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Quasi-Monte Carlo integration for feedback control under uncertainty

Philipp A. Guth, Peter Kritzer, Karl Kunisch

TL;DR

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 under moderate assumptions on the input random field.

Abstract

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 methods.

Quasi-Monte Carlo integration for feedback control under uncertainty

TL;DR

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 under moderate assumptions on the input random field.

Abstract

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 methods.
Paper Structure (22 sections, 13 theorems, 160 equations)

This paper contains 22 sections, 13 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Contents
  3. Notation
  4. Optimality conditions of the parameterized linear-quadratic control problem
  5. Parametric regularity analysis
  6. Affine parameter dependence
  7. Uniform bounded invertibility of $A$ and $G$
  8. Parametric regularity of $A$ and $G$
  9. Parametric regularity of the optimal cost
  10. Analytic dependence of the feedback law on the parameters
  11. Homogeneous constraint
  12. Nonhomogeneous case
  13. QMC approximation of the feedback $K$
  14. Randomly shifted lattice rules for smoothness one
  15. Folded lattice rules for smoothness one
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let $\mathbb{G}({\boldsymbol{\sigma}})$ be a $p$-analytic family of operators. Then, for every $f \in Y'$ and every ${\boldsymbol{\sigma}} \in {\mathfrak S}$, there is a unique solution $y({\boldsymbol{\sigma}})\in X$ of the parameterized operator equation Moreover, the parametric solution $y({\boldsymbol{\sigma}})$ depends analytically on the parameters, with for all ${\boldsymbol{\nu}} \in {\m

Theorems & Definitions (22)

  • Definition 3.1
  • Theorem 3.1: kunoth2013analytic
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • ...and 12 more