Aubrey Rhoden

Graduation Semester and Year

2013

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Jianzhong Su

Abstract

In our terminology "globally convergent numerical method" means a numerical method whose convergence to a good approximation of the correct solution is independent of the initial approximation in inverse problems. A numerical imaging algorithm has been proposed to solve a coecient inverse problem for an elliptic equation and then the algorithm is validated with the data generated by computer simulation. Previouswork in this eld was focused on the steady-state optical problem with multiple source positions moving along a straight line as well as the frequency domain problem with sweeping frequency. This work includes the steady-state thermal tomography problem with multiple source positions moving along a straight line as well as the time-dependent optical tomography problem using only two fixed source positions. Aconvergence analysis shows that this method converges globally assuming the smallness of the asymptotic solution (the so-called tail function). A heuristic approach for approximating the "new tail-function" has been utilized and verified in numerical experiments, so has the global convergence. Numerical experiments in the 2D timedependentoptical and steady-state thermal property reconstruction are presented.

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

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