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

• Published in: Numerische Mathematik Vol. 156; no. 2; pp. 565 - 608
• Main Authors: Guth, Philipp A., Kaarnioja, Vesa, Kuo, Frances Y., Schillings, Claudia, Sloan, Ian H.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024; Springer Nature B.V
• Subjects: Constraints; Diffusion coefficient; Fields (mathematics); Integrals; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Monte Carlo simulation; Numerical Analysis; Numerical and Computational Physics; Numerical methods; Optimal control; Parabolic differential equations; Partial differential equations; Risk; Simulation; Stochastic processes; Theoretical; Thermal diffusion; Uncertainty; 65D30; 49M41; 35R60; 65D32
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-024-01397-9

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.