Author

Madhu Gupta

ORCID Identifier(s)

0000-0002-7668-3114

Graduation Semester and Year

2022

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Souvik Roy

Second Advisor

Hristo Kojouharov

Abstract

In this thesis, we first discuss nonlinear optimization frameworks for the sparsity- based on nonlinear reconstruction of parameters in hybrid imaging modalities such as current density impedance imaging (CDII) and two-photon photoacoustic computed tomography (2P-PACT). The framework comprises minimizing an objective functional involving a least square fit and some regularization terms that promote sparsity patterns and enhance the edges to facilitate high contrast and resolution. Next, we show the construction and analysis of the second-order nonstandard finite difference methods (NSFD) scheme for theta methods and explicit Runge-Kutta method. Finally, we present an application of the NSFD scheme for Fokker-Planck (FP) frameworks in esophageal cancer. We study a stochastic model of calcium signaling dynamics in the deterministic setup using the FP framework and solve this PDE using the NSFD scheme. We also present a detailed analysis of the numerical solution. To demonstrate the effectiveness of the theoretical studies, we show various numerical experiments.

Keywords

Nonlinear optimization, Sparse hybrid imaging, Nonstandard finite difference method, Fokker-Planck framework

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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