Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Abstract

Abstract. The goal of this paper is the efficient numerical simulation of optimization problems governed by either steady-state or unsteady partial differential equations involving random coefficients. This class of problems often leads to prohibitively high dimensional saddle-point systems with tensor product structure, especially when discretized with the stochastic Galerkin finite element method. Here, we derive and analyze robust Schur complement-based block-diagonal preconditioners for solving the resulting stochastic optimality systems with all-at-once low-rank iterative solvers. Moreover, we illustrate the effectiveness of our solvers with numerical experiments.Key words. stochastic Galerkin system, iterative methods, PDE-constrained optimization, saddle-point system, low-rank solution, preconditioning, Schur complement AMS subject classifications. 35R60, 60H15, 60H35, 65N22, 65F10, 65F50 DOI. 10.1137/15M10185021. Introduction. Optimization problems constrained by deterministic steadystate partial differential equations (PDEs) are computationally challenging. This is even more so if the constraints are deterministic unsteady PDEs since one would then need to solve a system of PDEs coupled globally in time and space, and time-stepping methods quickly reach their limitations due to the enormous demand for storage [25]. Yet, more challenging than the aforementioned are problems constrained by unsteady PDEs involving (countably many) parametric or uncertain inputs. This class of problems often leads to prohibitively high dimensional linear systems with Kronecker product structure, especially when discretized with the stochastic Galerkin finite element method (SGFEM). Moreover, a typical model for an optimal control problem with stochastic inputs (SOCP) will usually be used for the quantification of the statistics of the system response-a task that could in turn result in additional enormous computational expense.Stochastic finite element-based solvers for a large range of PDEs with random data have been studied extensively [1,3,12,21,24]. However, optimization problems constrained by PDEs with random inputs have, in our opinion, not yet received adequate attention. Hence, this study is aimed at pushing the research frontier with respect to the numerical simulation of the latter class of stochastic problems (that is, SOCPs) toward larger and more challenging problems. Some of the papers on SOCPs include [12,13,24]. While [12] studies the existence and the uniqueness of solutions to control problems constrained by elliptic PDEs with random inputs, the emphasis in

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