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

Fairbanks, Hillary R.; Doostan, Alireza; Ketelsen, Christian; Iaccarino, Gianluca

doi:10.1016/j.jcp.2017.03.060

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

Journal Article · Fri Mar 31 00:00:00 EDT 2017 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2017.03.060· OSTI ID:1533956

Fairbanks, Hillary R. ^[1]; Doostan, Alireza ^[1]; Ketelsen, Christian ^[1]; Iaccarino, Gianluca ^[2]

University of Colorado, Boulder, CO (United States)
Stanford University, CA (United States)

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. Here 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.

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Research Organization:: Univ. of Colorado, Boulder, CO (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)

Grant/Contract Number:: SC0006402; CMMI-145460; PSAAP2

OSTI ID:: 1533956

Alternate ID(s):: OSTI ID: 1416591

Journal Information:: Journal of Computational Physics, Vol. 341, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 23 works

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97 MATHEMATICS AND COMPUTING
uncertainty quantification
stochastic PDEs
multilevel Monte Carlo
control variate
low-rank approximation
multifidelity
interpolative decomposition

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

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