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A multilevel approach towards unbiased sampling of random elliptic partial differential equations

Part of: Equilibrium statistical mechanics Probabilistic methods, simulation and stochastic differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  29 November 2018

Xiaoou Li ,
Jingchen Liu  and
Shun Xu
Xiaoou Li*
Affiliation:
University of Minnesota
Jingchen Liu*
Affiliation:
Columbia University
Shun Xu*
Affiliation:
Columbia University
*
* Postal address: School of Statistics, University of Minnesota, 224 Church Street SE, Minneapolis, MN 55455, USA. Email address: xiaoou@stat.columbia.edu
** Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, 10027, USA.
** Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, 10027, USA.
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Abstract

Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.

Keywords

Unbiased samplingpartial differential equations with random coefficientsMonte Carlo method

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations 82B80: Numerical methods (Monte Carlo, series resummation, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1007 - 1031
Copyright
Copyright © Applied Probability Trust 2018 

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