Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

Abstract

We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton-Picard approach which divides the variable space into slow and fast modes. The division is performed either classically with eigenspace methods or with a novel two-grid approach. We prove mesh-independent convergence for the classical Newton-Picard preconditioner, present a complexity analysis, and show numerical results for the classical and the two-grid preconditioners. Moreover, the preconditioners compare favorably with existing symmetric positive definite Schur complement preconditioners in a Krylov method context.

Robust Iterative Solution of a Class of Time‐Dependent Optimal Control Problems

“…A recently developing area of research in the area of optimal control is the construction of preconditioners for time-dependent PDE-constrained optimization problems [3,[9][10][11]. It is desirable to design these preconditioners such that the computation time of the corresponding iterative method grows close to linearly with the problem size, although constructing methods as such often results in a lack of robustness with respect to the regularization term involved in the formulation of the problem.…”

A sequential homotopy method for mathematical programming problems

“…Existing approaches often pass these difficulties on to the level of the quadratic subproblem solvers, which may fail to resolve these issues in a way that guarantees convergence of the overall nonlinear iteration. Our method can thus be used as a black-box globalization framework for any locally convergent optimization method that can be used within a continuation framework, e.g., methods of structureexploiting inexact Sequential Quadratic Programming (SQP) [32,52,48,49,27] or semismooth Newton methods [42,53,55,31,34,32,56,30]. The local methods are 7 even allowed to converge to maxima and saddle points.…”

A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control

“…The solution method in [18] for a parabolic partial differential constraint equation is based on the extension of a special decomposition of the solution operator and used both for the forward state equation and the backward in time adjoint equation. The problem is solved via a Newton-Picard fixed point iteration method, namely, via a truncated expansion as preconditioning operator.…”