APPROXIMATION OF HIGH-DIMENSIONAL PERIODIC FUNCTIONS WITH FOURIER-BASED METHODS

We propose an approximation method for high-dimensional 1-periodic functions based on the multivariate ANOVA decomposition. We provide analysis of classical ANOVA decomposition on the torus and prove some important properties, such as the inheritance of smoothness for Sobolev type spaces and the weighted Wiener algebra. We exploit special kinds of sparsity in the ANOVA decomposition with the aim of approximating a function in a scattered data or black-box approximation scenario. This method allows us to simultaneously achieve an importance ranking on dimensions and dimension interactions (referred to as an attribute ranking in some applications). In scattered data approximation we rely on a special algorithm based on the non-equispaced fast Fourier transform (or NFFT) for fast multiplication with arising Fourier matrices. For black-box approximation we choose the well-known rank-1 lattices as sampling schemes and show properties of the arising special lattices.

Automated summary

The proposed method utilizes multivariate ANOVA decomposition to approximate high-dimensional periodic functions, showing advantages in achieving importance ranking on dimensions and dimension interactions in scattered data or black-box approximation scenarios.