An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

Huicong Zhong and Xiaobing Feng ()
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Huicong Zhong: School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, China
Xiaobing Feng: Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA

Mathematics, 2023, vol. 11, issue 19, 1-26

Abstract: This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d -dimensional integral of a given function. The new algorithm, called the MDI-SG (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 ( d 3 N b ) ( b ≤ 2 ) 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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