A fast multilevel dimension iteration algorithm for high dimensional numerical integration

TL;DR

This paper introduces a multilevel dimension iteration algorithm that significantly reduces the computational complexity of high-dimensional numerical integration, making it feasible to perform integrations in many dimensions efficiently.

Contribution

The paper presents a novel multilevel iteration algorithm that reduces complexity from exponential to polynomial order for tensor product-based high-dimensional integrals.

Findings

Reduces computational complexity from O(N^d) to O(d^3N^2) or better.

Effectively circumvents the curse of dimensionality in high-dimensional integration.

Applicable to any quadrature rule with lattice-like integration points.

Abstract

In this paper, we propose and study a fast multilevel dimension iteration (MDI) algorithm for computing arbitrary $d$-dimensional integrals based on 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 {\color{black} $O(d^3N^2)$ 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 the 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.