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README.md

 # paper-2024-pde-greedy
 
+Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods
+=========================================================================================
 
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+This repository contains the supplementary material for the publication
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+Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods (2022)
+T. Wenzel, D. Winkle, G. Santin, B. Haasdonk
 ```
 
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+It was used to carry out the numerical experiments and generate the figures for the publication.
+The experiments were performed on Linux systems in 2024 and should work with Python versions of that time (e.g., `python3.9`).
 
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+## Installation
 
-## Description
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+To run the python experiments, create a virtual environment and install the [PDE-VKOGA](https://gitlab.mathematik.uni-stuttgart.de/pub/ians-anm/pde-vkoga) package.
+For convenience, this steps can be done with help of the `setup_python_experiments.sh` file.
+To activate the virtual environment, use `source venv/bin/activate`.
+In order to reproduce the experiments, run the files within the folder  experiments_pde_vkoga.
 
-## Badges
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-## Visuals
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+To runt he matlab experiments (i.e. to reproduce the FEM part of the convergence table within Section 6.1), please
+a) Install RBmatlab from the website www.morepas.org/software. The code can be run if RBmatlab from version 16.09 is installed.
+b) run the script fem_sector_example.m from this directory
 
-## Installation
-Within a particular ecosystem, there may be a common way of installing things, such as using Yarn, NuGet, or Homebrew. However, consider the possibility that whoever is reading your README is a novice and would like more guidance. Listing specific steps helps remove ambiguity and gets people to using your project as quickly as possible. If it only runs in a specific context like a particular programming language version or operating system or has dependencies that have to be installed manually, also add a Requirements subsection.
+Caution:
 
-## Usage
-Use examples liberally, and show the expected output if you can. It's helpful to have inline the smallest example of usage that you can demonstrate, while providing links to more sophisticated examples if they are too long to reasonably include in the README.
+The FEM calculation is very expensive (takes several hours) as hundred of
+thousands of point with global coordinates need to be searched and found
+in FEM meshes. This is required to be consistent in the error computation
+to other approximation techniques using these uniform test grids.
 
-## Support
-Tell people where they can go to for help. It can be any combination of an issue tracker, a chat room, an email address, etc.
 
-## Roadmap
-If you have ideas for releases in the future, it is a good idea to list them in the README.
 
-## Contributing
-State if you are open to contributions and what your requirements are for accepting them.
+## How to cite:
+If you use this code in your work, please cite the paper
 
-For people who want to make changes to your project, it's helpful to have some documentation on how to get started. Perhaps there is a script that they should run or some environment variables that they need to set. Make these steps explicit. These instructions could also be useful to your future self.
+> T. Wenzel, D. Winkle, G. Santin, and B. Haasdonk. Adaptive meshfree solution
+of linear partial differential equations with PDE-greedy kernel methods. ArXiv,
+(2207.13971), 2022. Submitted.
 
-You can also document commands to lint the code or run tests. These steps help to ensure high code quality and reduce the likelihood that the changes inadvertently break something. Having instructions for running tests is especially helpful if it requires external setup, such as starting a Selenium server for testing in a browser.
 
-## Authors and acknowledgment
-Show your appreciation to those who have contributed to the project.
+```bibtex:
+@article{wenzel2022adaptive,
+  doi = {10.48550/ARXIV.2207.13971},
+  url = {https://arxiv.org/abs/2207.13971},
+  author = {Wenzel, Tizian and Winkle, Daniel and Santin, Gabriele and Haasdonk, Bernard},
+  title = {Adaptive meshfree solution of linear partial differential equations with {PDE}-greedy kernel methods},
+  number = {2207.13971}, 
+  publisher = {arXiv},
+  year = {2022},
+  journal={ArXiv},
+  type={ArXiv},
+  note={Submitted}
+}
+}
+```
 
-## License
-For open source projects, say how it is licensed.
 
-## Project status
-If you have run out of energy or time for your project, put a note at the top of the README saying that development has slowed down or stopped completely. Someone may choose to fork your project or volunteer to step in as a maintainer or owner, allowing your project to keep going. You can also make an explicit request for maintainers.

experiments_fem/fem_sector_example.m

+function fem_sector_example(step)
+%function fem_sector_example(step)
+%
+% Demonstration of FEM error convergence for sector example.
+%
+% This operation is very expensive (takes several hours) as hundred of
+% thousands of point with global coordinates need to be searched and found
+% in FEM meshes. This is required to be consistent in the error computation
+% to other approximation techniques using these uniform test grids.
