An efficient and fast sparse grid algorithm for high-dimensional numerical integration

TL;DR

This paper introduces the MDI-SG algorithm, a fast and efficient sparse grid method for high-dimensional numerical integration that reduces computational complexity and mitigates the curse of dimensionality.

Contribution

The paper presents the MDI-SG algorithm, which improves sparse grid integration by reusing function evaluations and reducing complexity from exponential to polynomial order.

Findings

Computational complexity is polynomial, O(Nd^3), for the MDI-SG method.

MDI-SG effectively reduces the curse of dimensionality in high-dimensional integration.

Numerical results confirm the efficiency and speed of the proposed algorithm.

Abstract

This paper is concerned with developing an efficient numerical algorithm for fast implementation of the sparse grid method for computing the $d$-dimensional integral of a given function. The new algorithm, called the MDI-SG ({\em multilevel dimension iteration sparse grid}) method, implements the sparse grid method based on a dimension iteration/reduction procedure, it does not need to store the integration points, neither does it compute the function values independently at each integration point, instead, it re-uses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order $O(Nd^3 )$ or better, compared to the exponential order $O(N(\log N)^{d-1})$ for the standard sparse grid method, where $N$ denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.