Many mathematical and computational analyses have been performed for deterministic partial differential equations (PDEs) that have perfectly known input data. However, in reality, many physical and engineering problems involve some level of uncertainty in their input, e.g., unknown properties of the material, the lack of information on boundary data, etc. One effective and realistic means for modeling such uncertainty is through stochastic partial differential equations (SPDEs) using randomness for uncertainty. In fact, SPDEs are known to be effective tools for modeling complex physical and engineering phenomena. In this talk, we propose and analyze some optimal control problems for partial differential equations with random coefficients and forcing terms.

These input data are assumed to be dependent on a finite number of random variables. We set up three different kind of problems and prove existence of optimal solution and derive an optimality system. In the method, we use a Galerkin approximation in physical space and a sparse grid collocation in the probability space. We provide a comparison of these three cases for fully discrete solution using an appropriate norm and analyze the computational efficiency.