A Multilevel Approach towards Unbiased Sampling of Random Elliptic Partial Differential Equations

TL;DR

This paper introduces a novel Monte Carlo method combining multilevel and randomization techniques to produce unbiased estimators for expectations of solutions to random elliptic PDEs, accounting for measurement errors.

Contribution

It develops a multilevel, randomized Monte Carlo scheme that achieves unbiasedness with finite variance and computational cost for elliptic PDEs with uncertainties.

Findings

Unbiased estimators with finite variance and cost

Effective integration of multilevel and randomization methods

Applicability to PDEs with measurement errors

Abstract

Partial differential equation is a powerful tool to characterize various physics 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 multilevel Monte Carlo [Giles, 2008] and a randomization scheme proposed by [Rhee and Glynn, 2012, Rhee and Glynn, 2013]. Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher order approximations.