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dc.contributor.authorLessard, Jean-Philippeen_US
dc.date.accessioned2008-02-07T18:49:49Z
dc.date.available2008-02-07T18:49:49Z
dc.date.issued2007-08-07en_US
dc.identifier.urihttp://hdl.handle.net/1853/19861
dc.description.abstractStudying the zeros of a parameter dependent operator F defined on a Hilbert space H is a fundamental problem in mathematics. When the Hilbert space is finite dimensional, continuation provides, via predictor-corrector algorithms, efficient techniques to numerically follow the zeros of F as we move the parameter. In the case of infinite dimensional Hilbert spaces, this procedure must be applied to some finite dimensional approximation which of course raises the question of validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced zero for the finite dimensional system can be used to explicitly define a set which contains a unique zero for the infinite dimensional problem F: HxR->Im(F). We use this new validated continuation to study equilibrium solutions of partial differential equations, to prove the existence of chaos in ordinary differential equations and to follow branches of periodic solutions of delay differential equations. In the context of partial differential equations, we show that the cost of validated continuation is less than twice the cost of the standard continuation method alone.en_US
dc.publisherGeorgia Institute of Technologyen_US
dc.subjectContinuationen_US
dc.subjectEquilibria of PDEsen_US
dc.subjectChaos in ODEsen_US
dc.subjectPeriodic solutions of delay equationsen_US
dc.subject.lcshInfinite-dimensional manifolds
dc.subject.lcshStochastic partial differential equations
dc.subject.lcshContinuation methods
dc.subject.lcshDifferential equations, Partial
dc.titleValidated Continuation for Infinite Dimensional Problemsen_US
dc.typeDissertationen_US
dc.description.degreePh.D.en_US
dc.contributor.departmentMathematicsen_US
dc.description.advisorCommittee Chair: Konstantin Mischaikow; Committee Member: Chongchun Zeng; Committee Member: Erik Verriest; Committee Member: Guillermo H. Goldsztein; Committee Member: Hao-Min Zhouen_US


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