutes of a hobby task.

Then, suddenly, over a couple of weeks the bitcoin price exploded from 20 USD per bitcoin to over 1000 USD per bitcoin. I didn’t exactly become a millionaire, but my paltry fraction of a bitcoin now had the power to buy more things.

I made a few more purchases. A PDF copy of Julian Assange’s et al *Cypherpunks* book. A month of VPN access. Sent bitcoins to a kind stranger on the internet in exchange for digital albums from indie band *Nectarphonic*. When Gyft.com started selling gift cards to Amazon.com in exchange for bitcoins, I obtained my first physical product for bitcoins: a boxed set of Susan Collin’s *The Underland Chronicles*.

This really had just been my time getting used to bitcoin and how it works. The fluctuations in price, the cryptography behind it, the UI presented first by the “official” bitcoinqt client and understanding more how the alternative bitcoin client Electrum works. The security model, what it means to “own” bitcoins. Passwords and private keys. The workings of the blockchain. Using Tor for enhanced anonymity of transactions.

I finally got around to reading Satoshi Nakamoto’s whitepaper. Wow. I don’t know if this person or group is a genius or just a child of the times, but Nakamoto seems to have solved a cryptography protocol problem that nobody had solved before. Nakamoto didn’t invent the idea of cryptocurrencies, but merely built upon the ideas of others in order to build a decentralised system. I was duly impressed.

Then two days ago I saw that the first restaurant in Montréal was proclaiming to accept bitcoins. I could now buy Québec’s signature dish, poutine, with bitcoins! Excited, I started making plans.

I eagerly announced yesterday on IRC that I was planning to go to this restaurant to try out my bitcoins. Another chatter local to Montréal decided to tag along for fun. We made plans to go that very evening for a night of socialising, three couples. I eagerly explained to everyone my excitement over finally being able to pay someone face-to-face with bitcoins.

My fiancée and I got to Montréal Poutine in the Old Port a few minutes earlier than everyone else. It was a small location, hardly more than a greasy spoon, but par for the course for a poutine joint. There were signs all over the establishment announcing the possibility of paying with bitcoins and a wifi password on their chalkboards.

We eyed the menu, and I nervously made a few preliminary checks to ensure the transaction would go smoothly. Due to one of my many quirks, I do not own a pocket computer (“smartphones”, neither smart and hardly phones), so I was prepared to pay with my laptop and a webcam for scanning a QR code. I ensured that the internet connection was stable and that I could scan a QR code. As instructed in bitcoinaccepted.ca (or in payezenbitcoin.ca, because this is .ca), I proudly announced to my server that I was going to pay with bitcoins. “Oh, wow, it’s my first time,” she said in French. “Mine too,” I replied with a smile.

Our company arrived, the two other couples. We ordered various kinds of poutine, pizza, avocado fries, beer and mineral water. We chatted about many things, and soon predictable discussions about bitcoins ensued. Their method of operation, their security, their volatility but recent relative stability. Musings if they would ever work or not. The more techy in the bunch were showing signs of optimism, while those who found the idea more foreign were predicting the eventual doom for bitcoins.

After a very pleasant evening, it was time to pay the bill.

I announced that I had enough bitcoins to pay for all the food, but I asked everyone else to divvy up the drinks between them. I also told them that I had read that tips were not accepted in bitcoins yet, so that would have to be paid elsehow.

I readied my laptop and the webcam, to the growing amusement of my companions. The server came with one bill for food and another for drinks. I reminded her that I was going to pay with bitcoins, but I wasn’t quite sure what to expect. I had seen videos online of handheld machines that displayed a QR code with the bitcoin address to pay to, and that was my guess for what I would see. Instead, she pointed me to a static QR code pasted next to the cash register. Was this the bitcoin address to pay to? I walked with my baroque laptop-and-webcam rig over to the cash register, in good fun as I heard my giggling friends behind me.

I used Electrum to scan the QR code, looking for the address. Instead of a bitcoin address, this was a Bitpay URL. I wasn’t quite sure what to do with it, and I floundered for a few seconds. I opened a generic QR code scanner to get the URL. One of the guys with us helpfully walked over to help me scan the QR code, but I had already managed to get the URL loaded in Firefox by this time. The server was re-reading the instructions next to the QR code on how to pay.

At the Bitpay URL, there were two fields to fill out: an order number and the amount in CAD to pay. The server read the instructions again and said to leave the order number blank. I filled in the amount with taxes, showed it to her, and we agreed it was correct. I clicked enter. A bitcoin address showed up. I went back to Electrum to send money to that address. I stumbled when typing my wallet password. The server, getting the hang of the transaction, said bemusedly, “I cannot help you with that.” Finally I got it, and the familiar “this invoice has been paid” message showed up on the Bitpay website.

A few seconds passed, and the restaurant staff confirmed on another computer that they had received the payment. I showed everyone at the table the paid invoice on my laptop screen. A few other customers in the restaurant were showing some interest. I announced to everyone, “it’s ok, bitcoin has worked, I made the payment!”

Everyone relaxed and proceeded to tackle the more familiar problem of paying for the drinks with more conventional currency.

I don’t know if bitcoin is going to crash or not. I don’t understand enough economics to comment on what the value of bitcoins is, or if it really is different from the value that we give to any other currency. I have a rough understanding of the cryptography behind it and what makes the whole thing work. I know people are working on making bitcoins more accessible to people less enthusiastic than me, and I very much wish they succeed.

All I know is that so far bitcoins have worked for me. I look forward to getting better acquainted with this new way to conduct business. I wish good fortune to everyone else who is also trying to build a new currency for an internet-powered age.

So, we want to know if this is going to work out.

#yop-poll-container-2_yp53a1e08df377f { width:200px; background:#555; padding:10px; color:#fff; overflow:hidden; font-size:12px; } #yop-poll-name-2_yp53a1e08df377f { font-size:14px; font-weight:bold; } #yop-poll-question-2_yp53a1e08df377f { font-size:14px; margin:5px 0px; } #yop-poll-answers-2_yp53a1e08df377f { } #yop-poll-answers-2_yp53a1e08df377f ul { list-style: none outside none; margin: 0; padding: 0; } #yop-poll-answers-2_yp53a1e08df377f ul li { font-style:normal; margin:0px 0px 10px 0px; padding:0px; font-size:12px; } #yop-poll-answers-2_yp53a1e08df377f ul li input { margin:0px; float:none; } #yop-poll-answers-2_yp53a1e08df377f ul li label { margin:0px; font-style:normal; font-weight:normal; font-size:12px; float:none; } .yop-poll-results-2_yp53a1e08df377f { font-size: 12px; font-style: italic; font-weight: normal; margin-left: 15px; } #yop-poll-custom-2_yp53a1e08df377f { } #yop-poll-custom-2_yp53a1e08df377f ul { list-style: none outside none; margin: 0; padding: 0; } #yop-poll-custom-2_yp53a1e08df377f ul li { padding:0px; margin:0px; font-size:14px; } #yop-poll-container-2_yp53a1e08df377f input[type='text'] { margin:0px 0px 5px 0px; padding:2%; width:96%; text-indent:2%; font-size:12px; } #yop-poll-captcha-input-div-2_yp53a1e08df377f { margin-top:5px; } #yop-poll-captcha-helpers-div-2_yp53a1e08df377f { width:30px; float:left; margin-left:5px; height:0px; } #yop-poll-captcha-helpers-div-2_yp53a1e08df377f img { margin-bottom:2px; } #yop-poll-captcha-image-div-2_yp53a1e08df377f { margin-bottom:5px; } #yop_poll_captcha_image_2_yp53a1e08df377f { float:left; } .yop_poll_clear { clear:both; } #yop-poll-vote-2_yp53a1e08df377f { } .yop-poll-results-bar-2_yp53a1e08df377f { background:#f5f5f5; height:10px; } .yop-poll-results-bar-2_yp53a1e08df377f div { background:#333333; height:10px; } #yop-poll-vote-2_yp53a1e08df377f div#yop-poll-vote-2_yp53a1e08df377f button { float:left; } #yop-poll-vote-2_yp53a1e08df377f div#yop-poll-results-2_yp53a1e08df377f { float: right; margin-bottom: 20px; margin-top: -20px; width: auto; } #yop-poll-vote-2_yp53a1e08df377f div#yop-poll-results-2_yp53a1e08df377f a { color:#fff; text-decoration:underline; font-size:12px;} #yop-poll-vote-2_yp53a1e08df377f div#yop-poll-back-2_yp53a1e08df377f a { color:#fff; text-decoration:underline; font-size:12px;} #yop-poll-vote-2_yp53a1e08df377f div { float:left; width:100%; } #yop-poll-container-error-2_yp53a1e08df377f { font-size:12px; font-style:italic; color:red; text-transform:lowercase; } #yop-poll-container-success-2_yp53a1e08df377f { font-size:12px; font-style:italic; color:green; }
This is the link for the documentation file :

SOCIS 2013 - Documentation for new functions added to odepkg

This is the link for the implemented code :

OdePkg with my added functions

SOCIS 2013 - Documentation for new functions added to odepkg

This is the link for the implemented code :

OdePkg with my added functions

by Roberto Porcù (noreply@blogger.com) at January 05, 2014 06:25 AM

Constrained mechanical systems form an important class of differential equations on manifolds. For all the theory that I present here and I've used for the implementation I refer to "Geometric Numerical Integration" by Hairer, Lubich and Wanner.

Consider a mechanical system described by position coordinates \(q_1,\dots q_d\) and suppose that the motion is constrained to satisfy \(g(\mathbf q) = \mathbf 0\) where \(g:\mathbb R^d \rightarrow \mathbb R^m \), with \(m \lt d \). So that, the equations of motion governing the system become: $$ \left\{ \begin{array}{l} \dot{\mathbf q} = \frac{\partial H}{\partial \mathbf p} \\ \dot{\mathbf p} = -\frac{\partial H}{\partial \mathbf q} - G(q)^T\lambda \\ g(\mathbf q) = \mathbf 0 \end{array}\right. $$ where \(G(\mathbf q) = \dfrac{\partial g(\mathbf q)}{\partial \mathbf q} \).

