Dissertation Defense: Peter Rizzi, Mathematical Sciences

Committee Names
Dr. Thomas Wanner (chair)
Dr. Evelyn Sander
Dr. Tim Sauer
Dr. Juan Raul Cebral

Defense Title
On Validated Equilibria and Bifurcations in Materials Science and Stochastic Dynamics

Defense Abstract
Partial Differential Equations (PDEs) and their solutions provide a theoretically rich, widely applicable, and analytically challenging subfield of mathematics and physics. In the applied setting, analytical solutions are rarely known, and one must resort to numerical methods. These methods are, at their best, approximations to the exact solution of the PDE and quickly exceed the computational ability of humans making a computer necessary to execute them efficiently. While many of these methods, such as the Finite Element Method (FEM) and the Spectral Method, produce good quality approximations, problems exist where the solution to the numerical formulation is in fact not close to a solution of the PDE. The proliferation of computing in scientific applications demands exploration of verification methods for numerical solutions. We explore two special cases: the Fokker-Planck-Kolmogorov Equation and block copolymers.