MSc Math Defence: A collage-based approach to the Sturm-Liouville boundary value inverse problem with boundary data

CANDIDATE:  VICTORIA BROTT

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 its 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 a variety of forms in which data can be given, including functions, or 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.