Repository: Freie Universit├Ąt Berlin, Math Department

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

Guth, Philipp A. and Kaarnioja, Vesa and Kuo, Frances Y. and Sloan, Ian H. (2022) Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration. arXiv preprint arXiv:2208.02767 . (Submitted)

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Official URL: https://doi.org/10.48550/arXiv.2208.02767

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.

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:2983
Deposited By: Ulrike Eickers
Deposited On:22 May 2023 12:38
Last Modified:22 May 2023 12:38

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