Symplectic Euler method can be extended to constrained systems but we focus on **SHAKE** and **RATTLE** algorithms. SHAKE is a 2-steps algorithm so that, since I'm implementing only 1-step algorithms and the overall structure of solvers and integrators is made for 1-step solvers, I implemented just **RATTLE** algorithm.

The RATTLE algorithm implemented works with any general Hamiltonian \(H(\mathbf q,\mathbf p \) and is defined as follows: $$ \left\{\begin{array}{l} \mathbf p_{n+\frac{1}{2}} = \mathbf p_n -\frac{h}{2}\big(H_{\mathbf q}(\mathbf q_n,\mathbf p_{n+\frac{1}{2}}) + G(\mathbf q_n)^T \mathbf {\lambda}_n \big) \\ \mathbf q_{n+1} = \mathbf q_n +\frac{h}{2} \big( H_{\mathbf p}(\mathbf q_n,\mathbf p_{n+\frac{1}{2}}) + H_{\mathbf p}(\mathbf q_{n+1},\mathbf p_{n+\frac{1}{2}}) \big) \\ g(\mathbf q_{n+1}) = \mathbf 0 \\ \mathbf p_{n+1} = \mathbf p_{n+\frac{1}{2}} -\frac{h}{2}\big(H_{\mathbf q}(\mathbf q_{n+1},\mathbf p_{n+\frac{1}{2}}) + G(\mathbf q_{n+1})^T \mathbf{\mu}_n \big) \\ G(\mathbf q_{n+1}) H_{\mathbf p}(\mathbf q_{n+1},\mathbf p_{n+1}) = \mathbf 0 \end{array}\right. $$ where \( h=\Delta t=t_{k+1}-t_k\) and \(\mathbf{\mu}_n \),\( \mathbf{\mu}_n \) are Lagrangian multipliers nedded to impose the constraints.

It can be demonstrated that this numerical method is symmetric, symplectic and convergent of order 2.

The following code represent it's implementation:

`function [t_next,x_next,err]=rattle(f,t,x,dt,options)`

H = odeget(options,'HamiltonianHessFcn',[],'fast');

GG = odeget(options,'ConstraintHessFcn',[],'fast');

if( ~isempty(H) && ~isempty(GG) )

fsolve_opts = optimset('Jacobian','on');

else

fsolve_opts = optimset('Jacobian','off');

end

g = odeget(options,'ConstraintFcn',[],'fast');

G = odeget(options,'ConstraintGradFcn',[],'fast');

c_nb = odeget(options,'ConstraintsNb',[],'fast');

dim = length(x)/2;

q0 = x(1:dim);

p0 = x(dim+1:end);

RATTLE = @(y)constr_sys(y,dim,c_nb,f,H,g,G,GG,t,q0,p0,dt);

y0 = [q0;p0;p0;zeros(2*c_nb,1)];

y0 = fsolve(RATTLE,y0,fsolve_opts);

t_next = t+dt;

x_next = [y0(1:dim);y0(2*dim+1:3*dim)];

if(nargout==3)

dt = dt/2;

RATTLE = @(y)constr_sys(y,dim,c_nb,f,H,g,G,GG,t,q0,p0,dt);

y0 = [q0;p0;p0;zeros(2*c_nb,1)];

y0 = fsolve(RATTLE,y0,fsolve_opts);

q0 = y0(1:dim);

p0 = y0(2*dim+1:3*dim);

t = t+dt;

RATTLE = @(y)constr_sys(y,dim,c_nb,f,H,g,G,GG,t,q0,p0,dt);

y0 = [q0;p0;p0;zeros(2*c_nb,1)];

y0 = fsolve(RATTLE,y0,fsolve_opts);

x_est = [y0(1:dim);y0(2*dim+1:3*dim)];

err = norm(x_est-x_next,2);

end

end

function [F,J] = constr_sys(y,dim,c_nb,f,H,g,G,GG,t,q0,p0,dt)

F = zeros(3*dim+2*c_nb,1);

F(1:dim) = y(1:dim) - q0 - (dt/2).*(f(t,[q0; ...

y(dim+1:2*dim)])(1:dim) + f(t+dt,y(1:2*dim))(1:dim));

F(dim+1:2*dim) = y(dim+1:2*dim) - p0 - (dt/2).*(f(t,[q0; ...

y(dim+1:2*dim)])(dim+1:end) - G(q0)'*y(3*dim+1:3*dim+c_nb));

F(2*dim+1:3*dim) = y(2*dim+1:3*dim) - y(dim+1:2*dim) - ...

(dt/2)*(f(t+dt,y(1:2*dim))(dim+1:end) - ...

G(y(1:dim))'*y(3*dim+c_nb+1:end));

F(3*dim+1:3*dim+c_nb) = g(y(1:dim));

F(3*dim+c_nb+1:end) = G(y(1:dim))*(f(t+dt,[y(1:dim); ...

y(2*dim+1:3*dim)])(1:dim));

if( nargout==2 )

J = zeros(3*dim+2*c_nb,3*dim+2*c_nb);

J(1:dim,1:dim) = eye(dim) - ...

(dt/2)*(H(t+dt,y(1:2*dim))(dim+1:end,1:dim));

J(1:dim,dim+1:2*dim) = -(dt/2)*(H(t,[q0; ...

y(dim+1:2*dim)])(dim+1:end,dim+1:end) + ...

H(t+dt,y(1:2*dim))(dim+1:end,dim+1:end));

J(dim+1:2*dim,dim+1:2*dim) = eye(dim) + ...

(dt/2)*(H(t,[q0;y(dim+1:2*dim)])(1:dim,dim+1:end));

J(dim+1:2*dim,3*dim+1:3*dim+c_nb) = (dt/2)*G(q0)';

J(2*dim+1:3*dim,1:dim) = (dt/2)*(H(t+dt, ...

y(1:2*dim))(1:dim,1:dim));

for k = 1:1:c_nb

J(2*dim+1:3*dim,1:dim) = J(2*dim+1:3*dim,1:dim) - ...

(dt/2)*(y(3*dim+c_nb+k)*(GG(y(1:dim))(:,:,k)));

end

J(2*dim+1:3*dim,dim+1:2*dim) = -eye(dim) + ...

(dt/2)*(H(t+dt,y(1:2*dim))(1:dim,dim+1:end));

J(2*dim+1:3*dim,2*dim+1:3*dim) = eye(dim) + ...

(dt/2)*(H(t+dt,y(1:2*dim))(1:dim,dim+1:end));

J(2*dim+1:3*dim,3*dim+c_nb+1:end) = (dt/2)*G(y(1:dim))';

J(3*dim+1:3*dim+c_nb,1:dim) = G(y(1:dim));

J(3*dim+c_nb+1:end,1:dim) = G(y(1:dim))* ...

(H(t+dt,[y(1:dim);y(2*dim+1:3*dim)])(dim+1:end,1:dim));

for k = 1:1:c_nb

J(3*dim+c_nb+k,1:dim) = J(3*dim+c_nb+k,1:dim) + ...

((GG(y(1:dim))(:,:,k))*(f(t+dt,[y(1:dim); ...

y(2*dim+1:3*dim)])(1:dim)))';

end

J(3*dim+c_nb+1:end,2*dim+1:3*dim) = G(y(1:dim))* ...

(H(t+dt,[y(1:dim);y(2*dim+1:3*dim)]) ...

(dim+1:end,dim+1:end));

end

end

It works with any number of constraint, unless this is equal or greater to system dimension. As usual, all the source code is available at my public repository octave-odepkg. by Roberto Porcù (noreply@blogger.com) at December 15, 2013 03:47 PM

In this first post I want to explain the organization of the code that I'm going to implement. The main idea is to have a structured organization and to subdivide the code according to the most important operations which must be executed. So that should be easier to optimize the bottlenecks and to extend the code with new functionalities.

We will have two main actors: the **steppers** and the **integrate functions**.

A **stepper** will be a function and will represent the numerical method used for the integration. Its job is to execute just one integration step and its signature will be:

`[x_next,err] = stepper(f,x,t,dt)`

`x_next`

is the solution at the next step, `err`

is an estimation of the error (obtainable for example with Richardson Extrapolation), `f`

is a function handle representing the equations to be integrated, `x`

is the solution at the current time `t`

and finally `dt`

is the time step.

Inside this function I'll check if `err`

is really requested by means of `nargout`

keyword. The estimation of the error will be useful to determine the optimal `dt`

in adaptive integrators.

To set the parameters for both the numerical method and the solver I'll use `odeset`

and `odeget`

which are not embedded into Octave but are functions of Odepkg. My intent is to edit these functions in order to make them suitable for my parameters setting.

An **integrate function** will be the function executing the algorithm of integration on more steps. We will have different **integrate functions** depending on how we want to obtain the solution. For example we can have:

`integrate_const(stepper,f,x0,t0,t1,dt)`

integrating from t0 to t<=t1 with a fixed`dt`

;`integrate_n_steps(stepper,f,x0,t0,dt,n)`

integrating from t0 for n integration steps with fixed`dt`

;`integrate_adaptive(stepper,p,f,x0,t0,t1,dt)`

integrating from t0 to t1 with an adaptive timestep, where p is the order of the stepper.

In this part I'll show you an example of two simple steppers (`fe_richardson`

and `fe_heun`

). Both the steppers make use of the Forward Euler method to find the new solution: $$x_{k+1} = x_{k} + h f(t_{k},x_{k})\ .$$ They differ in the way the error is estimated: `fe_richardson`

makes use of Richardson Extrapolation while `fe_heun`

makes use of Heun-Euler method.

