Multigrid preconditioning of linear systems for semismooth Newton methods applied to optimization problems constrained by smoothing operators
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2013-11-07
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Andrei Drăgănescu, Jyoti Saraswat. (2016) Optimal-Order Preconditioners for Linear Systems Arising in the Semismooth Newton Solution of a Class of Control-Constrained Problems. SIAM Journal on Matrix Analysis and Applications 37:3, pages 1038-1070.
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This is a preprint of an article published by Taylor & Francis in Optimization Methods and Software on Nov. 20, 2013, available online: https://doi.org/10.1080/10556788.2013.854356
This is a preprint of an article published by Taylor & Francis in Optimization Methods and Software on Nov. 20, 2013, available online: https://doi.org/10.1080/10556788.2013.854356
Abstract
This article is concerned with the question of constructing effcient multigrid preconditioners for the linear systems arising when applying semismooth Newton methods to large-scale linear-quadratic optimization problems constrained by smoothing operators with box-constraints on the controls. It is shown that, for certain discretizations of the optimization problem, the linear systems to be solved at each semismooth Newton iteration reduce to inverting principal minors of the Hessian of the associated unconstrained problem. As in the case when box-constraints on the controls are absent, the multigrid preconditioner introduced here is shown to increase in quality as the mesh-size decreases, resulting in a number of iterations that decreases with mesh-size. However, unlike the unconstrained case, the spectral distance between the preconditioners and the Hessian is shown to be of suboptimal order in general.