No abstract

<p>We study methods that aim to reduce the dimension of a finite dimensional solution space, in which the solution corresponding to a certain parametrized Optimal Control Problems governed by environmental models, e.g. Quasi-Geostrophic flow, is sought. The parameter is modeled as a random variable to incorporate possible uncertainty, for example in parametric measurements. For such a reduction to be useful, it should be guaranteed, for every possible parameter value, that it results in an acceleration of the solution process while maintaining an accurate approximate solution. In order to do this, conditions are formulated, and under those conditions, several versions of a specific reduction method known as Proper Orthogonal Decomposition are implemented. We consider examples and show that a simplification of the general state of the art reduction method performs equally well.</p>

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.