+%
+% The code can be run if RBmatlab from version 16.09 is installed
+% as obtained from www.morepas.org/software 
+
+% B. Haasdonk 8.5.2024
+
+if nargin < 1
+  step = 4;
+end;
+
+disp('FEM error convergence study for sector geometry')
+disp('Note that the global to local coordinate search is extremely')
+disp('expensive due to the many test points, and will overall take')
+disp('several hours.')
+
+switch step
+  
+%  case 1 %  create scale of refined gridfiles
+%    fns = {'sectorg_alpha5over3','sectorg_alpha2over3'};
+%    for fni = 1:length(fns)
+%      fn = fns{fni}
+%      [p,e,t] = initmesh(fn);
+%      meshfile = [fn,'_r0.mat'];
+%      save(meshfile,'p','e','t');
+%      for i = 1:4
+%	[p,e,t] = refinemesh(fn,p,e,t);
+%	meshfile = [fn,'_r',num2str(i),'.mat'];
+%	save(meshfile,'p','e','t');  
+%      end;
+%    end;
+%    disp('mesh sequence generated and stored');
+
+%    % continue with error computation: 
+%    fem_sector_example(4);
+    
+ case 2 % load and plot mesh
+
+  load('sectorg_alpha5over3_r1','p','t');
+  grid = triagrid(p,t,[]);
+  plot(grid);
+  axis equal;
+
+  figure;
+  load('sectorg_alpha2over3_r1','p','t');
+  grid = triagrid(p,t,[]);
+  plot(grid);
+  axis equal;
+
+ case 3 % fem on mesh 
+  
+  params = [];
+%  params.solution_number = 1; % example with smooth solution 
+  params.solution_number = 2; % example with singular sol, inhom bv
+
+%  params.alpha = 5/3;
+%  params.grid_initfile = 'sectorg_alpha5over3_r1.mat';
+%  params.grid_initfile = 'sectorg_alpha5over3_r2.mat';
+%  params.grid_initfile = 'sectorg_alpha5over3_r3.mat';
+%  params.grid_initfile = 'sectorg_alpha5over3_r4.mat';
+  params.alpha = 2/3;
+  params.grid_initfile = 'sectorg_alpha2over3_r1.mat';
+%  params.grid_initfile = 'sectorg_alpha2over3_r2.mat';
+%  params.grid_initfile = 'sectorg_alpha2over3_r3.mat';
+%  params.grid_initfile = 'sectorg_alpha2over3_r4.mat';
+  model = pacman_model(params);
+  % generate grid and fem matrices:
+  model_data = gen_model_data(model);
+  figure, plot(model_data.grid);
+  axis equal; axis tight; title('FEM grid')  
+  sim_data = detailed_simulation(model, model_data); 
+  % plot results
+  figure, plot_sim_data(model,model_data,sim_data,[]);
+  title('FEM solution of -Laplace u = f')
+  axis equal;
+  axis tight;
+%  disp(['ndofs = ',num2str(sim_data.uh.df_info.ndofs)]);
+
+ case 4 % fem on mesh and convergence study 
+     
+  params = [];
+  params.solution_number = 1; % example with smooth solution 
+%  params.solution_number = 2; % example with singular sol, inhom bv
+
+    disp(' ');
+    disp('ndofs  |    L2-error    |     H1-error    |     infty-error    |   L2-error-grid  | infty-error-grid    |    t_CPU')
+    disp('---------------------------------------------------------------------------------------------------------------------')
+    
+  for i = 1:4
+    % params.alpha = 5/3;
+    % params.grid_initfile = ['sectorg_alpha5over3_r',num2str(i),'.mat'];
+    %  params.grid_initfile = 'sectorg_alpha5over3_r2.mat';
+    %  params.grid_initfile = 'sectorg_alpha5over3_r3.mat';
+    %  params.grid_initfile = 'sectorg_alpha5over3_r4.mat';
+    params.alpha = 2/3;
+    params.grid_initfile = ['sectorg_alpha2over3_r',num2str(i),'.mat'];
+    %  params.grid_initfile = 'sectorg_alpha2over3_r2.mat';
+    %  params.grid_initfile = 'sectorg_alpha2over3_r3.mat';
+    %  params.grid_initfile = 'sectorg_alpha2over3_r4.mat';
+    model = pacman_model(params);
+    % generate grid and fem matrices:
+    tic
+    model_data = gen_model_data(model);
+    sim_data = detailed_simulation(model, model_data); 
+    t = toc;
+    % error computation:  
+    % project exact solution onto higher degree polynomial fem func
+    par.pdeg = 4;
+    par.qdeg = 8;
+    par.dimrange = 1;    
+    p4_df_info = feminfo(par,model_data.grid);
+    
+    uexact_h = femdiscfunc([],p4_df_info);
+    uexact_h = fem_interpol_global(model.solution,uexact_h);
+    uh = femdiscfunc([],p4_df_info);
+    u_local_eval = @(grid,elids,lcoord,params) ...
