A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences

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