Matrix-valued radial basis functions: stability estimates and applications

Radial basis functions (RBFs) have found important applications in areas such as signal processing, medical imaging, and neural networks since the early 1980’s. Several applications require that certain physical properties are satisfied by the interpolant, for example being divergence free in case of incompressible data. In this paper we consider a class of customized (e.g. divergence-free) RBFs that are matrix valued and have compact support; these are matrix-valued analogues of the well-known Wendland functions. We obtain stability estimates for a wide class of interpolants based on matrix-valued RBFs, also taking into account the size of the compact support of the generating RBF. We conclude with an application based on an incompressible Navier-Stokes equation, namely the driven-cavity problem, where we use divergence-free RBFs to solve the underlying partial differential equation numerically. We discuss the impact of the size of the support of the basis function on the stability of the solution.