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
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Gupta, Madhu, "ON SOME PROBLEMS IN SPARSE HYBRID IMAGING, NON-STANDARD FINITE DIFFERENCE METHODS, AND FOKKER-PLANCK FRAMEWORKS IN ESOPHAGEAL CANCER" (2022). Mathematics Dissertations. 219.
https://mavmatrix.uta.edu/math_dissertations/219
Comments
Degree granted by The University of Texas at Arlington