Paper Info
Title
A Quasi-Monte Carlo Method For Optimal Control Under Uncertainty
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 nonconvex 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.
Year
DOI
Venue
2021
10.1137/19M1294952
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
Keywords
DocType
Volume
optimal control, uncertainty quantification, quasi-Monte Carlo method, PDE-constrained optimization with uncertain coefficients, optimization under uncertainty
Journal
9
Issue
ISSN
Citations 
2
2166-2525
0
PageRank 
References 
Authors
0.34
0
5
Name
Order
Citations
PageRank
Philipp A. Guth100.34
Vesa Kaarnioja200.68
Frances Y. Kuo300.34
Claudia Schillings400.34
Ian H. Sloan51180183.02