PUchebfun provides a way to represent multivariate functions (up to 3D) with a partition of unity approximation consisting of Chebyshev tensor product polynomials. These approximations are infinitely smooth, highly accurate and memory efficient. The overlapping domains are formed through recursive bisection, and are organized into a binary tree.
The paper describing the method can be found here.
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Matlab 2017a
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chebfun 5.5
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Kevin W. Aiton
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Tobin A. Driscoll
To construct an aproximation on the square [-1 1;-1 1] we can call:
F = PUchebfun(@(x,y) atan(50*(x+y.^2)));PUchebfun can be called this way for 1D-3D functions for the domain [-1 1]^d. To plot functions and see the subdomains, we call:
plot(F);
figure(), show(F);
We can also compute the norm of F, integrate F over [-1 1;-1 1], and compute F(x,y)d/dx with:
norm(F)
sum(F)
diff(F,1,1)Furthermore, PUchebfuns can be symboically combined like so:
F1 = PUchebfun(@(x,y) atan(100*(x+y.^2)));
F2 = PUchebfun(@(x,y) atan(100*(x.^2+y)));
F3 = F1 + F2;
plot(F3)
figure(), show(F3);
We have included some preliminary work with creating Chebyshev approximations with non-rectangular domains. We can build an approximation on two overlayed astroid curves like so:
F = PUFunLS(@(x,y)atan(x.^2+y),DoubleAstroid(),[-0.95 0.95;-0.95 0.95],'tol',1e-10)
