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

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

TL;DR

The high-dimensional integrals are computed numerically using specially designed QMC methods and 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.

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.

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

TL;DR

The high-dimensional integrals are computed numerically using specially designed QMC methods and 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.

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.
Paper Structure (20 sections, 25 theorems, 165 equations, 6 figures, 2 algorithms)

This paper contains 20 sections, 25 theorems, 165 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Function space setting
  4. Variational formulation
  5. Dual problem
  6. Parabolic optimal control problems under uncertainty with control constraints
  7. Linear risk measures, including the expected value
  8. The entropic risk measure
  9. Optimality conditions
  10. Parametric regularity of the adjoint state
  11. Regularity analysis for the entropic risk measure
  12. Error analysis
  13. Truncation error
  14. Quasi-Monte Carlo error
  15. Combined error bound
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $f=(z,u_0)\in \mathcal{Y}'$. For all ${\boldsymbol{\nu}}\in\mathscr{F}$ and all ${\boldsymbol{y}}\in U$, we have where $\beta_1$ is as described in eq:beta and the sequence ${\boldsymbol{b}} = (b_j)_{j\ge 1}$ is defined by

Figures (6)

  • Figure 1: The approximate dimension truncation errors corresponding to the state and adjoint PDEs.
  • Figure 2: The approximate dimension truncation errors corresponding to $\|S_{s'}-S_s\|_{L^2(V;I)}$ and $|T_{s'}-T_s|$.
  • Figure 3: Left: The approximate root-mean-square error for QMC approximation of the integrals $\int_{U_s}u_s^{\boldsymbol{y}}\,{\rm d}{\boldsymbol{y}}$ and $\int_{U_s}q_s^{\boldsymbol{y}}\,{\rm d}{\boldsymbol{y}}$. Right: The approximate root-mean-square error for QMC approximation of quantities $S_s$ and $T_s$. All computations were carried out using $R=16$ random shifts, $n=2^m$, $m\in\{4,\ldots,15\}$, and dimension $s=100$.
  • Figure 4: The inverse Riesz transform $R_V^{-1}z^*$ of the reconstructed optimal control $z^*$ using the entropic risk measure for several values of $t$ in the constrained setting.
  • Figure 5: Left: the inverse Riesz transform of the control at time $t=1$ in the constrained setting after 25 iterations of the projected gradient descent algorithm using the entropic risk measure. Right: The difference between the reconstruction obtained in the constrained setting and the corresponding solution in the unconstrained setting.
  • ...and 1 more figures

Theorems & Definitions (53)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • ...and 43 more