This is the implementation of the `fe_richardson`

stepper:

`function [t_next,x_next,err] = fwe_richardson(f,t,x,dt,options)`

x_next = x + dt.*f(t,x);

t_next = t+dt;

if(nargout == 3)

x1 = x + (.5*dt).*f(t,x);

x_est = x1 + (.5*dt).*f(t+ .5*dt,x1);

err = norm(x_next - x_est,2);

end

end

The **Heun-Euler method** combines the Heun method, which is of order 2, with the Euler method, which is of order 1. This combination allows to get an estimation of the local error. We can write the Heun method as $$\begin{aligned} \tilde{x}_{k+1} &= x_{k} + h f(t,x_{k}) \\ x_{k+1}^{*} &= x_{k} + \frac{h}{2} (f(t,x_{k}) + f(t+h,\tilde{x}_{k+1})) \end{aligned}$$ So that the error estimation is given by: $$ e_{k+1} = \tilde{x}_{k+1} - x^*_{k+1}=\frac{h}{2}(f(t+h,\tilde{x}_{k+1}) - f(t,x_{k}))$$ This is the implementation of the `fe_heun`

method:

`function [t_next,x_next,err]=fwe_heun(f,t,x,dt,options)`

x_next = x + dt.*f(t,x);

t_next = t+dt;

if(nargout == 3)

err = .5 * dt * norm(f(t+dt,x_next) - f(t,x),2);

end

end

Finally I'll describe how `integrate_adaptive`

is implemented. As i said this function does the integration on the total interval `[t0,t1]`

adapting the timestep at each iteration, by means of the error estimation returned by the stepper. This function takes as input:

`stepper`

: the stepper;`p`

: the order of the stepper;`f`

: the function to be integrated;`x0`

: the initial condition;`t0, t1`

: the extremes of the time interval;`dt`

: the first timestep.

`t1`

, the function substitutes the last element of the solution with a linear interpolation calculated in `t1`

between the last two elements of the solution. `function [t,x] = integrate_adaptive(stepper,order, ...`

func,tspan,x0,options)

t = tspan(1);

x = x0(:);

dt = odeget(options,'InitialStep',[],'fast');

tau = 1.e-6;

counter = 2;

while( t(end) (tspan(end)-1.e-13) )

[s,y,err] = stepper(func,t(end),x(:,end),dt,options);

if( s(end) > (tspan(counter)+1.e-13) )

z = [t(end);s];

u = [x(:,end),y];

i = 1;

while( i=length(z) )

if( abs(z(i)-tspan(counter)) 1.e-13 )

x = [x,y];

t = [t;s];

i_old = i;

i = length(z)+1;

elseif( z(i) > (tspan(counter)+1.e-13) )

x = [x(:,1:end-1),u(:,1:i-1),u(:,i-1)+ ...

((tspan(counter)-z(i-1))/(z(i)-z(i-1)))* ...

(u(:,i)-u(:,i-1)),u(:,i:end)];

t = [t(1:end-1);z(1:i-1);tspan(counter);z(i:end)];

i_old = i;

i = length(z)+1;

end

i++;

end

counter++;

else

x = [x,y];

t = [t;s];

end

err += eps;

dt = dt.*(tau/abs(err))^(1.0/(order+1));

end

if( counter > max(size(tspan)) )

k = (length(z)-i_old)+1;

x = x(:,1:end-k);

t = t(1:end-k);

end

x = x';

end

The method used to calculate the next timestep is taken from the book of Shampine, Gladwell and Thomson. `tau`

is a parameter which now is fixed but in the future will be setted by means of `odeset`

and `odeget`

functions.

At the end, functions like `ode45`

will be just aliases. For example, we define a function called `odefwe`

which does the integration by means of Forward Euler method, adapting the timestep with Richardson Extrapolation. The call to this function will just hide a call to `integrate_adaptive`

function in this way:

`odefwe(f,x0,t0,t1,dt) = ...`

integrate_adaptive(fe_richardson,1,f,x0,t0,t1,dt)

In the same way we'll define a function called `ode12`

which does the integration in an adaptive way using the Heun-Euler method. So that, now we will have: `ode12(f,x0,t0,t1,dt) = integrate_adaptive(fe_heun,1,f,x0,t0,t1,dt)`

by Roberto Porcù (noreply@blogger.com) at December 09, 2013 08:37 AM

**Odeset** and **odeget** functions allow to build a valid ODE options structure. They are already available in Octave odepkg but they are not perfectly compatible with MATLAB odeset and odeget functions. Furthermore, for geometric integrators like Spectral Variational Integrators, I will need new options into the ODE options structure, which now are not admitted in Octave odepkg.

So that I have written my own versions of **odeget.m**, **odeset.m** and the new function **ode_struct_value_check.m** in order to have full compatibility with MATLAB odeset and odeget (their same behaviour on the basis of their official documentation) and also the possibility to add new field names which i will need for future solvers of this project.

The new fields are the union of MATLAB ODE options and Octave ODE options, plus my new options (default values are in square brackets):

`' AbsTol: scalar or vector, >0, [1.e-6]'`

' BDF: binary, {on, [off]}'

' Events: '

' Explicit: binary, {yes, no, []}'

' InexactSolver: switch, {inexact_newton, []}'

' InitialSlope: vector, []'

' InitialStep: scalar, >0, []'

' Jacobian: matrix or function_handle, []'

' JConstant: binary, {on, [off]}'

' JPattern: sparse matrix, []'

' Mass: matrix or function_handle, []'

' MassConstant: binary, {on, [off]}'

' MassSingular: switch, {yes, [maybe], no}'

'MaxNewtonIterations: scalar, integer, >0, [1.e3]'

' MaxOrder: switch, {1, 2, 3, 4, [5]}'

' MaxStep: scalar, >0, []'

' MStateDependence: switch, {none, [weak], strong}'

' MvPattern: sparse matrix, []'

' NewtonTol: scalar or vector, >0, []'

' NonNegative: vector of integers, []'

' NormControl: binary, {on, [off]}'

' OutputFcn: function_handle, []'

' OutputSave: scalar, integer, >0, []'

' OutputSel: scalar or vector, []'

' PolynomialDegree: scalar or vector, integer, >0, []'

' QuadratureOrder: scalar or vector, integer, >0, []'

' Refine: scalar, integer, >0, []'

' RelTol: scalar, >0, [1.e-3]'

' Stats: binary, {on, [off]}'

' TimeStepNumber: scalar, integer, >0, []'

' TimeStepSize: scalar, >0, []'

' Vectorized: binary, {on, [off]}'

The usage of **odeset** will be one of the following:

`odeset`

opt = odeset()

opt = odeset(old_ODE_options,new_ODE_options)

opt = odeset(old_ODE_options,'field1',value1,'field2',value2,...)

opt = odeset('field1',value1,'field2',value2,'field3',value3,...)

The usage of **odeget** will be one of the following:

`option_value = odeget(ODE_structure,'field')`

option_value = odeget(ODE_structure,'field',default_value)

option_value = odeget(any_struct,'field',default_value,'fast')

The last usage is needed for MATLAB compatibility and represents a fast access to the given structure with no error checking on options names. The idea is that word cases are not relevant, if we want an exact matching then the tolerance will be 0, if no index is returned then it will definitely be a typing error, if one index is returned but words don't match exactly a warning will be displayed saying that the program is going on assuming that the user was intending the closest option name, otherwise all the fields whose distance is under the tolerance will be displayed and the user will be asked to insert the name again.

Signature of this function follows.

`res = fuzzy_compare(string1,string_set,correctness)`

**string1**must be a string and is the option name we're looking for;**string_set**must be a cell array of strings or a column vector of strings; it represents the set of option names;**correctness**is the tolerance we want. It is an optional input argument;

`tolerance = minimus + floor(length(string1)/4)*floor(minimus/3)`

There exist many definitions of distance between strings. I've chosen the Levenshtein distance. The Levenshtein distance is a string metric and is equal to the minimum number of single-characters edit (insertion, deletion and substitution) required to change one word into the other. This is the main algorithm of **levenshtein** function:

`function [dist,d] = levenshtein(string1,string2)`

m = length(string1);

n = length(string2);

d = zeros(m+1,n+1);

d(2:m+1,1) = [1:1:m];

d(1,2:n+1) = [1:1:n];

for j=2:1:n+1

for k=2:1:m+1

if(string1(k-1)==string2(j-1))

d(k,j)=d(k-1,j-1);

else

d(k,j)=min(d(k-1,j)+1,min(d(k,j-1)+1,d(k-1,j-1)+1));

end

end

end

dist = d(m+1,n+1);

end

All the code is available under the terms of GNU General Public License at my_octave-odepkg.by Roberto Porcù (noreply@blogger.com) at December 09, 2013 08:32 AM

The **backward Euler method** is one of the most common methods used to solve ODEs. It is an implicit method of order one but its strenght lies in its A-stability property.

Given a system of ODEs of the form: $$\frac{d\mathbf{y}}{dt} = \mathbf{f}(t,\mathbf{y}) \\ \mathbf{y}(t_0) = \mathbf{y}_0\ ,$$ the backward Euler method is written as: $$ \mathbf{y}_{k+1} = \mathbf{y}_k + \Delta t\ \mathbf{f}(t_{k+1},\mathbf{y}_{k+1})\ .$$

If we define **u** to be the exact solution then we have that: $$ \frac{d\mathbf{u}}{dt} = \mathbf{f}(t,\mathbf{u}) \\ \mathbf{u}(t_k) = \mathbf{y}_k $$ So that $$\mathbf u(t_{k+1}) = \mathbf u(t_k) + \int_{t_k}^{t_{k+1}} \mathbf{f}(t,\mathbf u) dt\ ,$$ which, under some conditions of regularity, can be rewritten as $$\mathbf u(t_{k+1}) = \Delta t\ \mathbf f(t_{k+1},\mathbf u(t_{k+1})) - \frac{{\Delta t}^2}{2} \mathbf f'(\xi)\ .$$

From these results it can be derived that the local truncation error (**LTE**) is $$ \mathbf e_{k+1} = \mathbf y_{k+1} - \mathbf u(t_{k+1}) = \mathbf o({\Delta t}^2)\ .$$

Now we have to solve the non-linear system built up with the backward Euler method. We want to use an **inexact Newton solver**. The classical Newton method is written as: given a non-linear functions system **F**(**x**) = **0** with a given initial condition, solve the linearized system: $$ \mathbf F'(\mathbf x_k)\mathbf s_k + \mathbf F(\mathbf x_k) = \mathbf 0 \\ \mathbf x_{k+1} = \mathbf x_k + \mathbf s_k\ .$$ Go on iterating this method until a given tolerance is reached.

