A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions

TL;DR

This paper investigates a hybrid numerical integration method combining quasi-Monte Carlo and Monte Carlo rules for high-dimensional problems, showing near-optimal error rates when the dimension for QMC scales linearly with the number of points.

Contribution

It introduces a concatenated quadrature rule that effectively combines QMC and Monte Carlo methods for high-dimensional integration, leveraging weight decay.

Findings

Achieves almost optimal mean squared worst-case error rates

Effective for extremely high-dimensional integrals

Applicable to PDEs with random coefficients

Abstract

In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the $s$-dimensional unit cube by using rank-1 lattice point sets only for the first $d\, (<s)$ coordinates and random points for the remaining $s-d$ coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as long as $d$ scales at most linearly with the number of points. This result might be useful for numerical integration in extremely high dimensions, such as partial differential equations with random coefficients for which even the standard fast component-by-component algorithm is considered computationally expensive.