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RUN_ME_eigenfunction.m
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RUN_ME_eigenfunction.m
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% This code numerically computes the first eigenfunction of the
% infinity Laplacian operator on different domains.
% @authors: Farid Bozorgnia, Leon Bungert, Daniel Tenbrinck
% @date: 10/01/22
% tidy up workspace
clear all;
close all;
% define problem
problem = 'eigenfunction';
% problem = 'infinity-harmonic';
% define domain
shape = 'square';
% shape = 'square-disk';
% shape = 'half-disk';
% shape = 'disk';
% shape = 'ellipsis';
% shape = 'L-shape';
% shape = 'trapez';
% shape = 'rectangle';
% shape = 'stadium';
% shape = 'triangle';
% shape = 'triangle2';
% shape = 'dumbell';
% shape = 'two-disks';
% shape = 'heart1';
% shape = 'heart2';
% define boundary conditions, mainly for harmonic equation
% f = @(x,y) (abs(x).^(4/3) - abs(y).^(4/3));
% f = @(x,y) x.^3-3.*x.*y.^2;
f = @(x,y) 0.*x.^0;
% f = @(x,y) (x==1).*(abs(y)<0.01);
% either enforce positive or negative peaks or do normalization
% WARNING! Normalization is not guaranteed to produce correct solution,
high_ridge = true;
normalization = false;
% define initialization
init = 'zero';
% init = 'random';
% init = 'distance';
% initialize necessary parameters
alpha = 1; % concave approximation parameter <= 1
TOL = 1*1e-7;
max_iterations = 5000;
max_broyden = 0;
broyden = 'bad';
samples = 1*50; % has to be even!!!!
stencil_shape = 'full'; % valid stencils: full, square
neighborhood_size = 9; % at least 3
save2disk = false; % save results to disk
visualize = true; % visualize the results
savingFreq = 10; % saving frequency
visFreq = 100; % visualization frequency
% catch normalization in case of inf-harmonic problem
if strcmpi(problem, 'infinity-harmonic')
high_ridge = true;
normalization = false;
end
% initialize grid on unit square [-1 1]^2 according to amount of sample
% points
step_size = 2 / samples;
[x,y] = meshgrid(linspace(-1,1,samples+1));
% define subdomain of the square domain
switch shape
case 'square'
domain = @(x,y) max(abs(x),abs(y)) <= 1;
case 'square-disk'
domain = @(x,y) (x.^2 + y.^2 < 1)|(x > 0);
closed_domain = @(x,y) (x.^2 + y.^2 <= 1)|(x >= 0);
case 'half-disk'
domain = @(x,y) (x.^2 + y.^2 < 1).*(x<0);
closed_domain = @(x,y) (x.^2 + y.^2 <= 1).*(x<=0);
case 'disk'
domain = @(x,y) x.^2 + y.^2 < 1;
closed_domain = @(x,y) x.^2 + y.^2 <= 1;
case 'ellipsis'
domain = @(x,y) x.^2 + (4*y).^2 < 1;
closed_domain = @(x,y) x.^2 + (4*y).^2 <= 1;
case 'L-shape'
domain = @(x,y) (x<0)|(y>0);
closed_domain = @(x,y) (x<=0)|(y>=0);
case 'trapez'
domain = @(x,y) ((x<0)|(y>0)).*(y<-x);
closed_domain = @(x,y) ((x<=0)|(y>=0)).*(y<=-x);
case 'rectangle'
domain = @(x,y) abs(y)<0.5;
closed_domain = @(x,y) abs(y)<=0.5;
case 'stadium'
domain = @(x,y) ((abs(y)<0.5).*(abs(x)<0.5))|((x-0.5).^2+y.^2<0.5^2)|((x+0.5).^2+y.^2<0.5^2);
closed_domain = @(x,y) ((abs(y)<=0.5).*(abs(x)<=0.5))|((x-0.5).^2+y.^2<=0.5^2)|((x+0.5).^2+y.^2<=0.5^2);
