A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences

TL;DR

This paper develops a weighted Proper Orthogonal Decomposition method for reduced basis approximations of PDE-constrained optimal control problems with random inputs, enabling efficient and general solutions applicable to environmental science applications.

Contribution

It introduces a weighted POD approach for parametrized PDE-constrained OCPs with random inputs, extending the adjoint method to non-linear and constrained cases, and explores the impact of aggregation steps.

Findings

The weighted POD method effectively approximates solutions across random parameter spaces.

Skipping the aggregation step reduces computational costs for noncoercive problems.

Numerical tests demonstrate the method's applicability to environmental science scenarios.

Abstract

Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.