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.