Sparse Quadrature for High-Dimensional Integration with Gaussian Measure

TL;DR

This paper demonstrates that sparse quadrature methods can achieve dimension-independent convergence rates for high-dimensional Gaussian integrals, and proposes practical schemes with numerical validation.

Contribution

The paper provides a theoretical analysis of dimension-independent convergence for sparse quadrature and introduces practical a-priori and a-posteriori construction schemes.

Findings

Convergence rate of $O(N^{-s})$ independent of parameter dimensions

Practical sparse quadrature schemes with demonstrated convergence

Numerical experiments confirming theoretical results

Abstract

In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions of the exactness and the boundedness of univariate quadrature rules as well as the regularity of the parametric functions with respect to the parameters, we obtain the convergence rate $O(N^{-s})$, where $N$ is the number of indices, and $s$ is independent of the number of the parameter dimensions. Moreover, we propose both an a-priori and an a-posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates.