Reconstruction and collocation of a class of non-periodic functions by sampling along tent-transformed rank-1 lattices

Spectral collocation and reconstruction methods have been widely studied for periodic functions using Fourier expansions.We investigate the use of cosine series for the approximation and collocation of multivariate non-periodic functions with frequency support mainly determined by hyperbolic crosses.We seek methods that work for an arbitrary number of dimensions.We show that applying the tent-transformation on rank-1 lattice points renders them suitable to be collocation/sampling points for the approximation of non-periodic functions with perfect numerical stability. Moreover, we show that the approximation degree—in the sense of approximating inner products of basis functions up to a certain degree exactly—of the tent-transformed lattice point set with respect to cosine series, is the same as the approximation degree of the original lattice point set with respect to Fourier series, although the error can still be reduced in the case of cosine series. A component-by-component algorithm is studied to construct such a point set. We are then able to reconstruct a non-periodic function from its samples and approximate the solutions to certain PDEs subject to Neumann and Dirichlet boundary conditions. Finally, we present some numerical results.

Publication year:2016

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Authors from:Higher Education

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