Date of Award

12-31-2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

David Shirokoff

Second Advisor

Cyrill B. Muratov

Third Advisor

Brittany Froese Hamfeldt

Fourth Advisor

Andrew J. Bernoff

Fifth Advisor

Travis Askham

Abstract

In this dissertation, the global minimization of a large deviations rate function (the Helmholtz free energy functional) for the Boltzmann distribution is discussed. The Helmholtz functional arises in large systems of interacting particles — which are widely used as models in computational chemistry and molecular dynamics. Global minimizers of the rate function (Helmholtz functional) characterize the asymptotics of the partition function and thereby determine many important physical properties such as self-assembly, or phase transitions. Finding and verifying local minima to the Helmholtz free energy functional is relatively straightforward. However, finding and verifying global minima is much more difficult since the Helmholtz energy is nonconvex and nonlocal. Instead of minimizing the original nonconvex functional, the approach in this dissertation is to find minimizers to a convex lower bound functional. The so-called relaxed problem consists of a linear variational problem with an infinite number of Fourier constraints, leading to a variety of computational challenges. A fast solver (for the relaxed problem) based on matrix-free interior-point algorithms is developed by exploiting the Fourier structure in the problem in conjunction with a new preconditioner.

Included in

Mathematics Commons

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