Repository: Freie Universit├Ąt Berlin, Math Department

A quasi-Monte Carlo method for an optimal control problem under uncertainty

Guth, Philipp A. and Kaarnioja, Vesa and Kuo, Frances Y. and Schillings, Claudia and Sloan, Ian H. (2019) A quasi-Monte Carlo method for an optimal control problem under uncertainty. ArXiv .

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Official URL: https://doi.org/10.48550/arXiv.1910.10022

Abstract

We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are non-convex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Deterministic and Stochastic PDEs Group
ID Code:2986
Deposited By: Ulrike Eickers
Deposited On:26 May 2023 08:09
Last Modified:26 May 2023 08:09

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