Optimal control under uncertainty with joint chance state constraints: almost-everywhere bounds, variance reduction, and application to (bi-)linear elliptic PDEs (2412.05125v2)

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.

• The paper presents a new theoretical framework with almost-everywhere bounds ensuring existence and uniqueness of solutions under joint chance state constraints.
• It employs spherical-radial decomposition combined with quasi-Monte Carlo sampling to achieve significant variance reduction and computational efficiency.
• Numerical experiments confirm substantial sample savings and accurate probability estimations for applications in linear and bilinear elliptic PDEs.

An Analysis of Optimal Control under Uncertainty with Joint Chance State Constraints

The paper addresses a complex problem in the domain of optimal control of partial differential equations (PDEs) under uncertainty, with a focus on joint chance state constraints. The authors, Henrion, Stadler, and Wechsung, explore the intricacies of controlling systems where the state variables are influenced by probabilistic elements due to random inputs embedded within the governing PDEs.

Theoretical Contributions

The authors provide a comprehensive theoretical framework to handle joint chance state constraints, which mandate that state distributions must adhere to specified bounds with certain probability levels. A significant contribution of the paper is the introduction of almost-everywhere bounds for the states, along with proving the existence and uniqueness of solutions for such control problems. The results extend known theorems by relaxing certain conditions, accommodating more general situations commonly encountered in real-world applications.

Methodological Advancements

The paper leverages the spherical-radial decomposition (SRD) methodology to improve the computational efficiency of solving chance-constrained problems. Notably, SRD facilitates variance reduction in the estimation of probability functions, a crucial step for numerical implementations of chance constraints. Numerical experiments demonstrate that SRD offers substantial gains in computational efficiency, particularly when used in conjunction with quasi-Monte Carlo (QMC) sampling. This approach requires significantly fewer samples compared to standard Monte Carlo (MC) methods, yielding faster computations for ensuring high accuracy in probability estimations.

Numerical Findings

The authors conduct extensive numerical experiments to assess the performance of the proposed SRD approach for linear and bilinear elliptic PDEs. Results indicate that the SRD-based QMC method often outperforms the traditional MC sampling, especially in scenarios with high or small probability constraints, confirming theoretical variance reduction findings. The experiments underscore the computational savings achieved, validating the theoretical claims regarding reduced sample complexity and improved numerical accuracy.

Practical Implications and Limitations

The methodologies proposed hold practical significance for systems governed by elliptic PDEs with uncertain parameters, including applications in fields such as aeronautics and gas transport optimization. The ability to handle high-dimensional random parameter spaces with joint constraints aligns well with the risk-averse requirements of many engineering applications.

The paper highlights some limitations of the SRD method, particularly in non-linear contexts where the control-to-state map loses linearity. The increased complexity of SRD in such settings may offset its variance reduction advantages due to computational overheads. The authors note that even though the SRD approach is inherently robust for linear cases, future extensions to non-linear problems require further refinement.

Future Directions

Speculating on future developments, the integration of SRD in non-linear paradigms and its application to broader classes of PDEs presents a potential avenue for research. Moreover, exploring adaptive strategies for optimizing the SRD-based computational framework could enhance the capability of handling even larger-scale problems with complex constraints. Continued advancements in variance reduction techniques may further bridge the gap between theoretical optimality and practical computational feasibility.

In conclusion, the paper significantly contributes to the field by addressing the complex interplay of uncertainty and control in PDEs, presenting both advancements in theoretical underpinnings and improvements in computational strategies. The insights offered lay a foundation for further exploration into robust optimization methodologies applicable in diverse scientific domains.