Numerical methods for solar tomography in the STEREO era

In this thesis, we propose several advances in the numerical and computational algorithms that are used to determine tomographic estimates of physical parameters in the solar corona. We focus on methods for both global dynamic estimation of the coronal electron density and estimation of local transient phenomena, such as coronal mass ejections, from empirical observations acquired by instruments onboard the STEREO spacecraft. We present a first look at tomographic reconstructions of the solar corona from multiple points-of-view, which motivates the developments in this thesis. In particular, we propose a method for linear equality constrained state estimation that leads toward more physical global dynamic solar tomography estimates. We also present a formulation of the local static estimation problem, i.e., the tomographic estimation of local events and structures like coronal mass ejections, that couples the tomographic imaging problem to a phase field based level set method. This formulation will render feasible the 3D tomography of coronal mass ejections from limited observations. Finally, we develop a scalable algorithm for ray tracing dense meshes, which allows efficient computation of many of the tomographic projection matrices needed for the applications in this thesis.

A study of computational methods to analyze gene expression data

The recent advent of new technologies has led to huge amounts of genomic data. With these data come new opportunities to understand biological cellular processes underlying hidden regulation mechanisms and to identify disease related biomarkers for informative diagnostics. However, extracting biological insights from the immense amounts of genomic data is a challenging task. Therefore, effective and efficient computational techniques are needed to analyze and interpret genomic data. In this thesis, novel computational methods are proposed to address such challenges: a Bayesian mixture model, an extended Bayesian mixture model, and an Eigen-brain approach. The Bayesian mixture framework involves integration of the Bayesian network and the Gaussian mixture model. Based on the proposed framework and its conjunction with K-means clustering and principal component analysis (PCA), biological insights are derived such as context specific/dependent relationships and nested structures within microarray where biological replicates are encapsulated. The Bayesian mixture framework is then extended to explore posterior distributions of network space by incorporating a Markov chain Monte Carlo (MCMC) model. The extended Bayesian mixture model summarizes the sampled network structures by extracting biologically meaningful features. Finally, an Eigen-brain approach is proposed to analyze in situ hybridization data for the identification of the cell-type specific genes, which can be useful for informative blood diagnostics. Computational results with region-based clustering reveals the critical evidence for the consistency with brain anatomical structure.