NDLI: Multilevel Monte Carlo Analysis for Optimal Control of Elliptic PDEs with Random Coefficients

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ullmann, Elisabeth Ali, Ahmad Ahmad Hinze, Michael
Copyright Year	2017
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.
Starting Page	466
Ending Page	492
Page Count	27
File Format	PDF
DOI	10.1137/16M109870X
e-ISSN	21662525
Journal	SIAM/ASA Journal on Uncertainty Quantification (SJUQA3)
Issue Number	1
Volume Number	5
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-04-27
Access Restriction	Subscribed
Subject Keyword	uncertainty quantification control constraints variational discretization Monte Carlo methods PDE-constrained optimization Optimal control problems involving partial differential equations lognormal random fields Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Computational methods for stochastic equations Partial differential equations with randomness, stochastic partial differential equations
Content Type	Text
Resource Type	Article

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