Validated Continuation for Infinite Dimensional Problems

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Title: Validated Continuation for Infinite Dimensional Problems
Author: Lessard, Jean-Philippe
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.
Type: Dissertation
Date: 2007-08-07
Publisher: Georgia Institute of Technology
Subject: Continuation
Equilibria of PDEs
Chaos in ODEs
Periodic solutions of delay equations
Infinite-dimensional manifolds
Stochastic partial differential equations
Continuation methods
Differential equations, Partial
Department: Mathematics
Advisor: Committee Chair: Konstantin Mischaikow; Committee Member: Chongchun Zeng; Committee Member: Erik Verriest; Committee Member: Guillermo H. Goldsztein; Committee Member: Hao-Min Zhou
Degree: Ph.D.

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