In many cases, especially when we don't have an explicit expression for the Jacobian or it can't be inverted, we can use an inexact Newton method: $$ || \mathbf F'(\mathbf x_k)\mathbf s_k + \mathbf F(\mathbf x_k) || \leq \eta_k || \mathbf F(\mathbf x_k)||\ ,$$ where \(\eta_k\) is said to be the forcing term. The linearized system can be solved with an iterative method such as **GMRES** or **preconditioned conjugate gradient**.

There are two possible choices for the forcing term:

- first choice $$ \eta_k = \frac{|| \mathbf F(\mathbf x_k)-\mathbf F(\mathbf x_{k-1}) -\mathbf F'(\mathbf x_{k-1})\mathbf s_{k-1} ||}{||\mathbf F(\mathbf x_{k-1}) ||}\ ;$$
- second choice $$ \eta_k = \gamma \Big(\frac{|| \mathbf F(\mathbf x_k) ||}{|| \mathbf F(\mathbf x_{k-1}) ||}\Big)^{\alpha}\ .$$

Choosing the forcing term with one of the two previous possibilities, rather than imposing a fixed tolerance, it is possible to avoid the phenomenon of **oversolving**. Another advantage is that if we use the GMRES as the linear solver it's not necessary to know the Jacobian nor to datermine it, because we can approximate the term $$ \mathbf F'(\mathbf x_{k})\mathbf s_k $$ with a first order finite differences formulae: $$ \mathbf F'(\mathbf x_k)\mathbf s_k \approx \frac{\mathbf F(\mathbf x_k + \delta\mathbf s_k) - \mathbf F(\mathbf x_k)}{\delta} \ .$$

The signature of **inexact_newton** is the same of `fsolve`:

`[x,fval,exitflag,output,jacobian]=inexact_newton(fun,x0,options)`

by Roberto Porcù (noreply@blogger.com) at December 09, 2013 08:28 AM

The **variational integrators** are a class of numerical methods for mechanical systems which comes from the discrete formulation of **Hamilton's principle of stationary action**. They can be applied to ODEs, PDEs and to both conservative and forced systems. In absence of forcing terms these methods preserve momenta related to symmetries of the problem and don't dissipate energy. So that they exhibit long-term stability and good long-term behaviour.

Considering a configuration manifold V, the **discrete Lagrangian** is a function from V to the real numbers space, which represents an approximation of the action between two fixed configurations: $$ L_d(\mathbf q_k,\mathbf q_{k+1}) \approx \int_{t_k}^{t_{k+1}} L(\mathbf q,\dot{\mathbf q};t) dt \hspace{0.5cm}\text{with}\hspace{0.4cm}\mathbf q_{k},\mathbf q_{k+1}\hspace{0.3cm}\text{fixed.}$$ From here, applying the Hamilton's principle, we can get the **Euler-Lagrange discrete equations**: $$D_2L_d(\mathbf q_{k-1},\mathbf q_k) + D_1 L_d(\mathbf q_k,\mathbf q_{k+1}) = 0\ ,$$ and thanks to the **discrete Legendre transforms** we get the **discrete Hamilton's equation of motion**: $$\left\{\begin{array}{l} \mathbf p_k = -D_1 L_d(\mathbf q_k,\mathbf q_{k+1}) \\ \mathbf p_{k+1} = D_2 L_d(\mathbf q_k,\mathbf q_{k+1}) \end{array}\right.\ ,$$ so that we pass from a 2-step system of order N to a 1-step system of order 2N. This system gives the updating map: $$ (\mathbf q_k,\mathbf p_k)\rightarrow(\mathbf q_{k+1},\mathbf p_{k+1})\ .$$ For all the theory behind this I refer to "Discrete mechanics and variational integrators" by Marsden and West.

To create a **spectral variational integrator** I considered a discretization of the configuration manifold on a n-dimensional functional space generated by the orthogonal basis of Legendre polynomials. So that, after rescaling the problem from \([t_k,t_{k+1}]\) (with \(t_{k+1}-t_k=h\)) to \([-1,1]\), we get: $$ \begin{array}{l} \mathbf q_k(z) = \sum_{i=0}^{n-1}\mathbf q_k^i l_i(z)\\ \dot{\mathbf q}_k(z) = \frac{2}{h} \sum_{i=0}^{n-1}\mathbf q_k^i \dot l_i(z) \end{array} \ .$$ Then I approximate the action using the Gaussian quadrature rule using \(m\) internal nodes, so putting all together we have: $$ \int_{t_k}^{t_{k+1}} L(\mathbf q,\dot{\mathbf q};t)dt\hspace{0.5cm} \approx\hspace{0.5cm} \frac{h}{2}\sum_{j=0}^{m-1} \omega_j L\big( \sum_{i=0}^{n-1}\mathbf q_k^i l_i , \frac{2}{h} \sum_{i=0}^{n-1}\mathbf q_k^i \dot l_i \big) $$ Now imposing Hamilton's principle of stationary action under the constraints: $$ \mathbf q_k = \sum_{i=0}^{n-1}\mathbf q_k^i l_i(-1) \hspace{1.5cm} \mathbf q_{k+1} = \sum_{i=0}^{n-1}\mathbf q_k^i l_i(1)\ ,$$ we obtain the system: $$ \left\{ \begin{array}{l} \sum_{j=0}^{m-1}\omega_j \bigg[ \mathbf p_j \dot{l}_s(z_j) - \frac{h}{2} l_s(z_j) \frac{\partial H}{\partial \mathbf q} \bigg ( \sum_{i=0}^{n-1}\mathbf q_k^i l_i(z_j),\mathbf p_j \bigg ) \bigg ] + l_s(-1)\mathbf p_k - l_s(1)\mathbf p_{k+1} = 0 \hspace{1cm} \forall s=0,\dots,n-1\\ \frac{\partial H}{\partial \mathbf p}\bigg (\sum_{i=0}^{n-1}\mathbf q_k^i l_i(z_r),\mathbf p_r\bigg ) -\frac{2}{h}\ \sum_{i=0}^{n-1} \mathbf q_k^i \dot{l}_i(z_r)=0 \hspace{1cm} \forall r=0,\dots,m-1 \\ \sum_{i=0}^{n-1} \mathbf q_k^i l_i(-1) - \mathbf q_k = 0\\ \sum_{i=0}^{n-1} \mathbf q_k^i l_i(1) - \mathbf q_{k+1} = 0 \end{array}\right. $$

Within **odeSPVI** I implemented a geometric integrator for Hamiltonian systems, like odeSE and odeSV, which uses spectral variational integrators with any order for polynomials of the basis and with any number of internal quadrature nodes, for both unidimensional and multidimensional problems.

`[T,Y]=odeSPVI(@hamilton_equations,time,initial_conditions,options)`

This solver just uses the stepper This below is the implementation of the stepper:

`function [t_next,x_next,err]=spectral_var_int(f,t,x,dt,options)`

fsolve_opts = optimset('Jacobian','on');

q_dofs = odeget(options,'Q_DoFs',[],'fast');

p_dofs = odeget(options,'P_DoFs',[],'fast');

nodes = odeget(options,'Nodes',[],'fast');

weights = odeget(options,'Weights',[],'fast');

leg = odeget(options,'Legendre',[],'fast');

deriv = odeget(options,'Derivatives',[],'fast');

extremes = odeget(options,'Extremes',[],'fast');

dim = length(x)/2;

N = q_dofs+p_dofs+2;

q0 = x(1:dim)(:)';

p0 = x(dim+1:end)(:)';

SPVI = @(y)svi_system(y,q_dofs,p_dofs,weights,leg, ...

deriv,extreme,dim,N,f,t,q0,p0,dt);

y0 = [q0;zeros(q_dofs-1,dim);ones(p_dofs,1)*p0;q0;p0];

y0 = reshape(y0,dim*N,1);

z = fsolve(SPVI,y0,fsolve_opts);

%z = inexact_newton(SVI,y0,new_opts) %new_opts must be set

z = reshape(z,N,dim);

y = [leg*z(1:q_dofs,1:dim),z((q_dofs+1):(end-2),1:dim)];

x_next = [y;z(end-1,:),z(end,:)]';

t_next = [t+(dt/2)*(nodes+1), t+dt]';

if(nargout==3)

q_dofs_err = odeget(options,'Q_DoFs_err',[],'fast');

p_dofs_err = odeget(options,'P_DoFs_err',[],'fast');

nodes_err = odeget(options,'Nodes_err',[],'fast');

weights_err = odeget(options,'Weights_err',[],'fast');

leg_err = odeget(options,'Legendre_err',[],'fast');

deriv_err = odeget(options,'Derivatives_err',[],'fast');

extreme_err = odeget(options,'Extremes_err',[],'fast');

N_err = q_dofs_err+p_dofs_err+2;

q1 = x_next(1:dim,end)(:)';

p1 = x_next(dim+1:end,end)(:)';

SPVI = @(y)svi_system(y,q_dofs_err,p_dofs_err, ...

weights_err,leg_err,deriv_err,extreme_err, ...

dim,N_err,f,t,q0,p0,dt);

p_interp = zeros(p_dofs_err,dim);

p_lower_order = [p0;z(q_dofs+1:q_dofs+p_dofs,:);p1];

for i=1:1:p_dofs_err

p_interp(i,:) = .5*(p_lower_order(i,:) + ...

p_lower_order(i+1,:));

end

y0 = [z(1:q_dofs,:);zeros(1,dim);p_interp;q1;p1];

y0 = reshape(y0,dim*N_err,1);

z = fsolve(SPVI,y0,fsolve_opts);

%z = inexact_newton(SVI,y0,new_opts) %new_opts must be set

z = reshape(z,N_err,dim);

x_est = [z(end-1,:),z(end,:)]';

err = norm(x_est-x_next(:,end),2);

end

end

At lines 22 and 56 I put a comment line to show that it is possible to solve the implicit system also with Another important aspect to point out is that in case of an adaptive timestep the error is estimated using a new solution on the same timestep but with polynomials one order higher and one more internal node for the quadrature rule. Furthermore, for this last solution, the **starting guess** for `fsolve` is chosen in a non-trivial way: at line 53 we see that `y0` has in the first `q_dofs_err-1` rows the same modal values calculated before for the new solution `x_next` and a row of zeros just below. Then the starting nodal values for the momenta are set (lines 47-51) as a **trivial average** of new solution nodal values. This can seem wrong but I empirically stated that the number of iterations done by fsolve are the same of cases in which the reinitialization of nodal values is more sophisticated, so that is **less computationally expensive** to do a trivial average.

