Work

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

Year: 2024

Type: article

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.

Source: Numerische Mathematik

DOI: 10.1007/s00211-024-01397-9

Authors: Philipp A. Guth, Vesa Kaarnioja, Frances Y. Kuo, Claudia Schillings, Ian H. Sloan

Institutions: Johann Radon Institute for Computational and Applied Mathematics, Freie Universität Berlin, UNSW Sydney

Language: English

Topic: Mathematical Approximation and Integration

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Deutsche Forschungsgemeinschaft, Freie Universität Berlin, Australian Research Council