Annals of Mathematical Sciences and Applications

Volume 8 (2023)

Number 3

Special Issue Dedicated to the Memory of Professor Roland Glowinski

Guest Editors: Annalisa Quaini, Xiaolong Qin, Xuecheng Tai, and Enrique Zuazua

A fast multilevel dimension iteration algorithm for high dimensional numerical integration

Pages: 427 – 460



Xiaobing Feng (Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)

Huicong Zhong (School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi, China)


In this paper we propose and study a fast multilevel dimension iteration (MDI) algorithm for computing arbitrary $d$-dimensional integrals based on the tensor product approximations. It reduces the computational complexity (in terms of the CPU time) of a tensor product method from the exponential order $O(N^d)$ to the polynomial order $O(d^3 N^2)$ or better, where $N$ stands for the number of quadrature points in each coordinate direction. As a result, the proposed MDI algorithm effectively circumvents the curse of the dimensionality of tensor product methods for high dimensional numerical integration. The main idea of the proposed MDI algorithm is to compute the function evaluations at all integration points in cluster and iteratively along each coordinate direction, so lots of computations for function evaluations can be reused in each iteration. This idea is also applicable to any quadrature rule whose integration points have a lattice-like structure.


multilevel dimension iteration (MDI), high dimensional integration, numerical quadrature rules, tensor product methods, Monte Carlo methods

2010 Mathematics Subject Classification

, 65Dxx. Primary 65D30. Secondary 65C05, 65N99.

Dedicated to the memory of Professor Roland Glowinski

The work of Xiaobing Feng was partially supported by NSF grants DMS-2012414 and DMS-2309626.

Received 22 May 2023

Accepted 26 July 2023

Published 14 November 2023