Core Concepts

The authors present a combination technique (CT) to efficiently solve optimal control problems constrained by random partial differential equations. The CT combines solutions computed on coarse spatial grids and with few quadrature points to obtain an accurate approximation, while drastically reducing the computational cost compared to standard approaches.

Abstract

The content presents a new framework to discretize and solve optimal control problems (OCPs) constrained by random partial differential equations (PDEs). The authors consider the following optimization problem:
min_u E[F(y(ζ))] + ν/2 ||u||^2_U
s.t. <e(y(ζ), ζ), v> = <φ + Bu, v> ∀v ∈ V, ρ-a.e. ζ ∈ Γ
where y(ζ) ∈ V solves the random PDE constraint, F is a convex quantity of interest, and u ∈ U is the control variable.
The authors propose a Combination Technique (CT) to efficiently solve this problem. The CT relies on a hierarchical representation of the optimal control specified by a set of multi-indices (α, β). Each multi-index corresponds to a level of discretization in the spatial variables and the stochastic parameters ζ.
The key aspects of the CT are:
It solves the OCP on several coarse tensor product grids and with few quadrature points, and then linearly combines the computed solutions.
Under suitable regularity assumptions, the CT can achieve the same accuracy as the full tensor product discretization, but at a greatly reduced computational cost.
The CT avoids the issues of sparse grids and multilevel Monte Carlo methods, as it only uses positive quadrature weights, preserving the convexity of the discretized OCP.
The authors propose an a-priori construction of the multi-index set based on a "profit" metric that balances the spatial and stochastic error contributions.
A theoretical complexity analysis shows that the asymptotic complexity of the CT depends only on the deterministic PDE solver, as in the MISC method for forward UQ problems.
Numerical experiments validate the effectiveness of the CT approach.

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Key Insights Distilled From

by Fabio Nobile... at **arxiv.org** 03-29-2024

Deeper Inquiries

The CT framework can be extended to handle infinite-dimensional random variables by adapting the hierarchical representation of the optimal control to the infinite-dimensional setting. In this case, the multi-indices would correspond to discretization levels in an infinite-dimensional space. The key challenge lies in ensuring that the error contributions and work contributions associated with the hierarchical surpluses are well-defined and converge appropriately in the infinite-dimensional space. This may require specialized techniques to handle the infinite-dimensional nature of the problem, such as functional analysis and operator theory. By carefully formulating the error and work contributions in the infinite-dimensional context, the CT framework can be effectively extended to handle infinite-dimensional random variables.

Deriving suitable a-posteriori error estimators for the CT approximation of OCP constrained by random PDEs poses several challenges. One of the main challenges is the complexity of the error estimation process, especially in high-dimensional and stochastic settings. The error estimators need to accurately capture the discrepancies between the CT approximation and the true solution while considering the spatial and stochastic discretizations. Additionally, ensuring the reliability and efficiency of the error estimators requires a deep understanding of the underlying mathematical principles and the interplay between the spatial and stochastic components of the problem. Developing robust and computationally efficient error estimators for the CT approximation of OCPs constrained by random PDEs is an active area of research that requires careful consideration and validation.

Yes, the CT approach can be combined with other multilevel/multi-index quadrature rules beyond stochastic collocation, such as multilevel/multi-index (Quasi-)Monte Carlo methods. By integrating these different quadrature rules into the CT framework, it is possible to enhance the computational efficiency and accuracy of the approximation. The combination of CT with multilevel/multi-index (Quasi-)Monte Carlo methods can provide a more flexible and adaptive approach to solving optimal control problems constrained by random PDEs. This integration allows for the exploitation of the strengths of each method, leading to improved performance in handling high-dimensional and stochastic problems. The key lies in carefully designing the combination strategy and ensuring the compatibility of the different quadrature rules within the CT framework.

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