Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients

TL;DR

This paper develops a multilevel Monte Carlo method to efficiently estimate the expected optimal controls in PDE-constrained problems with random coefficients, addressing challenges due to limited regularity and unbounded coefficients.

Contribution

It introduces a novel error analysis and convergence proof for multilevel Monte Carlo estimators applied to stochastic PDE optimal control problems with non-uniformly bounded coefficients.

Findings

Finite element error bounds for pathwise optimal controls.

Convergence of the multilevel Monte Carlo estimator.

Numerical validation of theoretical results.

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 2D space to confirm the convergence result and the complexity bound.