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Mathematics > Numerical Analysis
Title: Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration
(Submitted on 4 Aug 2022 (v1), last revised 27 Mar 2024 (this version, v2))
Abstract: We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem -- and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.
Submission history
From: Philipp A. Guth [view email][v1] Thu, 4 Aug 2022 16:58:02 GMT (2722kb)
[v2] Wed, 27 Mar 2024 13:38:31 GMT (5706kb,D)
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