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On the convergence of right transforming iterations for the numerical solution of PDE‐constrained optimization problems
Author(s) -
Linsenmann Christopher
Publication year - 2012
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.788
Subject(s) - karush–kuhn–tucker conditions , solver , mathematics , convergence (economics) , iterative method , mathematical optimization , contraction (grammar) , scheme (mathematics) , optimization problem , rate of convergence , computer science , mathematical analysis , medicine , channel (broadcasting) , computer network , economics , economic growth
SUMMARY We present an iterative solver, called right transforming iterations (or right transformations ), for linear systems with a certain structure in the system matrix, such as they typically arise in the framework of Karush–Kuhn–Tucker (KKT) conditions for optimization problems under PDE constraints. The construction of the right transforming scheme depends on an inner approximate solver for the underlying PDE subproblems. We give a rigorous convergence proof for the right transforming iterative scheme in dependence on the convergence properties of the inner solver. Provided that a fast subsolver is available, this iterative scheme represents an efficient way of solving first‐order optimality conditions. Numerical examples endorse the theoretically predicted contraction rates. Copyright © 2011 John Wiley & Sons, Ltd.