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A Collage-Based Approach to a Sturm-Liouville Boundary Value Inverse Problem with Boundary Data

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Title: A Collage-Based Approach to a Sturm-Liouville Boundary Value Inverse Problem with Boundary Data
Author: Brott, Victoria
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Kunze, Herb
Abstract: Techniques for solving inverse problems for ordinary differential equations (ODEs) are well-established throughout the literature. Classically, each of these techniques utilizes observational data on the interior of the domain to determine other elements of the ODE. A novel technique, the Collage method, was introduced in 1999 in [10]. Unlike it’s predecessors that minimize the approximation error directly, Collage methods bound the approximation error above by a function which is more readily minimizable. Here we focus on solving inverse problems for Sturm-Liouville BVPs using a collage-based approach. In addition to an inverse problem with interior measurements this thesis also extends to solving inverse problems, with a Collage coding framework, that utilize only boundary data. We explore data given in two forms: functions and discrete data. Finally, we investigate the use of multiple datasets and how this can affect our results. Particle Swarm Ant Colony Optimization is employed throughout this work in order to handle the ill-posedness encountered in an inverse problem with boundary data.
URI: http://hdl.handle.net/10214/14667
Date: 2018-12
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