In the following code is shown the implementation of the system to be solved:

`function [F,Jac] = svi_system(y,q_dofs,p_dofs,w,L,D,C, ...`

dim,N,f,t,q0,p0,dt)

F = zeros(N*dim,1);

V = zeros(p_dofs,dim*2);

X = zeros(dim*2,1);

W = reshape(y,N,dim);

for i = 1:1:p_dofs

X = [L(i,:)*W(1:q_dofs,:),W(i+q_dofs,:)]';

V(i,:) = f(t,X);

end

for i = 1:1:dim

F((1+N*(i-1)):(q_dofs+N*(i-1)),1) = (ones(1,p_dofs)* ...

(((w.*y((q_dofs+1+N*(i-1)):(q_dofs+p_dofs+N*(i-1))))* ...

ones(1,q_dofs)).*D + (((dt/2).*w.*V(:,i+dim))* ...

ones(1,q_dofs)).*L) + (p0(i)*ones(1,q_dofs)).*C(1,:) - ...

(y(N*i)*ones(1,q_dofs)).*C(2,:))';

F(1+N*(i-1)+q_dofs:N*(i-1)+q_dofs+p_dofs,1) = V(:,i) - ...

(2.0/dt)*(D*y((1+N*(i-1)):(q_dofs+N*(i-1))));

F(N*i-1) = C(2,:)*y((1+N*(i-1)):(q_dofs+N*(i-1)))-y(N*i-1);

F(N*i) = C(1,:)*y((1+N*(i-1)):(q_dofs+N*(i-1))) - q0(i);

end

if(nargout>1)

warning('off','Octave:broadcast');

flag = 0;

try

[~,H]=f(0,zeros(2*dim,1));

catch

flag = 1;

warning();

end

DV = zeros((dim*2)^2,p_dofs);

if( flag==1 )

for i = 1:1:p_dofs

X = [L(i,:)*W(1:q_dofs,:),W(i+q_dofs,:)]';

DV(:,i) = differential(f,t,X);

end

else

for i = 1:1:p_dofs

X = [L(i,:)*W(1:q_dofs,:),W(i+q_dofs,:)]';

[~,DV(:,i)] = f(t,X);

end

end

DV = DV';

Jac=zeros(N*dim,N*dim);

for u=1:1:dim

[...]

end

end

end

Here, at line 46, I don't show the implementation of the Jacobian of the implicit system because, with this visualization style, it may appear very chaotic. The important aspect is that the user can use the implemented Jacobian to `function Hamilt = differential(f,t,x)`

f_x = f(t,x);

dim_f = length(f_x);

dim_x = length(x);

if( dim_f ~= dim_x )

error('not implemented yet');

end

Hamilt = zeros(dim_f*dim_x,1);

delta = sqrt(eps);

for i = 1:1:dim_f

for j = i:1:dim_x

Hamilt(i+(j-1)*dim_f) = (1/delta)*(f(t,x+delta* ...

[zeros(j-1,1);1;zeros(dim_x-j,1)])(i) - f_x(i));

Hamilt(j+(i-1)*dim_x) = Hamilt(i+(j-1)*dim_f);

end

end

end

The Hessian of the Hamiltonian must be stored in a particular way (that I have to optimize yet, but the actual one works fine too) which is showed in the following example which is the definition of the Hamilton's equations for the armonic oscillator:

`function [dy,ddy] = armonic_oscillator(t,y)`

dy = zeros(size(y));

d = length(y)/2;

q = y(1:d);

p= y(d+1:end);

dy(1:d) = p;

dy(d+1:end) = -q;

H = eye(2*d);

ddy = transf(H);

end

function v = transf(H)

[r,c] = size(H);

v = zeros(r*c,1);

for i=1:1:r

for j=1:1:c

v(i+(j-1)*r) = H(i,j);

end

end

end

by Roberto Porcù (noreply@blogger.com) at December 09, 2013 08:20 AM

The **symplectic Euler** method is a modification of the Euler method and is useful to solve Hamilton's equation of motion, that is a system of ODE where the unknowns are the generalized coordinates **q** and the generalized momenta **p**. It is of first order but is better than the classical Euler method because it is a symplectic integrator, so that it yelds better results.

Given a Hamiltonian system with Hamiltonian **H=H(t;q,p)** then the system of ODE to solve writes: $$\left\{\begin{array}{l} \dot{q} = \frac{dH}{dp}(t;q,p) = f(t;q,p)\\ \dot{p}=-\frac{dH}{dq}(t;q,p) = g(t;q,p) \end{array}\right. $$

From E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration" we can state that the semi-implicit Euler scheme for previous ODEs writes: $$ \left\{\begin{array}{l} p_{k+1} = p_k + h g(t_k;q_k,p_{k+1}) \\ q_{k+1}=q_k + h f(t_k;q_k,p_{k+1}) \end{array}\right. $$ If g(t;q,p) does not depend on p, then this scheme will be totally explicit, otherwise the first equation will be implicit and the second will be explicit.

The function that I created to make use of this integrator is **odeSE.m** and it passes to the integrate functions the stepper defined in the function **symplectic_euler.m**. The signature of **odeSE** is similar to those of the other ode solvers:

`[t,y]=odeSE(@ode_fcn,tspan,y0)`

[t,y]=odeSE(@ode_fcn,tspan,y0,options)

It's important to know that **options** variable can be set with **odeset** and, if the system of ODE is explicit, the field options.Explicit can be set to 'yes' in order to speedup the computation. If **tspan** has only one element, then options.TimeStepNumber and options.TimeStepSize must not be empty, so that it will be used the integrate function **integrate_n_steps**.

This is the code of the stepper **symplectic_euler.m**:

`function [x_next,err] = symplectic_euler(f,t,x,dt,options)`

dim = length(x)/2;

q = x(1:dim);

p = x(dim+1:end);

if( strcmp(options.Explicit,'yes') )

p_next = p + dt*(f(t,[q; p])(dim+1:end));

q_next = q + dt*f(t,[q; p_next])(1:dim);

x_next = [q_next; p_next];

if(nargout == 2)

dt_new = dt/2;

p_next = p + dt_new*(f(t,[q; p])(dim+1:end));

q_next = q + dt_new*f(t,[q; p_next])(1:dim);

q = q_next;

p = p_next;

t = t+dt_new;

p_next = p + dt_new*(f(t,[q; p])(dim+1:end));

q_next = q + dt_new*f(t,[q; p_next])(1:dim);

x_est = [q_next;p_next];

err = norm(x_next-x_est,2);

end

else

SE1 = @(y) (y-p-dt*(f(t,[q; y])(dim+1:end)));

p_next = fsolve(SE1,zeros(size(p)));

q_next = q + dt*f(t,[q; p_next])(1:dim);

x_next = [q_next; p_next];

if(nargout == 2)

dt_new = dt/2;

SE1 = @(y) (y-p-dt_new*(f(t,[q; y])(dim+1:end)));

p_next = fsolve(SE1,zeros(size(p)));

q_next = q + dt_new*f(t,[q; p_next])(1:dim);

q = q_next;

p = p_next;

t = t+dt_new;

SE1 = @(y) (y-p-dt_new*(f(t,[q; y])(dim+1:end)));

p_next = fsolve(SE1,zeros(size(p)));

q_next = q + dt_new*f(t,[q; p_next])(1:dim);

x_est = [q_next;p_next];

err = norm(x_next-x_est,2);

end

end

end

The **velocity-Verlet** method is a numerical method used to integrate Newton's equations of motion. The Verlet integrator offers greater stability, as well as other properties that are important in physical systems such as preservation of the symplectic form on phase space, at no significant additional cost over the simple Euler method.

If we call **x** the coordinate, **v** the velocity and **a** the acceleration then the equations of motion write: $$\left\{ \begin{array}{l} \frac{dx}{dt} = v(t,x)\\ \frac{dv}{dt} = a(t,x) \end{array} \right. $$

So that, given the initial conditions (coordinates and velocities), the velocity-verlet scheme writes: $$ \left\{ \begin{array}{l} x_{k+1} = x_k + h v_k + 0.5 h^2 a_k\\ v_{k+1} = v_k + 0.5 h (a_k + a_{k+1}) \end{array}\right. $$ where $$ a_{k+1} = a(t_{k+1},x_{k+1})\ .$$

This method is one order better than the **symplectic Euler** method. The global error of this method is of order two. Furthermore, if the acceleration indeed results from the forces in a conservative mechanical or Hamiltonian system, the energy of the approximation essentially oscillates around the constant energy of the exactly solved system, with a global error bound again of order two.

The function that uses **velocity-Verlet** scheme is **odeVV.m** and the stepper called at each iteration is **velocity_verlet.m**. The signature of **odeVV** is the same of those of others ODE solvers:

`[t,y]=odeVV(@ode_fcn,tspan,y0)`

[t,y]=odeVV(@ode_fcn,tspan,y0,options)

The documentation of input arguments is the same descripted in the previous **symplectic Euler** section, but there is the difference that now the function **ode_fcn** must return a vector containing the velocities in its first half and the expression of the acceleration in the second half.

