Preconditioning KKT Systems
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2002-03-25
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This research presents new preconditioners for linear systems. We proceed fromthe most general case to the very specific problem area of sparse optimal control.In the first most general approach, we assume only that the coefficient matrix isnonsingular. We target highly indefinite, nonsymmetric problems that cause difficultiesfor preconditioned iterative solvers, and where standard preconditioners, likeincomplete factorizations, often fail. We experiment with nonsymmetric permutationsand scalings aimed at placing large entries on the diagonal in the context of preconditioningfor general sparse matrices. Our numerical experiments indicate that thereliability and performance of preconditioned iterative solvers are greatly enhancedby such preprocessing.Secondly, we present two new preconditioners for KKT systems. KKT systemsarise in areas such as quadratic programming, sparse optimal control, and mixedfinite element formulations. Our preconditioners approximate a constraint preconditionerwith incomplete factorizations for the normal equations. Numerical experimentscompare these two preconditioners with exact constraint preconditioning andthe approach described above of permuting large entries to the diagonal. Finally, we turn to a specific problem area: sparse optimal control. Many optimalcontrol problems are broken into several phases, and within a phase, mostvariables and constraints depend only on nearby variables and constraints. However, free initial and final times and time-independent parameters impact variables andconstraints throughout a phase, resulting in dense factored blocks in the KKT matrix.We drop fill due to these variables to reduce density within each phase. Theresulting preconditioner is tightly banded and nearly block tri-diagonal. Numericalexperiments demonstrate that the preconditioners are effective, with very little fill inthe factorization.
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PhD
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Applied Mathematics
