UC San Diego

The level-set method (LS) is a widely-used and powerful tool for capturing moving interfaces under complex dynamics in fields ranging from two-phase flow to image segmentation. It has recently been successfully applied, in the Variational Implicit Solvent Model (VISM), to find the “shape” of a biomolecule, the interface separating the solute atoms of a biomolecule from the surrounding solvent. In the introduces fast and efficient numerical methods for the application of the level-set method to VISM (LS-VISM), and can be boiled down to two major contributions. The first of these involves the implementation and analysis of a more discrete binary level-set method in LS-VISM that replaces traditional continuous level-set functions with binary ones, and traditional partial differential equation solvers with discrete “flips” that minimize the free energy of the system. This results in vast improvements in speed, with runtime decreasing from hours to seconds, which ultimately allowed for its application in Monte Carlo simulations of the protein binding process. The second contribution in my thesis involves the construction and analysis of the Compact Coupling Interface Method (CCIM), a finite difference method for elliptic interface problems with interfacial jump conditions. This method is able to robustly and accurately calculate values of not only the solution but its derivative as well, which ultimately allows for the accurate handling of electrostatic contributions of the solute and solvent in LS-VISM, which take this form as linearized Poisson-Boltzmann equations with discontinuous dielectric constants across the interface.