+	my_uh_local_eval(grid,elids,lcoord,params,sim_data.uh); 
+    uh = fem_interpol_local(u_local_eval,uh);
+    
+    err = uh - uexact_h;
+    if i == 1
+      plot(err);
+      title('error for i=1')
+      axis equal;
+      axis tight;
+    end;
+%    keyboard;
+    l2err = fem_l2_norm(err);
+    h1err = fem_h1_norm(err);
+    linftyerr = max(abs(err.dofs));
+    [l2err_uniformgrid, linftyerr_uniformgrid] = errors_uniformgrid(model,uh);
+%    l2err_uniformgrid = zeros(size(l2err));
+%    linftyerr_uniformgrid = zeros(size(l2err));
+    ndofs = sim_data.uh.df_info.ndofs;
+    disp([num2str(ndofs,'%10.4d'),'   |   ',...
+	  num2str(l2err,'%10.5e'),'   |   ',...
+	  num2str(h1err,'%10.5e'),'   |   ',...
+	  num2str(linftyerr,'%10.5e'),'   |   ',...
+	  num2str(l2err_uniformgrid,'%10.5e'),'   |   ',...
+	  num2str(linftyerr_uniformgrid,'%10.5e'),'   |   ',...
+      num2str(t,'%10.5e')]);
+  end;
+
+ otherwise
+  error('step number unknown');
+end;
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%    auxiliary functions
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+function [l2err_uniformgrid, linftyerr_uniformgrid] = errors_uniformgrid(model, uh);  
+% determine test points
+  h = 0.002;
+%  h = 0.002;
+%  h = 0.002;
+  x = -1:h:1;
+  [XX,YY] = meshgrid(x,x);  
+  i = find((XX.^2+YY.^2)<=1);
+  XX = XX(i);
+  YY = YY(i);
+  Phi = atan(YY./XX);
+  i = find(Phi<0 & XX>=0);
+  Phi(i) = Phi(i) + 2*pi;
+  i = find(XX<0);
+  Phi(i) = Phi(i) + pi;
+  i = find(Phi<model.alpha*pi);
+  XX = XX(i);
+  YY = YY(i);
+  Phi = Phi(i);
+  %scatter3(XX,YY,Phi)
+  test_sol = model.solution([XX,YY]);
+  test_uh = zeros(size(test_sol));  
+  % evaluate fem approximation in global points
+  % bad... : loop over points, should be vectorized...
+  f = waitbar(0,'Iterating over points');
+  for i = 1:length(Phi);
+    if mod(i,1000)==0;
+      waitbar(i/length(Phi),f,'Iterating over points');
+    end;
+    eind = find_triangle(uh.grid,[XX(i),YY(i)]);
+    if eind>0
+      lcoord = global2local(uh.grid,eind,[XX(i),YY(i)]);
+      test_uh(i) = fem_evaluate(uh,eind,lcoord);
+      % sanity check: transform local coordinate back to global:
+      p = local2global(uh.grid,eind,lcoord,[]);
+      d = p-[XX(i),YY(i)];
+%      if norm(d)>10*eps
+%	disp('norm of reconstructed point too large!, please inspect');%
+%	keyboard
+%      end;     
+    else
+      test_uh(i) = NaN;
+    end;
+  end;
+  close(f);
+  
+  linftyerr_uniformgrid = max(abs(test_sol-test_uh));
+  i = find(~isnan(test_uh));
+  l2err_uniformgrid = sqrt(h*h*sum((test_sol(i)-test_uh(i)).^2));
+
+  %figure;scatter3(XX,YY,double(isnan(test_uh)))
+%  if isnan(l2err_uniformgrid)
+%    disp('NaN in error!')
+%    keyboard;
+%  end;
+  
+function eind = find_triangle(grid,glob)
+% returns the index of a triangle in grid containing the global point glob
+% if no triangle is found, then -1 is returned.