case 'triangle'
domain = @(x,y) (x<y);
closed_domain = @(x,y) (x<=y);
case 'triangle2'
domain = @(x,y) (abs(x)<y);
closed_domain = @(x,y) (abs(x)<=y);
case 'dumbell'
domain = @(x,y) ((x+1).^2+y.^2<1)|((x-1).^2+y.^2<1)|(abs(y)<0.5);
closed_domain = @(x,y) ((x+1).^2+y.^2<=1)|((x-1).^2+y.^2<=1)|(abs(y)<=0.5);
case 'two-disks'
domain = @(x,y) ((x+.5).^2+y.^2<.5^2)|((x-.5).^2+y.^2<.5^2)|((abs(y)<0.2).*(abs(x)<0.5));
closed_domain = @(x,y) ((x+.5).^2+y.^2<=.5^2)|((x-.5).^2+y.^2<=.5^2)|((abs(y)<=0.2).*(abs(x)<=0.5));
case 'heart1'
domain = @(x,y) (((y+0.5).^2+(x).^2<0.5^2).*(x<=0.5))|(((y-0.5).^2+(x).^2<0.5^2).*(x<=0.5))|((x < 1-0.75*abs(y)).*(x>0).*(-0.75<=y).*(y<0.75));
closed_domain = @(x,y) (((y+0.5).^2+(x).^2<=0.5^2).*(x<=0.5))|(((y-0.5).^2+(x).^2<=0.5^2).*(x<=0.5))|((x <= 1-0.75*abs(y)).*(x>=0).*(-0.75<=y).*(y<=0.75));
case 'heart2'
domain = @(x,y) (((y+0.5).^2+(x+0.5).^2<0.48^2).*(x<=-0.25))|(((y-0.5).^2+(x+0.5).^2<0.48^2).*(x<=-0.25))|((x < 1-1.35*abs(y)).*(x>-0.25))|((x<0).*(-0.5<x).*(abs(y)<0.5));
closed_domain = @(x,y) (((y+0.5).^2+(x+0.5).^2<=0.48^2).*(x<=-0.25))|(((y-0.5).^2+(x+0.5).^2<=0.48^2).*(x<=-0.25))|((x <= 1-1.35*abs(y)).*(x>-0.25))|((x<=0).*(-0.5<=x).*(abs(y)<0.5));
end
if ~exist('closed_domain','var')
closed_domain = domain;
end
% set boundary values by defining function on whole domain and erasing
% inner part then
p = zeros(size(x));
p([1 end],:)=-1;
p(:,[1 end])=-1;
p(find(~domain(x,y)))=-1;
distance_function = double(bwdist(p < 0)) * step_size;
dist_inner = distance_function(2:end-1, 2:end-1);
[max_dist, ~] = max(dist_inner(:));
lambda = 1/max_dist;
max_idx = find(dist_inner == max_dist);
max_dist_tmp = max_dist;
max_dist = 1;
center_idx = (length(max_idx)-1)/2+1;
max_idx = max_idx(center_idx);
% max_idx = max_idx(1);
% max_idx = [max_idx(1), max_idx(end)];
% max_idx = 1105;
% max_idx = 4513;
% initialization
switch init
case 'random'
phi = rand(size(distance_function));
case 'distance'
phi = distance_function / max_dist_tmp;
case 'zero'
phi = zeros(size(distance_function));
end
u = phi(2:end-1,2:end-1);
u = u(:);
% building the distance function for a specified stencil
distance = generate_stencil(neighborhood_size, step_size, stencil_shape);
radius = floor(neighborhood_size/2);
% initialize boundary conditions on larger grid
bcinner = f(x,y);
bc = padarray(bcinner, [radius-1, radius-1], NaN);
% compute masks for the domain
xinn = x(2:end-1,2:end-1); yinn = y(2:end-1,2:end-1);
mask = domain(xinn, yinn);
mask = mask(:);
mask_vis = double(closed_domain(x, y));
mask_vis(mask_vis==0) = nan;
% initialize matrix of values
values = zeros(floor(neighborhood_size^2 /2)*neighborhood_size^2, numel(u));
% initialize matrix for values of ustar
ustar = zeros(size(values,2),1);
% initialize stopping criterions
rel_change = inf;
scheme_accuracy = inf;
iteration = 0;
% create output folder
if save2disk
foldername = [problem, '_', shape, '_', 'high_ridge_', num2str(high_ridge),...
