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

A. Potschka (potschka***at***iwr.uni-heidelberg.de)
M.S. Mommer (mario.mommer***at***iwr.uni-heidelberg.de)
J.P. Schlöder (j.schloeder***at***iwr.uni-heidelberg.de)
H.G. Bock (bock***at***iwr.uni-heidelberg.de)

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.

Keywords: periodic boundary condition, parabolic partial differential equation, Newton-Picard, symmetric indefinite, preconditioner

Category 1: Applications -- Science and Engineering (Optimization of Systems modeled by PDEs )

Category 2: Nonlinear Optimization (Quadratic Programming )

Citation: SIAM J. Sci. Comput., 34(2), A1214–A1239. (26 pages) doi: 10.1137/100807776

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Entry Submitted: 03/17/2010
Entry Accepted: 03/17/2010
Entry Last Modified: 06/19/2012

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