Title:
Validated Continuation for Infinite Dimensional Problems

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dc.contributor.advisor Mischaikow, Konstantin
dc.contributor.author Lessard, Jean-Philippe en_US
dc.contributor.committeeMember Chongchun Zeng
dc.contributor.committeeMember Erik Verriest
dc.contributor.committeeMember Guillermo H. Goldsztein
dc.contributor.committeeMember Hao-Min Zhou
dc.contributor.department Mathematics en_US
dc.date.accessioned 2008-02-07T18:49:49Z
dc.date.available 2008-02-07T18:49:49Z
dc.date.issued 2007-08-07 en_US
dc.description.abstract Studying 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.description.degree Ph.D. en_US
dc.identifier.uri http://hdl.handle.net/1853/19861
dc.publisher Georgia Institute of Technology en_US
dc.subject Continuation en_US
dc.subject Equilibria of PDEs en_US
dc.subject Chaos in ODEs en_US
dc.subject Periodic solutions of delay equations en_US
dc.subject.lcsh Infinite-dimensional manifolds
dc.subject.lcsh Stochastic partial differential equations
dc.subject.lcsh Continuation methods
dc.subject.lcsh Differential equations, Partial
dc.title Validated Continuation for Infinite Dimensional Problems en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
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