Multilevel sparse grids collocation for linear partial differential equations, with tensor product smooth basis functions
It provides an easy-to-implement algorithm for solving high-dimensional PDEs, but the results are incremental as they match rather than surpass existing methods.
The paper extends the MuSIK algorithm to solve linear PDEs via collocation using a sparse kernel basis, demonstrating accuracy comparable to existing methods for space-time problems in up to four dimensions.
Radial basis functions have become a popular tool for approximation and solution of partial differential equations (PDEs). The recently proposed multilevel sparse interpolation with kernels (MuSIK) algorithm proposed in \cite{Georgoulis} shows good convergence. In this paper we use a sparse kernel basis for the solution of PDEs by collocation. We will use the form of approximation proposed and developed by Kansa \cite{Kansa1986}. We will give numerical examples using a tensor product basis with the multiquadric (MQ) and Gaussian basis functions. This paper is novel in that we consider space-time PDEs in four dimensions using an easy-to-implement algorithm, with smooth approximations. The accuracy observed numerically is as good, with respect to the number of data points used, as other methods in the literature; see \cite{Langer1,Wang1}.