+inside = ones(grid.nelements,1);
+for j = 1:3 % check if point is "above" edge connecting point j to j+1
+  jp1 = mod(j,3)+1;
+  jp2 = mod(jp1,3)+1;
+  Xj = grid.X(grid.VI(:,j));
+  Yj = grid.Y(grid.VI(:,j));
+  Xjp1 = grid.X(grid.VI(:,jp1));
+  Yjp1 = grid.Y(grid.VI(:,jp1));
+  Vjjp1 = [Xjp1-Xj, Yjp1-Yj];
+  Vjglob = [glob(1)*ones(size(Xj)) - Xj, glob(2)*ones(size(Yj)) - Yj];
+  crossz = sign(Vjjp1(:,1).*Vjglob(:,2) - Vjjp1(:,2).*Vjglob(:,1));
+  if j==1
+    Xjp2 = grid.X(grid.VI(:,jp2));
+    Yjp2 = grid.Y(grid.VI(:,jp2));
+    Vjjp2 = [Xjp2-Xj, Yjp2-Yj];
+    crossz2 = sign(Vjjp1(:,1).*Vjjp2(:,2) - Vjjp1(:,2).*Vjjp2(:,1)); % orientation of trias
+  end;
+  i = find(crossz.*crossz2<0);
+  inside(i) = 0;
+end;
+eind = find(inside);
+if isempty(eind)
+  eind = -1;
+end
+if length(eind)>1
+  %  disp('length eind > 1, please check! ');
+  eind = eind(1);
+end;
+%if length(eind)==1
+%  disp('length eind == 1, nice :-)  ');
+%end;
+
+% settings for pacman model
+function model = pacman_model(params);
+  if ~isfield(params,'alpha')
+    alpha = 5/3;
+  else
+    alpha = params.alpha;
+  end;
+%  disp(['chosen alpha = ',num2str(alpha)]);
+  %alpha = 5/3; % hard coded in sectorg.m if changed here, change there!
+  model = poisson_model(params);
+  model.alpha = alpha;
+  model = rmfield(model,{'boundary_type','normals',...
+			 'xnumintervals','ynumintervals','xrange','yrange'});
+  model.has_reaction = 0;
+  model.has_advection = 0;
+  model.has_diffusivity = 1;
+  model.has_source = 1;
+  model.has_dirichlet_values = 1;
+  model.has_neumann_values = 0;
+  model.has_robin_values = 0;
+  model.compute_output_functional = 0;
+  switch params.solution_number
+    case 1 % smooth solution
+      model.solution = @(glob,params) ...
+			sum(glob.^2,2);
+      model.source = @(glob,params) ...
+		     - 4 * ones(size(glob,1),1);
+    case 2 % solution with singularity and inhomogeneous bnd val.
+      model.solution = @(glob,params) ...
+			pacman_exact_solution(glob',alpha);
+      model.source = @(glob,params) ...
+		      neg_Laplace_pacman_exact_solution(glob',alpha);
+      % case 3 % solution wiht singularity and homogeneous bnd val.
+      %  % someting seems to be buggy here, no convergence observed...
+      %  model.solution = @(glob,params) ...
+      %      pacman_exact_solution2(glob',alpha);
+      %  model.source = @(glob,params) ...
+      %      neg_Laplace_pacman_exact_solution2(glob',alpha);
+      %  error('please use solution_number=1 or 2.')
+  end;
+  model.diffusivity_tensor = @(glob,params) ...
+			      [ones(size(glob,1),1),...
+			       zeros(size(glob,1),1),...
+			       zeros(size(glob,1),1),...
+			       ones(size(glob,1),1)];
+  model.reaction = @(glob,params) zeros(size(glob,1),1);
+  model.dirichlet_values = @(glob,params) ...
+			    params.solution(glob,params);
+  model.grid_initfile = params.grid_initfile;
+  model.gridtype = 'triagrid';
+  model.pdeg = 1;
+  model.qdeg = 2;
+  model.dimrange = 1;
+  model = elliptic_discrete_model(model);
+	     %model.detailed_simulation = @pacman_detailed_simulation;
+
+function f = pacman_exact_solution(x,alpha);
+% function u(x) = |x|^(1/alpha)sin(phi(x)/alpha)
+% with phi(x) = atan(x2/x1).
+% which has -Laplace u = 0 on pacman shape
+% but non-homogeneous boundary values.
