Problem 4.5

meshfree interpolation with compactly supported RBFs

Contents

Gaussian and Wendland RBFs

f = @(x) 4*x.*(1-x);
phi = @(r) exp(-r.^2); ep = 1;                  % Gaussian
wld30 = @(r) max(1-r,0).^2;                     % Wendlands \phi_3,0
wld31 = @(r) max(1-r,0).^4.*(4*r+1);            % Wendlands \phi_3,1
wld32 = @(r) max(1-r,0).^6.*(35*r.^2+18*r+3)/3; % Wendlands \phi_3,2
wld33 = @(r) max(1-r,0).^8.*(32*r.^3+25*r.^2+8*r+1); % Wendlands \phi_3,3
x = linspace(0,2,100);
plot(x,phi(3*x),'-',x,wld30(x),'r-',x,wld31(x),'m-',x,wld32(x),'k-',x,wld33(x),'g-');
xlabel('x'); ylabel('\phi');
legend('Gaussians','Wendland 3,0','Wendland 3,1','Wendland 3,2','Wendland 3,3')

comparison

M = 2000;
X_eval = linspace(1/M,1-1/M,M)';
N = logspace(1,3,30);
for k=1:30,
    X = linspace(0,1,N(k))';
    y = f(X);
    D = distm(X,X);
    D1 = phi(3*ep*D); c1 = D1\y;
    D2 = wld30(ep*D); c2 = D2\y;
    D3 = wld31(ep*D); c3 = D3\y;
    D4 = wld32(ep*D); c4 = D4\y;
    D5 = wld33(ep*D); c5 = D5\y;
    D_e = distm(X_eval,X);
    D_e1 = phi(3*ep*D_e);
    D_e2 = wld30(ep*D_e);
    D_e3 = wld31(ep*D_e);
    D_e4 = wld32(ep*D_e);
    D_e5 = wld33(ep*D_e);
    y_eval = f(X_eval);
    err_1(k) = norm(D_e1*c1 - y_eval)/sqrt(M); cond1(k) = condest(D1);
    err_2(k) = norm(D_e2*c2 - y_eval)/sqrt(M); cond2(k) = condest(D2);
    err_3(k) = norm(D_e3*c3 - y_eval)/sqrt(M); cond3(k) = condest(D3);
    err_4(k) = norm(D_e4*c4 - y_eval)/sqrt(M); cond4(k) = condest(D4);
    err_5(k) = norm(D_e5*c5 - y_eval)/sqrt(M); cond5(k) = condest(D5);
end
clf; loglog(N,err_1,'o-',N,err_2,'ro-',N,err_3,'mo-',N,err_4,'ko-',N,err_5,'go-');
xlabel('N'); ylabel('errors');
legend('Gaussians','Wendland 3,0','Wendland 3,1','Wendland 3,2','Wendland 3,3')
 figure
 loglog(N,cond1,'o-',N,cond2,'ro-',N,cond3,'mo-',N,cond4,'ko-',N,cond5,'go-');
xlabel('N'); ylabel('condition');
legend('Gaussians','Wendland 3,0','Wendland 3,1','Wendland 3,2','Wendland 3,3')
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.639196e-17. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  2.701602e-18. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.887355e-18. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  9.970463e-19. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  8.791070e-19. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.434898e-18. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  7.670766e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.431458e-19. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.481175e-19. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  2.766846e-19. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  4.845894e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  9.031678e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.523491e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  9.472132e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  2.861944e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  6.092137e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  6.516461e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.775548e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.330198e-17. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.778575e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  4.111697e-19. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  8.530781e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.914603e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.051972e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.057953e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  3.711948e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.028919e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  2.671844e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.935918e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.258611e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  3.514171e-23. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  7.966385e-22. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  1.589675e-20. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  2.644357e-21. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  8.891769e-17. 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND =  5.019618e-21.