Kernel interpolation for partial differential equations

Abstract

Polynomial interpolation aims to construct a function, known as the interpolant, that approximates an unknown function based on its discrete data points. Despite the usefulness of this approach, it fails to perform effectively in higher dimensions, highlighting the need for a more suitable alternative. This necessitates the use of kernel interpolation, where positive definite functions - known as radial basis functions (RBFs) - are employed to perform the interpolation. Beyond its use for general function approximation, kernel interpolation is also applicable to finite element spaces. In this approach, a kernel function is obtained by numerically solving the partial differential equation (PDE) twice, forming the basis for approximating the solution to any type of PDE using discrete points associated with the problem.