+f1 = sum(x.^2,1).^(0.5/alpha);
+phi = atan(x(2,:)./x(1,:));
+i = find(phi<0 & x(1,:)>=0);
+phi(i) = phi(i) + 2*pi;
+i = find(x(1,:)<0);
+phi(i) = phi(i) + pi;
+f2 = sin(phi/alpha);
+i = find(isnan(f2));
+f2(i) = 0;
+f = f1.*f2;
+
+function f = neg_Laplace_pacman_exact_solution(x,alpha);
+f = zeros(size(x,2),1);
+
+function res = my_uh_local_eval(grid,elids,lcoord,params,df)
+% dummy function used for evaluating a discrete function at finer
+% lagrange-grid nodes
+res = fem_evaluate(df,elids,lcoord,[],[]);

experiments_fem/sectorg_alpha2over3.m

+function [x,y]=sectorg_alpha2over3(bs,s)
+
+nbs=3;
+alpha = 2/3 * pi;
+
+if nargin==0,
+  x=nbs; % number of boundary segments
+  return
+end
+
+d=[
+  0  0 1% start parameter value
+  1  alpha 0% end parameter value
+  1  1 1% left hand region
+  0  0 0% right hand region
+];
+
+bs1=bs(:)';
+
+if find(bs1<1 | bs1>nbs),
+  error('semicircleg:InvalidBs', 'Non existent boundary segment number.')
+end
+
+if nargin==1,
+  x=d(:,bs1);
+  return
+end
+
+x=zeros(size(s));
+y=zeros(size(s));
+[m,n]=size(bs);
+if m==1 && n==1,
+  bs=bs*ones(size(s)); % expand bs
+elseif m~=size(s,1) || n~=size(s,2),
+  error('semicircleg:SizeBs', 'bs must be scalar or of same size as s.');
+end
+
+if ~isempty(s),
+  % boundary segment 1
+  ii=find(bs==1);
+  x(ii) = s(ii);
+  y(ii) = zeros(size(ii));
+
+  % boundary segment 2
+  ii=find(bs==2);
+  x(ii) = cos(s(ii));
+  y(ii) = sin(s(ii));
+  
+  
+  % boundary segment 3
+  ii=find(bs==3);
+  x(ii) = s(ii)*cos(alpha);
+  y(ii) = s(ii)*sin(alpha);
+end
+

experiments_pde_vkoga/main_01_sector_examples.py

+# Code for the two sector domain examples of the paper "Adaptive meshfree approximation for linear
+# elliptic partial differential equations with PDE-greedy kernel methods" by T. Wenzel, D. Winkle, G. Santin, B. Haasdonk
+
+
+# Some imports
+import numpy as np
+from datetime import datetime
+from matplotlib import pyplot as plt
+
+from vkoga_pde.kernels_PDE import cubicMatern_laplace
+from vkoga_pde.vkoga_PDE import VKOGA_PDE
+from experiments_pde_vkoga.utilities_pacman import sample_domain_pacman, get_function_pacman
+
+
+np.random.seed(1)
+
+## Create some data: Circle without cone
+dim=2
+N1 = int(4e5)
+N2 = int(3e3)
+maxIter = 1305       # 341, 1305
+
+## Pick a kernel
+kernel = cubicMatern_laplace(dim=2)
+
+
+## Set up lists to loop over
+list_alpha = [2/3]
+list_str_function = ['singular']                # 'smooth'
+list_beta = [1]
+list_weight_dirichlet = [10**3]
+n_iter = len(list_alpha) * len(list_str_function) * len(list_beta) * len(list_weight_dirichlet)
+
+
+
+## Actually start the computations
+idx_counter = 0 
+dic_results = {}
+for alpha in list_alpha:
+    dic_results[alpha] = {}
+
+    # Sample the domain
+    X1, X2, X1_grid = sample_domain_pacman(N1, N2, alpha)
+
+    for str_function in list_str_function:
+
+        dic_results[alpha][str_function] = {}
+
+        # Define some functions: Example from Bernard
+        u, f = get_function_pacman(str_function, alpha)
+
+        # Compute solution on fine grid
+        u_X1_grid = u(X1_grid)
+
+        # Define right hand side values
+        y1, y2 = f(X1), u(X2)
+        y1_sol = u(X1)
+
+        for beta in list_beta:
+
+            dic_results[alpha][str_function][beta] = {}
+
+            for weight_dirichlet in list_weight_dirichlet:
+
+                dic_results[alpha][str_function][beta][weight_dirichlet] = {}
+
+                # Compute stuff
+                model_pde_vkoga = VKOGA_PDE(kernel=kernel, beta=beta, weight_dirichlet=weight_dirichlet, verbose=True)
+                _ = model_pde_vkoga.fit(X1, y1, X2, y2, maxIter=maxIter, y1_sol=y1_sol)
+
+                # Store values
+                dic_results[alpha][str_function][beta][weight_dirichlet]['f_sol'] = np.array(model_pde_vkoga.train_hist['f sol'])
+                dic_results[alpha][str_function][beta][weight_dirichlet]['f'] = np.array(model_pde_vkoga.train_hist['f'])
+                dic_results[alpha][str_function][beta][weight_dirichlet]['n_ctrs_bdry'] = len(model_pde_vkoga.ind_ctrs2)
+
+                idx_counter += 1
+                
+                # save dictionary
+                print(datetime.now().strftime("%H:%M:%S"), 'Computation {}/{} finished.'.format(idx_counter, n_iter))
+
+
+
+# Compute some test errors (to be sure that there is no overfitting!)
