A Low-rank Control Variate for Multilevel Monte Carlo Simulation of High-dimensional Uncertain Systems

TL;DR

This paper introduces a multilevel control variate method that leverages low-rank approximations to improve the efficiency of multilevel Monte Carlo simulations for high-dimensional uncertain systems.

Contribution

It extends MLMC by incorporating low-rank control variates based on coarser grid solutions, reducing computational cost in high-dimensional problems.

Findings

Significant variance reduction demonstrated in numerical examples.

Cost savings over standard MLMC when solutions admit low-rank structure.

Effective for high-dimensional systems with rapidly increasing simulation costs.

Abstract

Multilevel Monte Carlo (MLMC) is a recently proposed variation of Monte Carlo (MC) simulation that achieves variance reduction by simulating the governing equations on a series of spatial (or temporal) grids with increasing resolution. Instead of directly employing the fine grid solutions, MLMC estimates the expectation of the quantity of interest from the coarsest grid solutions as well as differences between each two consecutive grid solutions. When the differences corresponding to finer grids become smaller, hence less variable, fewer MC realizations of finer grid solutions are needed to compute the difference expectations, thus leading to a reduction in the overall work. This paper presents an extension of MLMC, referred to as multilevel control variates (MLCV), where a low-rank approximation to the solution on each grid, obtained primarily based on coarser grid solutions, is used as a control variate for estimating the expectations involved in MLMC. Cost estimates as well as numerical examples are presented to demonstrate the advantage of this new MLCV approach over the standard MLMC when the solution of interest admits a low-rank approximation and the cost of simulating finer grids grows fast.