Numerical Solution of Constrained Optimization Problems Governed by Partial Differential Equations with Uncertain Parameters

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This project will provide new mathematical algorithms and theoretical analyses for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain parameters. These problems arise in many science and engineering decision making applications, where a decision has to be made before the realization of uncertain inputs can be observed that will impact the outcome of the decision. The uncertainty can be incorporated into the optimization formulation using so-called risk measures, which typically involve the expected value of the quantity of interest and a measure of its deviation from the expected value. In principle, these formulations allow one to compute decisions that balance maximization of their expected outcome and minimization of the risk due to uncertainty. However, the numerical solution of these problems presents many theoretical and algorithmic challenges. For example, the numerical solution requires some sort of sampling of the random inputs, which can make these PDE constrained optimization problems extremely expensive to solve. To address several of the above mentioned challenges, this research will provide theoretical analyses of the well-posedness and of optimality conditions for a class of semilinear elliptic PDE constrained optimization problems, and it will derive discretization error bounds for sparse grid and quasi Monte Carlo discretizations applied to PDE constrained optimization. Furthermore, it will develop and analyze adaptive methods which reduce the total number of samples needed, or incorporate reduced order models. This research is at the interface between stochastic programming, deterministic PDE constrained optimization, and solution of PDEs with random inputs, and it will make algorithmic and theoretical contributions to these areas. The application of theories and numerical methods to example problems will serve as a model for other researchers and decision makers, and will lead to more efficient algorithms for important classes of decision making under uncertainty. Results will be disseminated through publication of algorithms and results. In addition, the results of the project will be used in regularly offered courses on the theory and applications of optimization as well as in special courses on PDE constrained optimization under uncertainty aiming at students in both mathematics and engineering.