+y_pred_s_ = model_pde_vkoga.predict_s(X1_grid)
+diff_s_ = y_pred_s_ - u(X1_grid)
+
+y_pred_Ls_ = model_pde_vkoga.predict_Ls(X1_grid)
+diff_Ls_ = y_pred_Ls_ - f(X1_grid)
+
+
+print('Max error on grid: ', np.max(np.abs(diff_s_)))
+print('L2 error on grid: ', np.linalg.norm(diff_s_)/np.sqrt(len(diff_s_)))
+
+
+
+## Plot the points
+plt.figure(11)
+plt.clf()
+plt.plot(model_pde_vkoga.ctrs_[model_pde_vkoga.ind_ctrs1][:, 0], model_pde_vkoga.ctrs_[model_pde_vkoga.ind_ctrs1][:, 1], 'ro', markersize=3)
+plt.plot(model_pde_vkoga.ctrs_[model_pde_vkoga.ind_ctrs2][:, 0], model_pde_vkoga.ctrs_[model_pde_vkoga.ind_ctrs2][:, 1], 'kx', markersize=10)
+# axis ratio equal
+plt.gca().set_aspect('equal')
+plt.show(block=False)
+
+
+## Plot the decay of the approximation errors
+plt.figure(12)
+plt.clf()
+plt.plot(model_pde_vkoga.train_hist['f sol'])
+# plot horizonal line at np.max(np.abs(diff_s))
+# plt.axhline(y=np.max(np.abs(diff_s_)), color='r', linestyle='-')
+plt.yscale('log')
+plt.xscale('log')
+plt.title('|u-s_n|')
+plt.show(block=False)
+
+
+plt.figure(13)
+plt.clf()
+plt.plot(np.array(model_pde_vkoga.train_hist['f'])**(1/2))
+# plot horizonal line at np.max(np.abs(diff_Ls))
+# plt.axhline(y=np.max(np.abs(diff_Ls_)), color='r', linestyle='-')
+plt.yscale('log')
+plt.xscale('log')
+plt.title('|L(u-s_n)|')
+plt.show(block=False)
+
+
+

experiments_pde_vkoga/main_02_highdim_example.py

+# Code for the high dimensional example of the paper "Adaptive meshfree approximation for linear
+# elliptic partial differential equations with PDE-greedy kernel methods" by T. Wenzel, D. Winkle, G. Santin, B. Haasdonk
+
+
+# Some imports
+import numpy as np
+from datetime import datetime
+from matplotlib import pyplot as plt
+
+from vkoga_pde.kernels_PDE import cubicMatern_laplace, Gaussian_laplace
+from vkoga_pde.vkoga_PDE import VKOGA_PDE
+
+
+np.random.seed(1)
+
+
+# Create data set: Square domain
+dim = 12
+
+scale_domain = 1/np.sqrt(dim)
+X1 = scale_domain * np.random.rand(int(1e5), dim)
+X2 = scale_domain * np.random.rand(2*dim*400, dim)
+
+# Set boundary values
+for idx_dim in range(dim):
+    X2[2*idx_dim*400 : (2*idx_dim+1)*400, idx_dim] = 0
+    X2[(2*idx_dim+1) * 400 : (2*idx_dim + 2) * 400, idx_dim] = scale_domain
+
+X1_test = scale_domain * np.random.rand(int(1e5), dim)
+
+
+u = lambda x: np.sum(x * x, axis=1, keepdims=True) + 1      # solution u(x) = 1/d norm(x)^2 => max in 0,1 on cube
+f = lambda x: -2 * dim * np.ones((x.shape[0], 1))           # f = -2 dim on unit cube
+
+
+# Define right hand side values
+y1 = f(X1)
+y1_sol = u(X1)
+y2 = u(X2)
+
+
+# Initialize and run models
+kernel = cubicMatern_laplace(dim=dim)
+maxIter = 1001
+
+list_beta = [1]
+list_weightings = [1e0, 1e3, 1e5]
+n_iter = len(list_beta) * len(list_weightings)
+
+
+## Actually start the computations
+idx_counter = 0 
+dic_results_train = {}
+list_models = []
+
+for beta in list_beta:
+
+    dic_results_train[beta] = {}
+
+    for weight_dirichlet in list_weightings:
+
+        dic_results_train[beta][weight_dirichlet] = {}
+        
+        # Compute stuff
+        model_pde_vkoga = VKOGA_PDE(kernel=kernel, beta=beta, weight_dirichlet=weight_dirichlet, verbose=True)
+        _ = model_pde_vkoga.fit(X1, y1, X2, y2, maxIter=maxIter, y1_sol=y1_sol)
+
+        list_models.append(model_pde_vkoga)
+
+        # Store values
+        dic_results_train[beta][weight_dirichlet]['f_sol'] = np.array(model_pde_vkoga.train_hist['f sol'])
+        dic_results_train[beta][weight_dirichlet]['f'] = np.array(model_pde_vkoga.train_hist['f'])
+        dic_results_train[beta][weight_dirichlet]['n_ctrs_bdry'] = len(model_pde_vkoga.ind_ctrs2)
+
+        idx_counter += 1
