Novel results for the anisotropic sparse grid quadrature

TL;DR

This paper introduces an improved anisotropic sparse grid quadrature method with dimension-independent error estimates and demonstrates its effectiveness through various numerical examples.

Contribution

It provides new theoretical error and cardinality estimates for anisotropic sparse grid quadrature, enhancing understanding and application in high-dimensional problems.

Findings

Dimension-independent error versus cost estimate.

Improved estimate for the cardinality of the index set.

Demonstrated rapid convergence in numerical examples.

Abstract

This article is dedicated to the anisotropic sparse grid quadrature for functions which are analytically extendable into an anisotropic tensor product domain. Taking into account this anisotropy, we end up with a dimension independent error versus cost estimate of the proposed quadrature. In addition, we provide a novel and improved estimate for the cardinality of the underlying anisotropic index set. To validate the theoretical findings, we present several examples ranging from simple quadrature problems to diffusion problems on random domains. These examples demonstrate the remarkable convergence behaviour of the anisotropic sparse grid quadrature in applications.