Multilevel sparse grids collocation for linear partial differential equations, with tensor product smooth basis functions

TL;DR

This paper introduces a multilevel sparse grid collocation method using tensor product smooth basis functions, specifically for solving four-dimensional space-time PDEs efficiently and accurately.

Contribution

It develops a novel collocation approach with sparse kernel basis functions for high-dimensional PDEs, extending previous methods to four-dimensional space-time problems.

Findings

Achieves accuracy comparable to existing methods

Efficiently handles four-dimensional space-time PDEs

Uses smooth multiquadric and Gaussian basis functions

Abstract

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}.