+
+        print(datetime.now().strftime("%H:%M:%S"), 'Computation {}/{} finished.'.format(idx_counter, n_iter))
+
+
+
+
+
+## For intermediate model sizes, compute test errors
+list_n_exp_size = list(np.geomspace(10, model_pde_vkoga.ctrs_.shape[0], 20, dtype=int))
+
+dic_results_test = {}
+for beta in list_beta:
+
+    dic_results_test[beta] = {}
+
+    for idx_model, model in enumerate(list_models):     # this corresponds to the different weightings!!
+
+        print(datetime.now().strftime("%H:%M:%S"), 'Computation for model {}/{} started.'.format(idx_model, len(list_models)))
+
+        dic_results_test[beta][list_weightings[idx_model]] = {}
+        dic_results_test[beta][list_weightings[idx_model]]['list_s_test_pred'] = []
+        dic_results_test[beta][list_weightings[idx_model]]['list_Ls_test_pred'] = []
+        
+
+        for n_expansion_size in list_n_exp_size:
+
+            print(datetime.now().strftime("%H:%M:%S"), 'Computation n_expansion_size = {} started.'.format(n_expansion_size))
+
+            # Restrict indices to intermediate model sizes
+            ind_ctrs1 = [idx for idx in model.ind_ctrs1 if idx < n_expansion_size]
+            ind_ctrs2 = [idx for idx in model.ind_ctrs2 if idx < n_expansion_size]
+            ind_ctrs3 = [idx for idx in model.ind_ctrs3 if idx < n_expansion_size]
+
+            # Compute coefficients for the restricted problem
+            coef_ = model.Cut_[:n_expansion_size, :n_expansion_size].transpose() @ model.c[:n_expansion_size]
+
+            # predict s (code taken from PDE VKOGA)
+            s_test_pred = model.kernel.d2_eval(X1_test, model.ctrs_[ind_ctrs1]) @ coef_[ind_ctrs1] \
+                            + model.kernel.eval(X1_test, model.ctrs_[ind_ctrs2]) @ coef_[ind_ctrs2] \
+                            + model.kernel.mixed_k_n(X1_test, model.ctrs_[ind_ctrs3], model.n_ctrs_[ind_ctrs3]) @ coef_[ind_ctrs3]
+            
+
+            # predict Ls (code taken from PDE VKOGA)
+            Ls_test_pred = model.kernel.dd_eval(X1_test, model.ctrs_[ind_ctrs1]) @ coef_[ind_ctrs1] \
+                            + model.kernel.d1_eval(X1_test, model.ctrs_[ind_ctrs2]) @ coef_[ind_ctrs2] \
+                            + model.kernel.mixed_L_n(X1_test, model.ctrs_[ind_ctrs3], model.n_ctrs_[ind_ctrs3]) @ coef_[ind_ctrs3]
+            
+
+            # Compute errors
+            diff_s = s_test_pred - u(X1_test)
+            diff_Ls = Ls_test_pred - f(X1_test)
+
+            # Append errors
+            dic_results_test[beta][list_weightings[idx_model]]['list_s_test_pred'].append(np.max(np.abs(diff_s)))
+            dic_results_test[beta][list_weightings[idx_model]]['list_Ls_test_pred'].append(np.max(np.abs(diff_Ls)))
+
+
+
+
+
+plt.figure(1001)
+plt.clf()
+for weight in list_weightings:
+    plt.plot(list_n_exp_size, dic_results_test[beta][weight]['list_s_test_pred'], 'x-', label='{:.1e}'.format(weight))
+plt.yscale('log')
+plt.xscale('log')
+plt.title('Max error on test set')
+plt.legend(loc='lower left')
+plt.show(block=False)
+
+
+
+plt.figure(1002)
+plt.clf()
+for weight in list_weightings:
+    plt.plot(list_n_exp_size, dic_results_test[beta][weight]['list_Ls_test_pred'], 'x-', label='{:.1e}'.format(weight))
+plt.yscale('log')
+plt.xscale('log')
+plt.title('Max L-error on test set')
+plt.legend(loc='lower left')
+plt.legend()
+plt.show(block=False)
+
+
+
+
+
+
+
+

experiments_pde_vkoga/utilities_pacman.py

+import numpy as np
+import math
+
+dim = 2
+
+
+def get_function_pacman(str_function, alpha=None):
+    if str_function == 'singular':
+        assert alpha is not None, 'Provide alpha value for str_function = singular!'
