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script_accuracy_test2.m
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script_accuracy_test2.m
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%% Script to investigate accuracy
%% Setting up the script
clc, clear
% free parameters
dim = 3; % dimension (1,2,3)
points = 'Halton'; % points (equid, uniform, Halton)
noise_level = 0; % 0, 10^(-6)
% fixed parameters
domain = 'ball'; % domain (cube, ball)
weightFun = 'sqrt(r)'; % weight function - 1, C2k, sqrt(r)
omega = generate_weightFun( weightFun, dim);
if dim == 1
volume = 2;
f = @(x) 1./(1 + x.^2 ) + sin(x); % test function
I_aux = -2^(-3/2)*( log(2+sqrt(2)) - log(2-sqrt(2)) - 2*atan(sqrt(2)+1) - 2*atan(sqrt(2)-1) );
n = 20;
n_max = 400;
elseif dim == 2
volume = pi;
f = @(x,y) 1./(1 + x.^2 + y.^2 ) + sin(x); % test function
I_aux = 2^(-3/2)*( -log(2+sqrt(2)) + log(2-sqrt(2)) + 2^(5/2) - 2*atan(sqrt(2)+1) - 2*atan(sqrt(2)-1) );
n = 4;
n_max = 40;
else
volume = 4*pi/3;
f = @(x,y,z) 1./(1 + x.^2 + y.^2 + z.^2 ) + sin(x); % test function
I_aux = 2^(-3/2)*( log(2+sqrt(2)) - log(2-sqrt(2)) ) + 2/3 -2^(-1/2)*( atan(sqrt(2)+1) + atan(sqrt(2)-1) );
n = 4;
n_max = 16;
end
I = I_aux*( 2*pi^(dim/2)/gamma(dim/2) ); % exact integral
NN_Leg = []; NN_MC = []; NN_LS = []; NN_l1 = []; % number of data points
err_Leg = []; err_MC = []; err_LS = []; err_l1 = []; % errors
while n <= n_max
% Legendre rule
example = matfile(['CFs/CF_Leg_dim=',num2str(dim),'_',domain,'_n=',num2str(n),'.mat']);
C = example.CF_Leg;
[ N, aux] = size(C);
NN_Leg = [NN_Leg; N];
X = C(:,1:dim); % data points
w = C(:,dim+1); % weights
% function values
f_values = zeros(N,1);
for m = 1:N
if dim == 1
f_values(m) = f( X(m,1) ).*omega( X(m,1) );
elseif dim == 2
f_values(m) = f( X(m,1), X(m,2) ).*omega( X(m,1), X(m,2) );
elseif dim == 3
f_values(m) = f( X(m,1), X(m,2) , X(m,3) ).*omega( X(m,1), X(m,2) , X(m,3) );
else
error('Desired dimension not yet implemented!')
end
end
% generate and add uniform noise
noise = noise_level*(2*rand(N,1)-1);
f_values = f_values + noise;
CF = dot( w, f_values ); % value of the CF
err_Leg = [err_Leg; abs( I - CF )]; % absolute error
% LS rule
example = matfile(['CFs/CF_LS_dim=',num2str(dim),'_',domain,'_',weightFun,'_',points,'_n=',num2str(n),'.mat']);
C = example.CF_LS;
[ N, aux] = size(C);
NN_LS = [NN_LS; N];
X = C(:,1:dim); % data points
w = C(:,dim+1); % weights
% function values
f_values = zeros(N,1);
for m = 1:N
if dim == 1
f_values(m) = f( X(m,1) );
elseif dim == 2
f_values(m) = f( X(m,1), X(m,2) );
elseif dim == 3
f_values(m) = f( X(m,1), X(m,2) , X(m,3) );
else
error('Desired dimension not yet implemented!')
end
end
% generate and add uniform noise
noise = noise_level*(2*rand(N,1)-1);
f_values = f_values + noise;
CF = dot( w, f_values ); % value of the CF
err_LS = [err_LS; abs( I - CF )]; % absolute error
% l1 rule
example = matfile(['CFs/CF_l1_dim=',num2str(dim),'_',domain,'_',weightFun,'_',points,'_n=',num2str(n),'.mat']);
C = example.CF_l1;
[ N, aux] = size(C);
NN_l1 = [NN_l1; N];
X = C(:,1:dim); % data points
w = C(:,dim+1); % weights
% function values
f_values = zeros(N,1);
for m = 1:N
if dim == 1
f_values(m) = f( X(m,1) );
elseif dim == 2
f_values(m) = f( X(m,1), X(m,2) );
elseif dim == 3
f_values(m) = f( X(m,1), X(m,2) , X(m,3) );
else
error('Desired dimension not yet implemented!')
end
end
% generate and add uniform noise
noise = noise_level*(2*rand(N,1)-1);
f_values = f_values + noise;
CF = dot( w, f_values ); % value of the CF
err_l1 = [err_l1; abs( I - CF )]; % absolute error
% MC integration
NN_MC = [NN_MC; N];
% MC weights
for m = 1:N
if dim == 1
w(m) = volume*omega( X(m,1) )/N;
elseif dim == 2
w(m) = volume*omega( X(m,1), X(m,2) )/N;
elseif dim == 3
w(m) = volume*omega( X(m,1), X(m,2) , X(m,3) )/N;
else
error('Desired dimension not yet implemented!')
end
end
CF = dot( w, f_values ); % value of the CF
err_MC = [err_MC; abs( I - CF )]; % absolute error
% increase n
if dim == 1
n = n + 20;
elseif dim == 2
n = n + 2;
else
n = n + 1;
end
end
% Plot the results
figure(1)
p = plot( NN_MC,err_MC,'ms', NN_LS,err_LS,'r+', NN_l1,err_l1,'b^', NN_Leg,err_Leg,'ko');
set(p, 'LineWidth',1.5)
set(p, 'markersize',8)
set(gca, 'FontSize', 20) % Increasing ticks fontsize
xlim([ max([NN_Leg(1);NN_LS(1)]), min([NN_Leg(end);NN_LS(end)]) ])
ylim([ min([err_MC;err_LS;err_l1;err_Leg])/10, max([err_MC;err_LS;err_l1;err_Leg])*10 ])
xlabel('$N$','Interpreter','latex')
ylabel('$|C[f] - I[f]|$','Interpreter','latex')
set(gca, 'XScale', 'log')
set(gca, 'YScale', 'log')
if strcmp( points, 'Halton')
id = legend('QMC','LS','$\ell^1$','Legendre','Interpreter','latex','Location','southwest');
else
id = legend('MC','LS','$\ell^1$','Legendre','Interpreter','latex','Location','southwest');
end
set(id, 'Interpreter','latex', 'FontSize',26)
grid on
str = sprintf( ['plots_accuracy/accuracy_test2_dim=',num2str(dim),'_',points,'.fig'] );
%savefig(str);