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An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration
Version 1
: Received: 21 August 2023 / Approved: 22 August 2023 / Online: 23 August 2023 (07:16:18 CEST)
A peer-reviewed article of this Preprint also exists.
Zhong, H.; Feng, X. An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration. Mathematics 2023, 11, 4191. Zhong, H.; Feng, X. An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration. Mathematics 2023, 11, 4191.
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, nor does it compute the function values independently at each integration point, instead, it reuses 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.
Keywords
Sparse grid (SG) method; multilevel dimension iteration (MDI); high-dimensional integration; numerical quadrature rules; curse of dimensionality
Subject
Computer Science and Mathematics, Computational Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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