This work is concerned with analysis and refinement for a class of island dynamics models for epitaxial growth of crystalline thin films. An island dynamics model consists of evolution equations for step edges (or island boundaries), coupled with a diffusion equation for the adatom density, on an epitaxial surface. The island dynamics model with irreversible aggregation is confirmed to be mathematically ill-posed, with a growth rate that is approximately linear for large wavenumbers. By including a kinetic model for the structure and evolution of step edges, the island dynamics model is made mathematically well-posed. In the limit of small edge Peclet number, the edge kinetics model reduces to a set of boundary conditions, involving line tension and one-dimensional surface diffusion, for the adatom density. Finally, in the infinitely fast terrace diffusion limit, a simplified model of one-dimensional surface diffusion and kink convection is derived and found to be linearly stable.

Optimization problems with orthogonality constraints have many applications in science and engineering.

In these applications, one often deals with large-scale problems which are ill-conditioned near the optimum.

Consequently, there is a need for first-order optimization methods which deal with orthogonality constraints,

converge rapidly even when the objective is not well-conditioned, and are robust.

In this dissertation we develop a generalization of Nesterov's accelerated gradient descent algorithm for optimization on the

manifold of orthonormal matrices. The performance of the algorithm scales with the square root of the condition number.

As a result, our method outperforms existing state-of-the-art algorithms on large, ill-conditioned problems.

We discuss applications of the method to electronic structure calculations and to the calculation of compressed modes.

In plasma physics, the direct simulation of inter-particle Coulomb collisions is often necessary to capture the relevant physics, but presents a computational bottleneck because of the complexity of the process. In this thesis, we derive, test and discuss two methods for accelerating the simulation of collisions in plasmas in certain scenarios.

The first is a hybrid fluid-Monte Carlo scheme that reduces the number of collisions that must be simulated. Coupling between the fluid and particle components of the scheme is accomplished by assigning to each particle a passive scalar approximating the relative entropy between its distribution of velocities and the fluid distribution. When this quantity is sufficiently small, the particle is moved into the fluid so its associated collisions need not be simulated.

The second method is an adaptation of the multilevel Monte Carlo method. Instead of a single time step, this method introduces a hierarchy of time steps - i.e. levels - and uses the interplay between adjacent levels for variance reduction. We present new applications to plasmas, a method for eliminating the cost of the coarsest level calculation, and an alternative method for achieving the optimal overall computational complexity.

Throughout, we discuss applications beyond plasma physics, including rarefied gases and chemical reaction networks.

The first chapter is based on applying the Poisson summation formula to a constrained optimization problem. Motivated by Shannon sampling theorem and results on shift-invariant subspaces, we establish a compatible framework for the two key factors: the accuracy constraint, which is described in the frequency space, and the efficiency function, which is expressed in the regular space. We derive the optimal wavelet, denoted as the double-sinc function, that obtains the smallest support while remaining first order accuracy. Based on this wavelet, we further improve its accuracy by loosen up the constraint in support and manage to achieve nearly optimal efficiency.

The goal of the second chapter is to recover the underlying signal from its superposed randomly-shifted noisy measurement, motivated by multi-reference alignment and Cryo-EM problem. The general setting is that we observe samples from noisy signal that is acted by a random group action, and we would like eliminate those noises, one type at a time. In our particular setting, rational Fourier monomials and total Fourier product are invariants under the group action and hence partially remove the effect of the noise from the group action. We then apply central limit theorem to eliminate the Gaussian noise. Finally, we apply the split Bregman algorithm in compressed sensing to obtain an explicit solution assuming that the signal is compactly supported.

The third chapter is dedicated to applying a variation principle to the Euler-Poisson equation for periodic flow in a diode to optimize the flux difference, which could potentially exceed the Child-Langmuir limit. We derive a set of dual equations and boundary conditions and use upwind method to solve both the forward and backward equations. In our numerical experiment we derive a periodic solution whose flux goes above the CL limit before the physical setting or the method of characteristics breaks down.