'_norm_', num2str(normalization), '_init_', init,...
'_nbrs_', num2str(neighborhood_size)];
if alpha < 1
foldername = [foldername, '_alpha_', num2str(alpha)];
end
outputfolder = ['results','/',foldername];
if ~exist('results','dir')
mkdir('results')
end
if ~exist(outputfolder, 'dir')
mkdir(outputfolder)
end
end
% initialize
if high_ridge
u(max_idx) = max_dist;
end
if normalization
u = max(u,0);
u = max_dist * u / norm(u,'inf');
end
B = speye(length(u));
umm = u;
um = u;
fm = zeros(size(u));
fmm = zeros(size(u));
%% iterate until convergence or maximum number of iterations is reached
fig = figure;
fig.PaperOrientation = 'landscape';
set(groot,'defaultAxesTickLabelInterpreter','latex');
set(groot,'defaulttextinterpreter','latex');
set(groot,'defaultLegendInterpreter','latex');
set(groot,'defaultAxesFontSize',20);
% start time measurement
tic
while max(rel_change,scheme_accuracy) > TOL && iteration <= max_iterations
% generate patch rows
pad_u = bc;
pad_u(radius+1:end-radius,radius+1:end-radius) = reshape(u, size(xinn));
patches = im2col(pad_u, [neighborhood_size neighborhood_size], 'sliding');
% compute values abs( (v(k) - v(l) )/( d(k)+d(l) )
for i = 1:floor(neighborhood_size^2 /2)
values( 1+(i-1)*neighborhood_size^2:i*neighborhood_size^2, :) = abs(patches - circshift(patches,[i,0])) ./ ...
repmat(distance' + circshift(distance',[i,0]), [1 size(patches,2)] );
end
% compute maximum values in each column
[~, indices] = max(values, [], 1);
% determine corresponding index pair within neighborhood
index_pairs = zeros(2,size(indices,2));
index_pair1 = mod(indices,neighborhood_size^2);
index_pair1(index_pair1 == 0) = neighborhood_size^2;
index_pair2 = mod(index_pair1 - ceil(indices / neighborhood_size^2) , neighborhood_size^2);
index_pair2(index_pair2 == 0) = neighborhood_size^2;
index_pairs(1,:) = index_pair1;
index_pairs(2,:) = index_pair2;
% compute new values of p1 as hh = (v(k1)*d(k2)+ v(k2)*d(k1))/(d(k1)+d(k2))
% -> this corresponds to u* in the Oberman discretization
for i = 1:size(patches,2)
ustar(i) = (patches(index_pairs(1,i),i) * distance(index_pairs(2,i)) + ...
patches(index_pairs(2,i),i) * distance(index_pairs(1,i)) ) / ...
(distance(index_pairs(1,i)) + distance(index_pairs(2,i)));
end
F2 = u - ustar;
if strcmpi(problem, 'infinity-harmonic')
scheme = F2;
elseif strcmpi(problem, 'eigenfunction')
% upwind gradients with general stencil
slopes = ( u - patches' ) ./ distance;
[max_slopes, local_idx] = max(slopes, [], 2);
distances = distance(local_idx)';
[local_subs_i, local_subs_j] = ind2sub([neighborhood_size, neighborhood_size],local_idx);
[center_subs_i, center_subs_j] = ind2sub(size(xinn), 1:numel(u));
global_subs_i = center_subs_i' + local_subs_i - 1;
global_subs_j = center_subs_j' + local_subs_j - 1;
global_idx = sub2ind(size(xinn) + [2*radius 2*radius] , global_subs_i, global_subs_j);
val = pad_u(global_idx);
F1 = ((u - val ) - lambda.*distances.*sign(u).*abs(u).^alpha) ;
scheme = min( F1 , F2);
else
error('Unknown problem!')
