A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Abstract

Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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

“…The following existence and uniqueness result of the solution y to (13) follows from the Lax-Milgram lemma; see, e.g., [12]. Theorem 1.…”

“…Let (14) and (15) be satisfied and let α = 0 in (8). Then there exists a unique optimal solution (y, u, f ) to the SOCP (8) and (13) satisfying the stochastic optimality conditions [24].…”

“…To discretize the SOCP given by (2) and (13) using the SGFEM, consider first the constraint (13). Given a basis for (4)) of the random field a satisfying (15), we now seek a finite-dimensional (19) ∀v ∈ W hn .…”

“…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…”

Source identification in acoustics and structural mechanics using Sierra/SD.

“…The source localization challenge is often cast within the mathematical framework of an inverse problem in which we are tasked with identifying a hidden driving force given the measured response of a system. Important examples of this class of problems include earthquake source localization [1,2], damage or defect identification from acoustic emission [3,4,5], odor or contaminant localization [6], acoustics [14], and source identification in electromagnetics [7], among others. In spite of the different physics involved in the latter examples, their mathematical structures share many common features, allowing us to develop methods that are applicable to a wide range of problems.…”