Optimal control with stochastic PDE constraints and uncertain controls

Abstract

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby obviating the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented

THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

“…There have been also papers by other authors published on the subject of optimal control with SPDE constraints (e.g., see recent three publications [32,33,34] and references therein). In [32], the authors proved the uniqueness of the optimal solution to the stochastic saddle problem after showing that it is equivalent to their optimality system.…”

“…In [32], the authors proved the uniqueness of the optimal solution to the stochastic saddle problem after showing that it is equivalent to their optimality system. In [33], the computational solutions of optimal control problems constrained by SPDEs with uncertain controls were investigated, demonstrating the application of their methods via numerical examples. In the work [34], the authors examined the use of stochastic collocation for the numerical solution of optimal control problems subject to SPDEs, discussing generalized polynomial chaos thoroughly and presenting computational examples to show the performance of their method.…”

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with 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.…”

“…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…”