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Towards Matrix‐Free AD‐Based Preconditioning of KKT Systems in PDE‐Constrained Optimization
Author(s) -
Griesse Roland,
Walther Andrea
Publication year - 2005
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200510013
Subject(s) - karush–kuhn–tucker conditions , jacobian matrix and determinant , hessian matrix , multigrid method , discretization , mathematical optimization , mathematics , partial differential equation , newton's method , optimization problem , constraint (computer aided design) , computer science , automatic differentiation , computation , nonlinear system , algorithm , mathematical analysis , geometry , quantum mechanics , physics
The presented approach aims at solving an equality constrained, finite‐dimensional optimization problem, where the constraints arise from the discretization of some partial differential equation (PDE) on a given space grid. For this purpose, a stationary point of the Lagrangian is computed using Newton's method, which requires the repeated solution of KKT systems. The proposed algorithm focuses on two topics: Firstly, Algorithmic Differentiation (AD) will be used to evaluate the necessary computations of gradients, Jacobian‐vector products, and Hessian‐vector products, so that only the objective f ( y , u ) and the PDE constraint e ( y , u ) = 0 have to be specified by the user. Secondly, we solve the KKT system iteratively using the QMR algorithm, with preconditioning provided by a multigrid approach. We wish to explore whether the Jacobian‐vector products provided by AD are sufficient to construct suitable multigrid preconditioners. Our approach is then embedded into a globalized optimization routine. Numerical results for optimization problems involving a nonlinear reaction‐diffusion model will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)