A Stochastic Gradient Method with Mesh Refinement for PDE Constrained Optimization under Uncertainty

摘要: Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization a convex and smooth tracking-type functional subject to linear partial differential equation with coefficients box constraints. The approach take is based stochastic approximation where, place true gradient, gradient chosen using one sample from known probability distribution. Feasibility maintained by performing projection at each iteration. application method optimization under uncertainty, new challenges arise. We observe discretization error made approximating finite elements. Analyzing interplay between PDE error, develop mesh refinement strategy coupled decreasing step sizes. Additionally, for modified algorithm iterate averaging larger effectiveness demonstrated numerically different field choices.