ON SOME PROBLEMS IN SPARSE HYBRID IMAGING, NON-STANDARD FINITE DIFFERENCE METHODS, AND FOKKER-PLANCK FRAMEWORKS IN ESOPHAGEAL CANCER
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.