This is the code of the stepper **symplectic_euler.m**:

`function [x_next,err] = velocity_verlet(f,t,x,dt,options)`

dim = length(x)/2;

q = x(1:dim);

v = x(dim+1:end);

a = f(t,x);

q_next = q + dt*v + .5*dt*dt*a(dim+1:end);

v_next = v + .5*dt*((a+f(t+dt,[q_next; v]))(dim+1:end));

x_next = [q_next; v_next];

if(nargout == 2)

dt_new = dt/2;

q_next = q + dt_new*v + .5*dt_new*dt_new*a(dim+1:end);

v_next = v + .5*dt_new*((a + ...

f(t+dt_new,[q_next;v]))(dim+1:end));

t = t+dt_new;

q = q_next;

v = v_next;

a = f(t,[q; v]);

q_next = q + dt_new*v + .5*dt_new*dt_new*a(dim+1:end);

v_next = v + .5*dt_new*((a + ...

f(t+dt_new,[q_next;v]))(dim+1:end));

x_est = [q_next; v_next];

err = norm(x_next - x_est,2);

end

end

The **Stormer-Verlet** scheme is a symplectic integrator of order two, useful to integrate Hamiltonian systems in the form described in the previous **symplectic Euler** section. It is characterized by the approximation of the integral defining the discrete Lagrangian $$ L_d(q_k,q_{k+1})\approx\int_{t_k}^{t_{k+1}}L(t;q,\dot{q})dt $$ with the trapezoidal rule. So that $$ L_d(q_k,q_{k+1}) = \frac{h}{2}\bigg( L\Big(q_k,\frac{q_{k+1}-q_k}{h}\Big) + L\Big(q_{k+1},\frac{q_{k+1}-q_k}{h}\Big) \bigg)\ .$$

Considering again the system: $$ \left\{\begin{array}{l} \dot{q} = \frac{dH}{dp}(t;q,p) = f(t;q,p)\\ \dot{p}=-\frac{dH}{dq}(t;q,p) = g(t;q,p) \end{array}\right. \ ,$$ the scheme implemented for this integrator (described in E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration") writes as follows: $$ \left\{\begin{array}{l} p_{k+\frac{1}{2}} = p_k + \frac{h}{2}g(t_k;q_k,p_{k+\frac{1}{2}}) \\ q_{k+1}=q_k+\frac{h}{2}\Big( f(t_k;q_k,p_{k+\frac{1}{2}}) + f(t_{k+1};q_{k+1},p_{k+\frac{1}{2}})\Big) \\ p_{k+1} = p_{k+\frac{1}{2}} + \frac{h}{2} g(t_{k+\frac{1}{2}};q_{k+1},p_{k+\frac{1}{2}}) \end{array}\right. $$

The function in which this method is implemented is **odeSV.m** and it calls the stepper **stormer_verlet.m**. The documentation is the same of **odeSE.m**. Its implementation is the following:

`function [x_next,err] = stormer_verlet(f,t,x,dt,options)`

dim = length(x)/2;

q = x(1:dim);

p = x(dim+1:end);

if( strcmp(options.Explicit,'yes') )

p_mid = p + .5*dt*(f(t,[q; p])(dim+1:end));

q_next = q + .5*dt*((f(t,[q; p_mid])(1:dim))+ ...

(f(t+dt,[q;p_mid])(1:dim)));

p_next = p_mid +.5*dt*(f(t+dt,[q_next;p_mid])(dim+1:end));

x_next = [q_next; p_next];

if(nargout == 2)

dt_new = dt/2;

p_mid = p + .5*dt_new*(f(t,[q; p])(dim+1:end));

q_next = q + .5*dt_new*((f(t,[q; p_mid])(1:dim))+ ...

(f(t+dt_new,[q;p_mid])(1:dim)));

p_next = p_mid + .5*dt_new* ...

(f(t+dt_new,[q_next;p_mid])(dim+1:end));

q = q_next;

p = p_next;

t = t+dt_new;

p_mid = p + .5*dt_new*(f(t,[q; p])(dim+1:end));

q_next = q + .5*dt_new*((f(t,[q; p_mid])(1:dim))+ ...

(f(t+dt_new,[q;p_mid])(1:dim)));

p_next = p_mid + .5*dt_new* ...

(f(t+dt_new,[q_next;p_mid])(dim+1:end));

x_est = [q_next; p_next];

err = norm(x_est - x_next,2);

end

else

SV1 = @(y) (y - p - .5*dt*(f(t,[q;y])(dim+1:end)));

p_mid = fsolve(SV1,zeros(size(p)));

SV2 = @(y) (y - q - .5*dt*((f(t,[q;p_mid])(1:dim))+ ...

(f(t+dt,[y;p_mid])(1:dim))));

q_next = fsolve(SV2,q);

p_next = p_mid + .5*dt* ...

f(t+dt,[q_next;p_mid])(dim+1:end);

x_next = [q_next;p_next];

if(nargout == 2)

dt_new = dt/2;

SV1 = @(y) (y - p - .5*dt_new*(f(t,[q;y])(dim+1:end)));

p_mid = fsolve(SV1,zeros(size(p)));

SV2 = @(y) (y - q - .5*dt_new* ...

((f(t,[q;p_mid])(1:dim))+(f(t+dt_new,[y;p_mid])(1:dim))));

q_next = fsolve(SV2,q);

p_next = p_mid + .5*dt_new* ...

f(t+dt_new,[q_next;p_mid])(dim+1:end);

q = q_next;

p = p_next;

t = t+dt_new;

SV1 = @(y) (y - p - .5*dt_new*(f(t,[q;y])(dim+1:end)));

p_mid = fsolve(SV1,zeros(size(p)));

SV2 = @(y) (y - q - .5*dt_new* ...

((f(t,[q;p_mid])(1:dim))+(f(t+dt_new,[y;p_mid])(1:dim))));

q_next = fsolve(SV2,q);

p_next = p_mid + .5*dt_new* ...

f(t+dt_new,[q_next;p_mid])(dim+1:end);

x_est = [q_next; p_next];

err = norm(x_next-x_est,2);

end

end

end

by Roberto Porcù (noreply@blogger.com) at December 02, 2013 05:15 AM

What I want to do at this moment of the project is to add to all the solvers I've written since now, the ability to manage all the possible options of the ODE structure (defined previously in **odeset** structure) that are already managed in corresponding MATLAB ode solvers; so that the solvers I've written will be totally MATLAB-compatible and this will be the same also for the new solvers I'm going to implement.

by Roberto Porcù (noreply@blogger.com) at November 15, 2013 08:13 AM

The image package has accumulated a lot of changes since its last release and I’m hoping to make a new release soon (to match the release of Octave 3.8.0). I have prepared a snapshot tarball (version 2.1.1) with the current development status which can be installed easily with `pkg`

. Would be great if users of the image package could give it a try and report any problems that the many changes may have caused.

It is important to note that this is not a final release. To make this clear, and to avoid that this becomes distributed as if it was, I made it dependent on an yet unreleased version of Octave (it is dependent on the development version of Octave anyway), and made it print a warning about it every time the package is loaded. After reading this, download the tarball and install it with `pkg install -nodeps /path/to/tarball`

I am partial to the changes in the new version, so these are the ones that I paid more attention to:

- complete rewrite of
`imdilate`

and`imerode`

with a performance improvement and many compatibility bugs fixed; - implementation of the
`@strel`

class and support for it in the image package functions; - support for N-dimensional matrices in several functions;
- rewrite of the block processing functions which in some cases performs 1000x faster.

There are also a lot of bug fixes and new functions. Some will break backwards compatibility really bad but needed to be done for the sake of Matlab compatibility. For example, `bwconncomp`

was returning indices for object perimeter but should have been returning indices for all elements in the objects. So do take a look at the NEWS file or use `news image`

after installing the package.

After this release, I plan to follow the Octave core release method of keeping two development branches: a stable branch for minor releases with small bug fixes and regressions, and a default branch for big changes. Hopefully that will allow for more frequent releases as things of different levels get ready.

Please report any problems you have found either in Octave’s bug or patch trackers.

The following list explains the main points of the timeline for my SOCIS project:

- adapt
**odeset**and**odeget**from Octave odepkg to be completely MATLAB-compatible: all option names supported by MATLAB should be available; furthermore we want the possibility to add new option names that will be useful for example for symplectic integrators; - implement a new code structure for one-step integrators
**ode12**,**ode23**,**ode45**,**odefwe**(forward Euler),**odebwe**(backward Euler) and**inexact solvers**, subdividing the execution into 'blocks' in order to be able to do an easier optimization and an easier extension to new functionalities; in particular implement the**steppers**which will be the functions called to do one integration step and then implement the**integrate functions**which will integrate, in different ways, over the whole time interval, calling the respective steppers. Afterwards adapt the interface of`ode45`to be completely MATLAB-compatible; - implement MATLAB-compatible version of
**deval**; - implement the following geometric integrators as new steppers:
- symplectic Euler;
- Stormer-Verlet;
- velocity-Verlet;
- spectral variational integrators;
- SHAKE;
- RATTLE.

- optimization, error checking and documentation of all the code written.

by Roberto Porcù (noreply@blogger.com) at November 11, 2013 02:17 PM

The implementation of **integrate_adaptive** function takes (for now) the first timestep as last input argument. Obviously the choice of the first timestep made by the user may be not the best, even though the adaptive technique will adjust the step at the second iteration. So that there exist many techniques to find out a good timestep for the first iteration.

Here I present the one I implemented, which was proposed by Gladwell, Shampine and Brankin in 1987. It is based on the hypothesis that:$$ \text{local error}\approx Ch^{p+1}y^{(p+1)}(x_0) $$ The algorithm is the following, and usually gives a good guess for the initial timestep.

