Quasi-Monte Carlo integration for feedback control under uncertainty

作     者：Guth, Philipp A. Kritzer, Peter Kunisch, Karl 

作者机构：Johann Radon Institute for Computational and Applied Mathematics ÖAW Altenbergerstrasse 69 Linz4040 Austria Institute of Mathematics and Scientific Computing Karl-Franzens University Graz Heinrichstrasse 36 Graz8010 Austria 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Stochastic systems 

摘      要：A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of uncertain parameters. It is shown that the Riccati-based feedback operator depends analytically on the parameters provided that the system operator depends analytically on the parameters, as is the case, for instance, in diffusion problems when the diffusion coefficient is parameterized by a Karhunen–Loève expansion. These novel parametric regularity results allow the application of quasi-Monte Carlo (QMC) methods to efficiently compute an a-priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input random field, QMC methods achieve superior error rates compared to ordinary Monte Carlo methods, independently of the stochastic dimension of the problem. Indeed, our paper for the first time studies Banach-space-valued integration by higher-order QMC *** Codes 35R60, 49N10, 65D30, 65D32, 93B52 © 2024, CC BY.