end
rho = 0.9;
if and(iteration >= 2, iteration <= max_broyden)
du = um-umm;
df = fm-fmm;
if strcmpi(broyden, 'good')
B = B + ((df - B * du) * du')/norm(du)^2;
unew = u - B \ scheme;
elseif strcmpi(broyden, 'bad')
B = B + ((du - B * df) * df')/norm(df)^2;
unew = (u - B * scheme);
else
error('Unkown Broyden method')
end
else
unew = (u - rho .* scheme);
end
fmm = fm;
fm = scheme;
umm = um;
um = u;
if high_ridge
unew(max_idx) = max_dist;
end
if normalization
unew = max(unew,0);
unew = max_dist * unew / norm(unew,'inf');
end
% compute difference between two iterations
diff = abs(u-unew).*mask;
% update u and pad boundary values
if strcmpi(problem, 'infinity-harmonic')
scheme(max_idx) = 0;
end
u = unew.*mask;
pad_u = bcinner;
pad_u(2:end-1, 2:end-1) = reshape(um, size(xinn));
solution = pad_u;
scheme_exp = padarray(reshape(scheme.*mask, size(xinn)),[1,1]);
% compute stopping criteria
rel_change = norm(diff, 'inf') / norm(unew.*mask, 'inf');
scheme_accuracy = norm(scheme.*mask, 'inf');
% save to disk
if save2disk && mod(iteration, savingFreq) == 0
filename = ['solution_', 'grid_size_', strrep(num2str(step_size),'.','-'),...
'_iteration_', num2str(iteration)];
save([outputfolder,'/',filename,'.mat'],'solution','scheme_exp')
end
% visualize current solution
if visualize && mod(iteration, visFreq) == 0
fig;
subplot(1,2,1);
surf(x,y,solution.*mask_vis); title(['Solution at iteration ',num2str(iteration)]);
axis equal;
view(-30,20); drawnow;
subplot(1,2,2);
surf(x,y,scheme_exp.*mask_vis); title(['Scheme at iteration ',num2str(iteration)]);
view(-30,20);
drawnow;
end
% give some output
disp(['Current iteration: ' num2str(iteration) ...
', relative change ' num2str(rel_change) ', accuracy ' num2str(scheme_accuracy)]);
% increase iteration counter
iteration = iteration + 1;
end
toc
%%
if save2disk
filename = ['solution_', 'grid_size_', strrep(num2str(step_size),'.','-'),...
'_iteration_', num2str(iteration-1)];
save([outputfolder,'/',filename,'.mat'],'solution','scheme_exp')
end
if visualize
solution = max_dist_tmp * solution;
scheme_exp = max_dist_tmp * scheme_exp;
subplot(1,2,1);
surf(x,y,solution.*mask_vis); title(['Solution at iteration ',num2str(iteration)]);
axis equal;
view(-30,20); drawnow;
subplot(1,2,2);
surf(x,y,scheme_exp.*mask_vis); title(['Scheme at iteration ',num2str(iteration)]);
view(-30,20); drawnow;
if save2disk
print([outputfolder,'/',filename],'-dpdf','-r300','-fillpage')
end
fig2 = figure(2);
fig2.PaperOrientation = 'landscape';
surf(x,y,solution.*mask_vis);
axis equal;
% zlim([0, max_dist]);
% axis([-1, 1, -1, 1, 0, max_dist]);
view(-30,20); drawnow;
if save2disk
print([outputfolder,'/','final_surf_',filename],'-dpdf','-r300','-fillpage')
end
fig3 = figure(3);
fig3.PaperOrientation = 'landscape';
contour(x.*closed_domain(x,y),y.*closed_domain(x,y),solution);
axis equal;
drawnow;
if save2disk
print([outputfolder,'/','final_contour_',filename],'-dpdf','-r300','-fillpage')
end
end