Preconditioning PDE-constrained optimization with L1-sparsity and control constraints. (1st September 2017)

• Record Type: Journal Article
• Authors: Porcelli, Margherita; Simoncini, Valeria; Stoll, Martin
• Is Part Of: Computers & mathematics with applications. Volume 74:issue 5(2017)
• Page Start: 1059
• Page End: 1075
• Publication Date: 2017-09-01
• Subjects: PDE-constrained optimization -- Saddle point systems -- Preconditioning -- Krylov subspace solver -- Sparsity -- Semismooth Newton method; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2017.04.033
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including nonsmooth L 1 sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L 1 -terms as well as additional inclusion of box control constraints require the use of semismooth Newton methods. We propose robust preconditioners for different formulations of the Newton equation. With the inclusion of a line-search strategy and an inexact approach for the solution of the linear systems, the resulting semismooth Newton's method is reliable for practical problems. Our results are underpinned by a theoretical analysis of the preconditioned matrix. Numerical experiments illustrate the robustness of the proposed scheme.