Multilevel Monte Carlo Analysis for Optimal Control of Elliptic PDEs with Random Coefficients
Abstract
This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control problem constrained by a diffusion equation with random coefficient together with box constraints for the control. For each realization of the diffusion coefficient we solve an optimal control problem using the variational discretization [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45--61]. Our framework allows for lognormal coefficients whose realizations are not uniformly bounded away from zero and infinity. We establish finite element error bounds for the pathwise optimal controls. This analysis is nontrivial due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the diffusion operator. We apply the error bounds to prove convergence of a multilevel Monte Carlo estimator for the expected value of the pathwise optimal controls. In addition we analyze the computational complexity of the multilevel estimator. We perform numerical experiments in two-dimensional space to confirm the convergence result and the complexity bound.
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Information & Authors
Information
Published In
SIAM/ASA Journal on Uncertainty Quantification
Pages: 466 - 492
DOI: 10.1137/16M109870X
ISSN (online): 2166-2525
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 13 October 2016
Accepted: 17 January 2017
Published online: 27 April 2017
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