Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

Proteins in silico-modeling and sampling

Proteins are linear chain molecules made out of amino acids. Only when they fold to their native states, they become functional. This dissertation aims to model the solvent (environment) effect and to develop & implement enhanced sampling methods that enable a reliable study of the protein folding problem in silico. We have developed an enhanced solvation model based on the solution to the Poisson-Boltzmann equation in order to describe the solvent effect. Following the quantum mechanical Polarizable Continuum Model (PCM), we decomposed net solvation free energy into three physical terms– Polarization, Dispersion and Cavitation. All the terms were implemented, analyzed and parametrized individually to obtain a high level of accuracy. In order to describe the thermodynamics of proteins, their conformational space needs to be sampled thoroughly. Simulations of proteins are hampered by slow relaxation due to their rugged free-energy landscape, with the barriers between minima being higher than the thermal energy at physiological temperatures. In order to overcome this problem a number of approaches have been proposed of which replica exchange method (REM) is the most popular. In this dissertation we describe a new variant of canonical replica exchange method in the context of molecular dynamic simulation. The advantage of this new method is the easily tunable high acceptance rate for the replica exchange. We call our method Microcanonical Replica Exchange Molecular Dynamic (MREMD). We have described the theoretical frame work, comment on its actual implementation, and its application to Trp-cage mini-protein in implicit solvent. We have been able to correctly predict the folding thermodynamics of this protein using our approach.

THE DISPERSION AND SELECTIVE FLOCCULATION OF HEMATITE ORE

Iron ore is one of the most important ores in the world. Over the past century, most mining of iron ore has been focused on magnetite (Fe3O4). As the name suggests, magnetite is magnetic in nature and is easily separated from gangue (unwanted) minerals through magnetic separation processes. Unfortunately, the magnetite ore bodies are diminishing. Because of this, there has been a recent drive to pursue technology that can economically separate hematite (Fe2O3) from its gangue minerals as hematite is a much more abundant source of iron. Most hematite ore has a very small liberation size that is frequently less than 25μm. Beneficiation of any ore with this fine of a liberation size requires advanced processing methods and is seldom pursued. A single process, known as selective flocculation and dispersion, has been successfully implemented at a plant scale for the beneficiation of fine liberation size hematite ore. Very little is known about this process as it was discovered by the U.S. Bureau of Mines by accident. The process is driven by water chemistry and surface chemistry modifications that enhance the separation of the hematite from its gangue minerals. This dissertation focuses on the role of water chemistry and process reagents in this hematite beneficiation process. It has been shown that certain ions, including calcium and magnesium, play a significant role in the process. These ions have a significant effect on the surface chemistry as reported by zeta potential studies. It was shown that magnesium ions within the process water have a more significant impact on surface chemistry than calcium ions due to steric hindrance effects at the hematite surface. It has also been shown that polyacrylic acid dispersants, if used in the process, can increase product quality (increase iron content, decrease phosphorus content, decrease silica content) substantially. Water, surface and reagent chemistry experiments were performed at a laboratory, pilot, and full plant scale during the course of this work. Many of the conclusions developed in the laboratory and pilot scale were found to be true at the full plant scale as well. These studies are the first published in history to develop theories of water chemistry and surface chemistry interactions at a full plant scale.