Radial basis functions interpolation in 1D ( derivatives)

How one can obtain the first and second derivatives (with a velocities and accelerations at the beginning and at the end supposed null conditions "V_init = V_final = 0 and A_int = A_final = 0") of the function f (x) modeled by RBF interpolation in 1D. This function is presented in the following link: https://www.mathworks.com/matlabcentral/fileexchange/10056-scattered-data-interpolation-and-approximation-using-radial-base-functions

help me, thank you

Answer 1

[Shishir Reddy]

Hi Nanou

To compute the first and second derivatives (velocity and acceleration) of a 1D function interpolated with RBFs and enforce:

• V_init = V_final = 0
• A_init=A_final=0

you can modify the interpolation system to include these as additional constraints. Below is a minimal example using Gaussian RBFs.

x = linspace(0, 1, 10)';        
y = sin(2*pi*x);            

% RBF setup (Gaussian)
epsilon = 5;
phi     = @(r) exp(-(epsilon*r).^2);
dphi    = @(r) -2*epsilon^2 * r .* exp(-(epsilon*r).^2);
ddphi   = @(r) 2*epsilon^2 * (2*epsilon^2 * r.^2 - 1) .* exp(-(epsilon*r).^2);

N = length(x);
A = zeros(N, N);
for i = 1:N
    for j = 1:N
        A(i,j) = phi(abs(x(i) - x(j)));
    end
end

% Boundary points
x0 = x(1); x1 = x(end);

% Derivative constraint rows (1 x N)
drow1  = dphi(x0 - x)' .* sign(x0 - x)';
drow2  = dphi(x1 - x)' .* sign(x1 - x)';
ddrow1 = ddphi(x0 - x)';
ddrow2 = ddphi(x1 - x)';

% Extend system
A_ext = [A; drow1; drow2; ddrow1; ddrow2];
y_ext = [y; 0; 0; 0; 0];

% Solve for RBF weights
lambda = A_ext \ y_ext;

% Evaluate interpolation and derivatives
xx = linspace(0, 1, 200)';
f = zeros(size(xx));
df = zeros(size(xx));
ddf = zeros(size(xx));

for i = 1:N
    r = abs(xx - x(i));
    s = sign(xx - x(i));
    f   = f   + lambda(i) * phi(r);
    df  = df  + lambda(i) * dphi(r) .* s;
    ddf = ddf + lambda(i) * ddphi(r);
end

% Plot
plot(xx, f, 'b', xx, df, 'r--', xx, ddf, 'g-.', x, y, 'ko');
legend('f(x)', 'f''(x)', 'f''''(x)', 'data points');
title('1D RBF Interpolation with Velocity and Acceleration Constraints');
grid on;

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I hope this helps.