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dc.contributor.authorBattermann, Astriden_US
dc.date.accessioned2011-08-05T14:49:48Z
dc.date.available2011-08-05T14:49:48Z
dc.date.issued1996-06-14en_US
dc.identifier.otheretd-12164379662151en_US
dc.identifier.urihttp://hdl.handle.net/10919/9579
dc.description.abstractThis work is concerned with the construction of preconditioners for indefinite linear systems. The systems under investigation arise in the numerical solution of quadratic programming problems, for example in the form of Karush--Kuhn--Tucker (KKT) optimality conditions or in interior--point methods. Therefore, the system matrix is referred to as a KKT matrix. It is not the purpose of this thesis to investigate systems arising from general quadratic programming problems, but to study systems arising in linear quadratic control problems governed by partial differential equations. The KKT matrix is symmetric, nonsingular, and indefinite. For the solution of the linear systems generalizations of the conjugate gradient method, MINRES and SYMMLQ, are used. The performance of these iterative solution methods depends on the eigenvalue distribution of the matrix and of the cost of the multiplication of the system matrix with a vector. To increase the performance of these methods, one tries to transform the system to favorably change its eigenvalue distribution. This is called preconditioning and the nonsingular transformation matrices are called preconditioners. Since the overall performance of the iterative methods also depends on the cost of matrix--vector multiplications, the preconditioner has to be constructed so that it can be applied efficiently. The preconditioners designed in this thesis are positive definite and they maintain the symmetry of the system. For the construction of the preconditioners we strongly exploit the structure of the underlying system. The preconditioners are composed of preconditioners for the submatrices in the KKT system. Therefore, known efficient preconditioners can be readily adapted to this context. The derivation of the preconditioners is motivated by the properties of the KKT matrices arising in optimal control problems. An analysis of the preconditioners is given and various cases which are important for interior point methods are treated separately. The preconditioners are tested on a typical problem, a Neumann boundary control for an elliptic equation. In many important situations the preconditioners substantially reduce the number of iterations needed by the solvers. In some cases, it can even be shown that the number of iterations for the preconditioned system is independent of the refinement of the discretization of the partial differential equation.en_US
dc.format.mediumETDen_US
dc.publisherVirginia Techen_US
dc.relation.haspartetd.pdfen_US
dc.rightsI hereby grant to Virginia Tech or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University Libraries in all forms of media, now or hereafter known. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation.en_US
dc.subjectpreconditioningen_US
dc.subjectoptimal controlen_US
dc.subjectquadratic programmingen_US
dc.subjectkarush-kuhn-tucker systemsen_US
dc.subjectindefinite systemsen_US
dc.titlePreconditioning of Karush--Kuhn--Tucker Systems arising in Optimal Control Problemsen_US
dc.typeThesisen_US
dc.contributor.departmentMathematicsen_US
dc.description.degreeMSen_US
thesis.degree.nameMaster of Scienceen_US
thesis.degree.levelmastersen_US
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen_US
dc.contributor.committeechairHeinkenschloss, Matthiasen_US
dc.contributor.committeememberBurns, John A.en_US
dc.contributor.committeememberBeattie, Christopher A.en_US
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-12164379662151en_US
dc.date.sdate1998-07-11en_US


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