Optimization of PDEs with uncertain inputs (Q2419399)

In the work the authors review a set of stochastic programming tools for formulating and solving optimization problems constrained by PDEs with uncertain coefficients. At first, they provide a general stochastic problem formulation and, under certain assumptions, show the existence of minimizers as well as first-order necessary optimality conditions. They demonstrate these results on linear control problems constrained by steady (i.e., stationary) PDEs. Then the authors discuss specific problem formulations including risk measures, probabilistic functions, and distributionally robust optimization. In addition to problem formulation, the challenges associated with the numerical solution of such problems are discussed. The authors consider three classical approaches for approximating and solving stochastic optimization problems: stochastic approximation, sample average and quadrature approximation, and the progressive hedging algorithm. They briefly discuss convergence of these methods and conclude with a numerical demonstration. For the entire collection see [Zbl 1402.49004].