Reduced Basis Methods for Optimal Control Problems with Random Inputs in Environmental Science

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.