In this talk, we present algorithms for the approximation of multivariate functions. We start with the approximation by trigonometric polynomials based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. The proposed algorithm based mainly on a one-dimensional fast Fourier transform, and the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. After a detailed introduction we will focus on the following questions in more detail.
• We discuss methods where the support of the trigonometric polynomial is unknown.
• We describes an extension of approximation methods for nonperiodic functions via a multivariate change of variables.
• We present a method based on the analysis of variance (ANOVA) decomposition that aims to detect the structure of the function, i.e., find out which dimension and dimension interactions are important. This information is then utilized in obtaining an approximation for the function.
This talk based on joint work with Robert Nasdala and Michael Schmischke.