Radial basis function interpolation for black-box multi-physics simulations

Abstract

Interpolation based on radial basis functions (RBF) is a standard data map- ping method used in multi-physics coupling. It works on scattered data without requiring additional mesh topology or neighborhood information of support points. However, sys- tem matrices of the equations for the coefficients tend to be ill-conditioned. In this work, we illustrate the problem by a simple example and discuss possible remedies. Furthermore, we investigate the numerical performance of this method on uniform and non-uniform meshes with a particular focus on the coupling of black-box components where typically no information about the underlying discretization can be extracted. Radial basis func- tion interpolation usually uses an enhancement of the radial basis functions by a global polynomial in order to properly capture constant components and linear trends in the given data. We present a method that determines this polynomial independent from the radial basis function ansatz, which substantially improves the condition number of the remaining RBF system. Furthermore, we show that a rescaling approach can be used to either increase the accuracy or improve the condition number even further by choosing radial basis functions with a smaller support radius. The results represent an intermediate state with the aim to be integrated into the multi-physics coupling library preCICE.