Many scientific and engineering applications require the solution of large systems of initial value problems arising from method of lines discretization of partial differential equations. For systems with widely varying time scales, or with complex physical dynamics, implicit time integration schemes are preferred due to their superior stability properties. However, for very large systems accurate solution of the implicit terms can be impractical. For this reason approximations are widely used in the implementation of such methods. The primary focus of this work is on the development of novel ``lightly-implicit'' time integration methodologies. These methods consider the time integration and the solution of the implicit terms as a single computational process. We propose several classes of lightly-implicit methods that can be constructed to allow for different, specific approximations. Rosenbrock-Krylov and exponential-Krylov methods are designed to permit low accuracy Krylov based approximations of the implicit terms, while maintaining full order of convergence. These methods are matrix free, have low memory requirements, and are particularly well suited to parallel architectures. Linear stability analysis of K-methods is leveraged to construct implementation improvements for both Rosenbrock-Krylov and exponential-Krylov methods. Linearly-implicit Runge-Kutta-W methods are designed to permit arbitrary, time dependent, and stage varying approximations of the linear stiff dynamics of the initial value problem. The methods presented here are constructed with approximate matrix factorization in mind, though the framework is flexible and can be extended to many other approximations. The flexibility of lightly-implicit methods, and their ability to leverage computationally favorable approximations makes them an ideal alternative to standard explicit and implicit schemes for large parallel applications. / Ph. D.

abstract: Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain. This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges. Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018

This thesis presents new numerical methods for solving differential equations with periodicity. Spectral methods for solving linear and nonlinear ODEs, linear ODE eigenvalue problems and linear time-dependent PDEs on a periodic interval are reviewed, and a novel approach for computing multiplication matrices is presented. Choreographies, periodic solutions of the n-body problem that share a common orbit, are computed for the first time to high accuracy using an algorithm based on approximation by trigonometric polynomials and optimization techniques with exact gradient and exact Hessian matrix. New choreographies in spaces of constant curvature are found. Exponential integrators for solving periodic semilinear stiff PDEs in 1D, 2D and 3D periodic domains are reviewed, and 30 exponential integrators are compared on 11 PDEs. It is shown that the complicated fifth-, sixth- and seventh-order methods do not really outperform one of the simplest exponential integrators, the fourth-order ETDRK4 scheme of Cox and Matthews. Finally, algorithms for solving semilinear stiff PDEs on the sphere with spectral accuracy in space and fourth-order accuracy in time are proposed. These are based on a new variant of the double Fourier sphere method in coefficient space and standard implicit-explicit time-stepping schemes. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform better. The algorithms described in each chapter of this thesis have been implemented in MATLAB and made available as part of Chebfun.