A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems

JOURNAL OF COMPUTATIONAL PHYSICS

卷:341

Article

Fairbanks, Hillary R.¹  Doostan, Alireza²  Ketelsen, Christian³  Laccarino, Gianluca⁴ 

[1] Univ Colorado, Appl Math & Stat Dept, Boulder, CO 80309 USA

[2] Univ Colorado, Dept Aerosp Engn Sci, Boulder, CO 80309 USA

[3] Univ Colorado, Dept Comp Sci, Boulder, CO 80309 USA

[4] Stanford Univ, Dept Mech Engn, Stanford, CA 94305 USA

关键词: Uncertainty quantification;    Stochastic PDEs;    Multilevel Monte Carlo;    Control variate;    Low-rank approximation;    Multifidelity;    Interpolative decomposition;   

DOI  :  10.1016/j.jcp.2017.03.060

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free