Paper Info
Title
A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems.
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.
Year
DOI
Venue
2017
10.1016/j.jcp.2017.03.060
Journal of Computational Physics
Keywords
Field
DocType
Uncertainty quantification,Stochastic PDEs,Multilevel Monte Carlo,Control variate,Low-rank approximation,Multifidelity,Interpolative decomposition
Monte Carlo method,Mathematical optimization,Interpolative decomposition,Uncertainty quantification,Control variates,Cost estimate,Low-rank approximation,Variance reduction,Mathematics,Grid
Journal
Volume
ISSN
Citations 
341
0021-9991
2
PageRank 
References 
Authors
0.40
10
4
Name
Order
Citations
PageRank
Hillary R. Fairbanks130.77
Alireza Doostan218815.57
C. Ketelsen3203.04
Gianluca Iaccarino422923.37