`function h = starting_stepsize(f,x0,t0,p)`

d0 = norm(x0,2);

d1 = norm(f(t0,x0),2);

if( d01.e-5 || d11.e-5 )

h0 = 1.e-6;

else

h0 = .01*(d0/d1);

end

x1 = x0 + h0.*f(t0,x0);

d2 = (1.0/h0)*norm(f(t0+h0,x1)-f(t0,x0),2);

if( max(d1,d2)= 1.e-15 )

h1 = max(1.e-6,h0*1.e-3);

else

h1 = (1.e-2 / max(d1,d2))^(1/(p+1));

end

h = min(100*h0,h1);

end

So that, what I will do is include this algorithm into **integrate_adaptive** function which will also have a new signature, because the initial timestep is no more needed:

`function [x,t] = integrate_adaptive(stepper,order,f,x0,t0,t1)`

by Roberto Porcù (noreply@blogger.com) at October 25, 2013 08:26 AM

Another useless encryption scheme devised by yours truly. While my last one (pi
code) was primarily a substitution cipher, this one transcends
all standard classifications; it's almost like a transposition cipher, but not
*really*. The main idea here is the use of certain numbers in a particular
Sudoku grid. The strength (if any) in this method lies in its unexpected nature;
it certainly takes quite a leap of the imagination to correctly deduce the
method from the ciphertext. Of course, once the method has been discovered,
deciphering merely involves solving a Sudoku grid and then figuring out the
substitution cipher used, meaning that the key is easy to determine and so this
method kind of just looks at Kerckhoffs's Principle and then keeps walking. But
that's okay, that's why this is filed under **useless**.

The first step is to find a Sudoku puzzle. I happened to have a Sudoku game installed, so I just ran that and started a random puzzle, which looked like this:

5 | 1 | 7 | 8 | |||||

3 | 1 | 9 | 5 | 4 | ||||

3 | 1 | 5 | ||||||

2 | 8 | |||||||

4 | 7 | 5 | 9 | |||||

9 | 4 | |||||||

6 | 5 | 2 | ||||||

2 | 6 | 5 | 4 | 8 | ||||

4 | 3 | 7 | 2 |

The (unique) solution to the above puzzle looks like this:

5 | 2 | 4 | 1 | 7 | 9 | 3 | 8 | 6 |

3 | 1 | 9 | 6 | 5 | 8 | 4 | 2 | 7 |

8 | 6 | 7 | 3 | 2 | 4 | 9 | 1 | 5 |

2 | 9 | 5 | 4 | 8 | 3 | 6 | 7 | 1 |

4 | 8 | 6 | 7 | 1 | 5 | 2 | 3 | 9 |

1 | 7 | 3 | 2 | 9 | 6 | 8 | 5 | 4 |

6 | 5 | 1 | 8 | 4 | 2 | 7 | 9 | 3 |

7 | 3 | 2 | 9 | 6 | 1 | 5 | 4 | 8 |

9 | 4 | 8 | 5 | 3 | 7 | 1 | 6 | 2 |

Clearly, each solved grid can be represented by a sequence of 81 numbers. Furthermore, due to the uniqueness of the solution, the initial grid and the solved grid are equivalent in the sense that the initial grid uniquely determines the solved grid. So we represent the initial grid as follows (with 0s standing for spaces):

`500170080319050400000300015200080000400705009000090004650002000002060548040037002`

This results in the following solution grid:

`524179386319658427867324915295483671486715239173296854651842793732961548948537162`

Now, depending on the length of the plaintext, we can choose somewhere between one and five numbers to use as the "holes" in the grid. For example, if we had a plaintext of 45 characters or fewer, we could choose the numbers 1, 2, 5, 6 and 9. Removing those numbers from the grid would leave us with this:

4 | 7 | 3 | 8 | |||||

3 | 8 | 4 | 7 | |||||

8 | 7 | 3 | 4 | |||||

4 | 8 | 3 | 7 | 1 | ||||

4 | 8 | 7 | 3 | |||||

7 | 3 | 8 | 4 | |||||

8 | 4 | 7 | 3 | |||||

7 | 3 | 6 | 4 | 8 | ||||

4 | 8 | 3 | 7 |

This has an entirely different arrangement of empty spaces than the initial grid. We can represent it using the following string:

`50017008031905040000030001520008000040070500900009000465000200000206054804003700212569`

We can then use the empty spaces to write our message, and add characters in the rest of the spaces to try and flatten frequencies. A simple substitution cipher should be used as well to prevent the ciphertext from looking too much like a transposition cipher. The alphabetic characters can then be interspersed with the numerals above, with the resulting string separated into chunks of 32 characters (with extra alphabetic characters appended as necessary, since only 81 are needed) so as to superficially resemble MD5 (because that's just fun).

Optimally, the solutions to the possible Sudoku grids being used should be stored in a dictionary somewhere, so that each grid only has to be solved once. The point of this encryption scheme is to be diabolically tricky but solvable, although perhaps only if you figure out the Sudoku aspect and know a crib or two.

Let's say you want to encode:

`admiral yamamoto to visit solomon islands eight am`

First, remove spaces:

`admiralyamamototovisitsolomonislandseightam`

This is 43 characters, so let's aim for one grid with 5 numbers. This means there will be 2 extra invalid characters and 36 nonsense characters. Since some spaces will be fairly large, it's a good idea to use a simple substitution cipher as well (or even a less simple one like Vigenère for additional fiendishness). In this case, we'll use the reverse alphabet cipher:

`abcdefghijklm`

`zyxwvutsrqpon`

So the encrypted plaintext becomes

`admiralyamamototovisitsolomonislandseightam`

`zwnrizobznznlglglerhrghlolnlmrhozmwhvrtsgzn`

Now put it in the grid:

z | w | 4 | n | 7 | r | 3 | 8 | i |

3 | z | o | b | z | 8 | 4 | n | 7 |

8 | z | 7 | 3 | n | 4 | l | g | l |

g | l | e | 4 | 8 | 3 | r | 7 | h |

4 | 8 | r | 7 | g | h | l | 3 | o |

l | 7 | 3 | n | l | m | 8 | r | 4 |

h | o | z | 8 | 4 | m | 7 | w | 3 |

7 | 3 | h | v | r | t | s | 4 | 8 |

g | 4 | 8 | z | 3 | 7 | n | c | a |

(The last two characters are the extra invalid characters used to fill up the empty spaces.)

Now replace the remaining numbers with random letters, chosen so as to flatten the letter frequencies as much as possible:

z | w | e | n | a | r | o | t | i |

q | z | o | b | z | a | e | n | s |

f | z | t | r | n | m | l | g | l |

g | l | e | e | a | s | r | h | h |

t | r | r | a | g | h | l | m | o |

l | c | c | n | l | m | z | r | v |

h | o | z | l | p | m | p | w | p |

a | e | h | v | r | t | s | b | d |

g | g | f | z | e | q | n | c | a |

Written as one line, the above grid would look like:

`zwenarotiqzobzaensfztrnmlglgleeasrhhtrraghlmolccnlmzrvhozlpmpwpaehvrtsbdggfzeqnca`

Recall that the original grid looks as follows:

`500170080319050400000300015200080000400705009000090004650002000002060548040037002`

With the numbers used added to the end of the grid, it might look like this:

`50017008031905040000030001520008000040070500900009000465000200000206054804003700215926`

Now, we randomly combine the two: (split across two lines here due to design considerations)

`z50w0ena1r7o008tiq0zo3bz1ae9nsf0z5t0r4n0mlgl0g0le0e0as3rhh00tr0rag1h5lm2ol00ccn0l8mz`

`r0v0ho0z0400lp7mpwp0a500eh9v0r0t0sbd090gg0046f500ze02000q0020n6054c80400a3700215926`

We could even put in extra characters at the end, since only the first 81 characters will be used:

`z50w0ena1r7o008tiq0zo3bz1ae9nsf0z5t0r4n0mlgl0g0le0e0as3rhh00tr0rag1h5lm2ol00ccn0l8mzr0v`

`0ho0z0400lp7mpwp0a500eh9v0r0t0sbd090gg0046f500ze02000q0020n6054c80400a3f70gg02a1e59f26x`

Here's how it would look if we split it up into 32-character chunks:

`z50w0ena1r7o008tiq0zo3bz1ae9nsf0`

`z5t0r4n0mlgl0g0le0e0as3rhh00tr0r`

`ag1h5lm2ol00ccn0l8mzr0v0ho0z0400`

`lp7mpwp0a500eh9v0r0t0sbd090gg004`

`6f500ze02000q0020n6054c80400a3f7`

`0gsssg02ea1e5dllg9famezelr2ezz6x`

And there you go. Not indecipherable but certainly very misleading. To decipher, first separate the numerals from the alphabetic characters, then write out the first 81 characters within a 9x9 grid. Then fill another 9x9 grid with the first 81 numerals (one digit per box), omitting the zeros, and solve the resulting Sudoku puzzle. Take the remaining (non-zero) digits and highlight the positions in the grid corresponding to those digits. The characters corresponding to the non-highlighted positions can be discarded. Now it only remains to reverse the substitution cipher and insert spaces where necessary.

