High-dimensional approximation with kernel-based multilevel methods on sparse grids

Abstract

Moderately high-dimensional approximation problems can successfully be solved by combining univariate approximation processes using an intelligent combination technique. While this has so far predominantly been done with either polynomials or splines, we suggest to employ a multilevel kernel-based approximation scheme. In contrast to those schemes built upon polynomials and splines, this new method is capable of combining arbitrary low-dimensional domains instead of just intervals and arbitrarily distributed points in these low-dimensional domains. We introduce the method and analyse its convergence in the so-called isotropic and anisotropic cases.