Interpolation lattices for hyperbolic cross trigonometric polynomials

Abstract

Sparse grid discretisations allow for a severe decrease in the number of degrees of freedom for high-dimensional problems. Recently, the corresponding hyperbolic cross fast Fourier transform has been shown to exhibit numerical instabilities already for moderate problem sizes. In contrast to standard sparse grids as spatial discretisation, we propose the use of oversampled lattice rules known from multivariate numerical integration. This allows for the highly efficient and perfectly stable evaluation and reconstruction of trigonometric polynomials using only one ordinary FFT. Moreover, we give numerical evidence that reasonable small lattices exist such that our new method outperforms the sparse grid based hyperbolic cross FFT for realistic problem sizes.