Challenge:

`dszb5c0ygo01ize7c0u0n80s31rwq9m0`

`5f0400ll0r00mw30atdx00r15gzs2000`

`800jq0z0n400bz7j05obo00rtbt90u0v`

`f0u0oo9r0v00pmfvzoxxmg4650f00g20`

`0mnz000o20rgo605ub48ubzxxo040a03`

`70kteo02osz3dfle4eraosme2tn5sddq`

All the relevant code is available in the repository for this website, under code/sudokucode. The 3 main files are shown below.

```
import json
import re
import random
import string
A_ORD = ord('a')
NOT_LETTERS_RE = re.compile('[^a-z]')
LETTERS_RE = re.compile('[a-z]')
NUMBERS_RE = re.compile('[0-9]')
POSSIBLE_DIGITS = map(str, range(1, 10))
# Does the reverse alphabet cipher thing on a string`
def reverse_cipher(plaintext):
real_letters = []
for letter in plaintext:
real_letters.append(chr(A_ORD + 25 - (ord(letter) - A_ORD)))
# whatever it works don't h8
return ''.join(real_letters)
def contains_only_letters(plaintext):
return NOT_LETTERS_RE.search(plaintext) is None
def sometimes():
return random.choice((True, False))
class SudokuCoder:
def __init__(self):
try:
self.grids = json.load(open('grids.json'))
except IOError:
# No grids in memory - can only encode, not decode.
self.grids = {}
def encode(self, plaintext):
plaintext = plaintext.lower()
num_letters = len(plaintext)
letters = reverse_cipher(plaintext)
if not contains_only_letters(plaintext):
raise CannotEncodeError('Can only contain alphabetic characters.')
if num_letters > 45:
raise CannotEncodeError('Maximum plaintext length: 45 characters.')
# Randomly choose a grid.
initial_grid = random.choice(self.grids.keys())
grid = self.grids[initial_grid]
# Randomly choose some digits to use as the holes.
num_digits = num_letters / 9 + 1
digits = random.sample(POSSIBLE_DIGITS, num_digits)
letter_index = 0
new_grid = []
# Now replace all the hole digits with the plaintext.
for digit in grid:
if digit in digits and letter_index < num_letters:
new_grid.append(letters[letter_index])
letter_index += 1
else:
# Choose a random character
# For both the extra ones and the nonsense ones
new_grid.append(random.choice(string.lowercase))
# Add extra characters depending on the number of digits
for digit in xrange(num_digits):
new_grid.append(random.choice(string.lowercase))
total_digits = initial_grid + ''.join(digits)
# Now randomly combine them.
grid_length = len(new_grid)
total_length = grid_length * 2
ciphertext = []
letter_index = 0
digit_index = 0
while letter_index < grid_length or digit_index < len(total_digits):
if ((sometimes() and letter_index < grid_length) or
digit_index == len(total_digits)):
ciphertext.append(new_grid[letter_index])
letter_index += 1
else:
ciphertext.append(total_digits[digit_index])
digit_index += 1
return ''.join(ciphertext)
def decode(self, ciphertext):
ciphertext = ciphertext.lower()
# Get the grid numbers (the first 81 digits).
all_numbers = NUMBERS_RE.findall(ciphertext)
initial_grid = ''.join(all_numbers[:81])
hole_numbers = all_numbers[81:]
all_letters = LETTERS_RE.findall(ciphertext)
grid_letters = all_letters[:81]
# Check if the solution to this initial grid exists.
if not initial_grid in self.grids:
raise GridNotFoundError
# Get the list indices of the hole numbers.
solution_grid = self.grids[initial_grid]
hole_indices = []
for i in xrange(len(solution_grid)):
if solution_grid[i] in hole_numbers:
hole_indices.append(i)
hole_letters = [grid_letters[index] for index in hole_indices]
plaintext = reverse_cipher(hole_letters)
return plaintext
def add_grid(self, initial, solution):
if not initial in self.grids:
self.grids[initial] = solution
json.dump(self.grids, open('grids.json', 'w'))
else:
raise GridAlreadyExists
class GridNotFoundError(Exception):
pass
class CannotEncodeError(Exception):
pass
class GridAlreadyExists(Exception):
pass
```

```
import sys
import utils
if len(sys.argv) == 2:
sudoku_coder = utils.SudokuCoder()
print sudoku_coder.encode(sys.argv[1])
else:
print "Usage: python encode.py [plaintext]"
```

```
import sys
import utils
if len(sys.argv) == 2:
sudoku_coder = utils.SudokuCoder()
print sudoku_coder.decode(sys.argv[1])
else:
print "Usage: python decode.py [ciphertext]"
```

Instead of encrypting the Sudoku puzzle itself, use a standard one. for example, if you and your imaginary recipient both have access to the New York Times, which we will pretend has a daily Sudoku puzzle if it doesn't actually, then just use that. Then, the only numbers you'd have to send along with the plaintext (and the extra characters) are the hole digits. This means that the key itself (or at least the prelude to the key) does not have to be distributed along with the message, resulting in a somewhat more secure scheme (for a given value of "secure"). This will work as long as you ensure that the Sudoku puzzle that you choose always has a unique solution.

Since I deviated so much from the original project while fixing the image IO functions in Octave core, I decided to only focus on optimizing the imerode and imdilate in the final phase of GSoC. The reason is that these are at the core of many functions in the image package.

On the original project it was planned to do all the work by expanding the __spatial_filtering__ function and that’s where I previously started. While doing so, it became evident that a complete rewrite was necessary. The convn() which could be used in most cases of binary images was performing much faster even though it was performing a more complex operation. As such, performing at least as fast as convn() became the target which was achieved:

octave> se = [0 1 0; 1 1 1; 0 1 0]; octave> im = rand (1000) > 0.5; octave> t = cputime (); for i = 1:100, a = imerode (im, se, "same"); endfor; cputime() - t ans = 2.1281 octave> t = cputime (); for i = 1:100, a = convn (im, se, "same") == nnz (se); endfor; cputime() - t ans = 2.7802

This implementation could be reused in __spatial_filtering__ to also speed up functions such as medfilt2, ordfilt2, and ordfiltn but there are specific algorithms for those cases which should be used instead.

I have tested the new implementation of imerode against the last release (version 2.0.0) and the last development version that was still making use of __spatial_filtering__ (cset 6db5e3c6759b). The tests are very simple (test imerode), just random matrices with different number of dimensions. The new implementation seems to perform much faster in all cases, and shows a performance increase between 1.5-30x (output of test imerode). The differences are bigger for grayscale images (since imerode was already using convn for binary cases), and larger structuring elements (SE) with multiple dimensions.

A couple of things:

- in the latest release of the image package (version 2.0.0), there was no erosion for multidimensional binary images (only grayscale);
- both development versions make use of the new strel class. One of the things that it does, its to decompose structuring elements automatically, hence why the tests use a cross rather than a square for SE;
- I’m only testing with the shape “same” since version 2.0.0 only had that one;
- when using binary images I test with different percentages of true values since the new implementation is sensitive to it;
- I do not compare the results since I know the new implementations also fix some bugs, specially related to the border pixels.
- imdilate uses exactly the same code and so I’m assuming that the differences from imerode are the same.

version 2.0.0 (old) | cset 6db5e3c6759b (dev) | latest development (new) | Performance old/new | Performance dev/new | |
---|---|---|---|---|---|

2D binary image (90%) | 0.009081 | 0.024602 | 0.006240 | 1.4551 | 3.9423 |

2D binary image (10%) | 0.007360 | 0.022881 | 0.004160 | 1.7692 | 5.5000 |

3D binary image with 2D SE | NO! | 0.481470 | 0.079125 | n/a | 6.0849 |

3D binary image with 3D SE | NO! | 0.518032 | 0.075605 | n/a | 6.8519 |

7D binary image with 5D SE | NO! | 13.940071 | 0.463229 | n/a | 30.093 |

2D uint8 image | 0.062324 | 0.043403 | 0.029322 | 2.1255 | 1.4802 |

3D uint8 image with 2D SE | NO | NO | 0.430347 | n/a | n/a |

3D uint8 image with 3D SE | 3.061951 | 1.725628 | 0.791569 | 3.8682 | 2.1800 |

7D uint8 image with 3D SE | NO | NO | 2.005325 | n/a | n/a |

7D uint8 image with 7D SE | 4.091456 | 2.940984 | 0.541634 | 7.5539 | 5.4298 |

2D uint8 image with a larger SE | 0.610678 | 0.305579 | 0.087445 | 6.9835 | 3.4945 |

The imageIO section of the original project was much shorter than this. Originally it was limited to implement imformats, and then expand it to do the writing and reading of multipage images in

a Matlab compatible way which is just implementing the options Index and Frames in imread, and WriteMode in imwrite. The date of that last commit was 16th of July. However, I didn’t stop there and tried to fix other incompatibilities with Matlab and add new options.

Here’s a list of the main changes:

- dealing with transparency: alpha channel is now returned as the third output of imread and only accepted with the Alpha option of imwrite. Previously, the alpha channel would be considered the 2nd or 4th element on the 3rd dimension for grayscale and RGB images respectively. This prevented the distinction between a RGB with transparency and a CMYK image. Also, since GraphicsMagick has the transparency values inverted from most other applications (Matlab inclusive), we were doing the same. This has been fixed.
- CMYK images: implemented reading and writing of images in the CMYK colorspace.
- imread options: in addition to Index and Frames, also implemented PixelRegion, which may speedup reading when only parts of the image are of interest, and Info, for compatibility only, which doesn’t have any effect at the moment.
- imwrite options: in addition to WriteMode, implemented Alpha (as mentioned above).
- indexed images: implement writing of indexed images. Previously, indexed images would be converted to RGB which would then be saved. An indexed image will now always return indexes independently if a colormap was requested as output (previously, a raster image would be returned in such cases).
- floating point and int32 images: GraphicsMagick is integer exclusively and we can’t differentiate between an image saved with int32 or floating point. Since the latter are far more common Octave will now return a floating point image when GraphicsMagick reports a bitdepth of 32. Previously, images with more than a reported bitdepth of 16 would not be read at all (note that this is dependent on the GM build options. If GM’s quantum depth was 16 or 8, the most common, GM would report a file with 32 bitdepth with the quantum depth it was built with so it would read files with a bitdepth of 32 as long as it was not built with quantum depth 32).
- bitdepth: imread now returns the actual bitdepth of the image in the file rather than trying to optimize it. This is still not perfect but we’re making a better guess at it than before (this should be very simple but with GraphicsMagick it’s not).
- imfinfo: some new fields, specially reading of Exif and GPS data which allowed to deprecate readexif in the image package.

In the end, I recognize that there’s some incompatibilities left to fix but I don’t know how anymore. As I mentioned on my previous post, GraphicsMagick is too smart for our use. I predict that we will eventually have to move to something different, if for nothing else, to read and write floating point images without trouble. I only had a quick glance on alternatives but FreeImage would seem like a good bet. Of course, there’s also the possibility to write our library wrapper around the format specific libraries.

Would be nice if users could throw some images to imread and imwrite and report any bugs. I’m specially afraid of regressions since there was so much stuff changed and we don’t really have tests for these functions yet.