Numerical Optimal Control for Problems with Random Forced SPDE Constraints

A numerical algorithm for solving optimization problems with stochastic diffusion equation as a constraint is proposed. First, separation of random and deterministic variables is done via Karhunen-Loeve expansion. Then, the problem is discretized, in spatial part, using the finite element method and the polynomial chaos expansion in the stochastic part of the problem. This process leads to the optimal control problem with a large scale system in its constraint. To overcome these difficulties the adjoint technique for derivative computation to implementation of the optimal control issue in preconditioned Newton’s conjugate gradient method is used. By some numerical simulation, it is shown that this hybrid approach is efficient and simple to implement.

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