Optimal control under uncertainty with joint chance state constraints: almost-everywhere bounds, variance reduction, and application to (bi-)linear elliptic PDEs

TL;DR

This work develops a framework for optimal control of elliptic PDEs under uncertainty with joint chance-state constraints. It leverages the spherical-radial decomposition (SRD) to represent elliptical uncertainty, enabling substantial variance reduction in probability estimation and providing explicit gradient expressions for the probability function, which accelerates optimization. The authors establish existence and convexity results for the joint-constraint setting, demonstrate innate dimension reduction of the random space via state-driven KL-type expansions, and validate the approach through comprehensive numerical experiments on linear and bilinear PDEs, showing that SRD-based sampling (especially SRD-QMC) achieves major efficiency gains over standard Monte Carlo. The results illuminate both theoretical and practical aspects of integrating SRD into PDE-constrained optimization under uncertainty, with clear guidance on discretization, gradient computation, and algorithmic choices. The practical impact lies in faster, more reliable solutions to risk-aware PDE control problems in engineering and physics where uniform state-constraints must be satisfied with high probability.

Abstract

We study optimal control of PDEs under uncertainty with the state variable subject to joint chance constraints. The controls are deterministic, but the states are probabilistic due to random variables in the governing equation. Joint chance constraints ensure that the random state variable meets pointwise bounds with high probability. For linear governing PDEs and elliptically distributed random parameters, we prove existence and uniqueness results for almost-everywhere state bounds. Using the spherical-radial decomposition (SRD) of the uncertain variable, we prove that when the probability is very large or small, the resulting Monte Carlo estimator for the chance constraint probability exhibits substantially reduced variance compared to the standard Monte Carlo estimator. We further illustrate how the SRD can be leveraged to efficiently compute derivatives of the probability function, and discuss different expansions of the uncertain variable in the governing equation. Numerical examples for linear and bilinear PDEs compare the performance of Monte Carlo and quasi-Monte Carlo sampling methods, examining probability estimation convergence as the number of samples increases. We also study how the accuracy of the probabilities depends on the truncation of the random variable expansion, and numerically illustrate the variance reduction of the SRD.

Figures (6)

• Figure 1: The left figure shows samples from the distribution of the random boundary data $\xi(\omega)$. The middle figure shows samples from the distribution of $y(\omega)-y_{0}^u$, the state with zero mean. The right figure shows the normalized (i.e., $\lambda_1$ is scaled to $1$) eigenvalue factors in the KL expansion \ref{['eq:kl-xi']} of $\xi$ (green, solid), and in the KL expansion \ref{['eq:kl-xi']} of $y$ (dark pink, dashed). \newlabelfig:C00
• Figure 1: Optimal controls $u$ for $p=0.82, 0.84, 0.86$ (from left to right) for bilinear example. Note that the optimal controls develop spikes near the points where the state distribution is close to the upper bound.
• Figure 2: \newlabelfig:chance-conv0 Root mean squared error (RMSE) of probability estimation is shown on $y$-axis for different sampling methods and different number $N$ of MC samples ($x$-axis). The left plot is for $\underline y\equiv -0.3$, $\bar{y}\equiv 0.3$, corresponding to the probability $p\approx 0.6496$. The right plot uses the wider bounds $\underline y\equiv -0.7$, $\bar{y}\equiv 0.7$, resulting in $p\approx 0.9848$.
• Figure 2: Left: Value of objective $\mathcal{J}$ at the optimal control $u$ as $p$ in the chance constraint is increased. Right: Mean of state corresponding to optimal control for $p=0.84$ shown in the middle in \ref{['fig:bilinear:u']}.
• Figure 3: Left: QMC sampling with SRD for different number $K$ of KL modes. Shown is the RMSE of the probability estimate for the same setup also used on the left of \ref{['fig:chance-conv']}, i.e., with $\underline y\equiv -0.3$, $\bar{y}\equiv 0.3$. Right: Variance normalized by $(1-p)$ using standard and SRD MC samples for the problem described in \ref{['sec:example-linear']}. The difference in $p$ is due to using different bounds $\bar{y} = -\underline y\equiv 0.5,0.6,0.7,0.8,0.9$ (from left to right, going from more to less restrictive). The variance is estimated using 100 MC simulations, each using $N=500$ samples. \newlabelfig:chance-conv-KL0

Theorems & Definitions (21)

• Example 1.1: Tracking-type objective governed by linear PDE with uncertain Neumann data
• Example 1.2: Bilinear control with uncertain right hand side
• Theorem 2.1
• Proof 1
• Theorem 2.2
• Proof 2
• Example 3.1
• Lemma 3.2
• Proof 3
• Lemma 3.3