+
+        u = lambda x: np.linalg.norm(x, axis=1, keepdims=True) ** (1 / alpha) \
+                      * np.sin(my_arctan(x[:, [0]], x[:, [1]]) / alpha) + 1
+        f = lambda x: np.zeros((x.shape[0], 1))
+    elif str_function == 'smooth':
+        u = lambda x: np.linalg.norm(x, axis=1, keepdims=True) ** 2 + 1
+        f = lambda x: -4 * np.ones((x.shape[0], 1))
+
+    return u, f
+
+def my_arctan(x1, x2):
+    # Returns angle in the interval [0, 2pi]
+
+    phi = np.arctan2(x2, x1)
+    phi += (phi < 0) * 2 * math.pi
+
+    return phi
+
+
+def sample_domain_pacman(n1, n2, alpha):
+
+    assert 0 <= alpha <= 2, 'alpha must satisfy 0 <= alpha * pi <= 2pi'
+
+    # Create interior
+    X1 = 2*np.random.rand(n1, dim) - 1
+    X1 = X1[np.linalg.norm(X1, axis=1) < 1]
+    X1 = X1[my_arctan(X1[:, 0], X1[:, 1]) < math.pi * alpha, :]      # this is not 100% correct due to points with y=0
+
+    # Create boundary
+    array_linspace = np.linspace(0, 1, n2).reshape(-1, 1)
+    X2_Gamma1 = np.concatenate((array_linspace, np.zeros_like(array_linspace)), axis=1)
+    X2_Gamma2 = np.concatenate((np.cos(array_linspace * alpha * math.pi), np.sin(array_linspace  * alpha * math.pi)), axis=1)
+    X2_Gamma3 = np.concatenate((np.cos(alpha * math.pi) * array_linspace,
+                                np.sin(alpha * math.pi) * array_linspace), axis=1)
+
+    X2 = np.concatenate((X2_Gamma1, X2_Gamma2, X2_Gamma3), axis=0)
+
+
+    # Create a meshgrid using numpy.meshgrid
+    X1_grid0_, X1_grid1_ = np.meshgrid(np.linspace(-1, 1, 1001), np.linspace(-1, 1, 1001))
+    X1_grid = np.concatenate((X1_grid0_.reshape(-1, 1), X1_grid1_.reshape(-1, 1)), axis=1)
+    X1_grid = X1_grid[np.linalg.norm(X1_grid, axis=1) < 1, :]                           # shrink to circle
+    X1_grid = X1_grid[my_arctan(X1_grid[:, 0], X1_grid[:, 1]) < alpha * math.pi, :]     # shrink to angle
+
+    return X1, X2, X1_grid
\ No newline at end of file

setup_python_experiments.sh

+set -e
+export BASEDIR="$(cd "$(dirname ${BASH_SOURCE[0]})" ;  pwd -P)"
+cd "${BASEDIR}"
+
+# Create and source virtualenv
+if [ -e "${BASEDIR}/venv/bin/activate" ]; then
+	echo "using existing virtualenv"
+else	
+	echo "creating virtualenv ..."
+	virtualenv --python=python3 venv
+fi
+
+source venv/bin/activate
+
+# Upgrade pip and install libraries (dependencies are also installed)
+pip install --upgrade pip
+pip install git+https://gitlab.mathematik.uni-stuttgart.de/pub/ians-anm/pde-vkoga@v0.1.1
+