How to handle the differentiation of non-analytic smooth functions (the bump function)

[Lin]

Dear there,

I am wondering if casADi can handle the differentiation of the so-called bump function? Wiki link for bump function. It is a smooth function, but it's non-analytical.

If the answer is yes, can you provide a Matlab example of how to code the function?

Thanks a lot!

Best,

Lin

P.S. The following code doesn't work.

fm=zeros(size(x));

fm(abs(x-R)<R) = exp(1./((x(abs(x-R)<R)-R).^2-R^2));

[Joel Andersson]

For radius 1, you can try:

x = casadi.SX.sym('x');

x2 = x*x;

phi = if_else(x2<1, exp(-1/(1-x2)), 0);

 Working with non-differentiable functions in CasADi is risky, lots of things can go wrong. So I wouldn't recommend it unless you are already pretty familiar with the tool.

Joel

[Bruno M L]

Interesting function!

Dear Joel, for what I understand the bump function is differentiable (smooth) but it is divided in two parts. Each part is analytic and differentiable so I believe if_else can work.

Dear Lin, if it works, please share with us.

Bruno

[Lin]

Dear Joel,

The if_else function works perfectly.

Thanks for the help!

Best,

Lin

[Lin]

Dear Bruno, 

The if_else function works perfectly.

Here is the plot of the casadi calculated derivatives.

The Matlab script:

function yout = testBump

clc

clear

x= casadi.SX.sym('x',1);

molli= casadi.Function('molli',{x},{Mollifier(x)});

deriva = casadi.Function('deriva',{x},{jacobian(molli(x),x)});

%%

figure

xx= -0.5:0.001:2.5;

subplot(2,1,1)

plot(xx, full(molli(xx)))

legend('Bump function')

grid on

subplot(2,1,2)

yout = full(deriva(xx));

plot(xx, full(deriva(xx)))

legend('derivative of bump function')

grid on

end

function y = Mollifier(x)

    R     = 1;

    m     = 1;

    x2    = (x-R).^2;

    y     = if_else(x2<R^2, exp(m./(x2 - R^2)), 0);      

end