published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

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Very accurate methods, based on boundary integral techniques, are developed for the study of multiple, interacting fluid interfaces in an Eulerian fluid. These methods are applied to the evolution of a thin, periodic layer of constant vorticity embedded in irrotational fluid. Numerical regularity experiments are conducted and suggest that the interfaces of the layer develop a curvature singularity in infinite time. This is to be contrasted with the more singular vorticity distribution of a vortex sheet developing such a singularity in a finite time.

No description available.

No description available.

No description available.

Recent research has shown that compressible signals can be recovered from a very limited number of measurements by minimizing nonconvex functions that closely resemble the L0-norm function. These functions have sparse minimizers and, therefore, are called sparsity-promoting functions (SPFs). Recovery is achieved by solving a nonconvex optimization problem when using these SPFs. Contemporary methods for the solution of such difficult problems are inefficient and not supported by robust convergence theorems. New signal-recovery methods for compressive sensing that can be used to solve nonconvex problems efficiently are proposed. Two categories of methods are considered, namely, sequential convex formulation (SCF) and proximal-point (PP) based methods. In SCF methods, quadratic or piecewise-linear approximations of the SPF are employed. Recovery is achieved by solving a sequence of convex optimization problems efficiently with state-of-the-art solvers. Convex problems are formulated as regularized least-squares, second-order cone programming, and weighted L1-norm minimization problems. In PP based methods, SPFs that entail rich optimization properties are employed. Recovery is achieved by iteratively performing two fundamental operations, namely, computation of the PP of the SPF and projection of the PP onto a convex set. The first operation is performed analytically or numerically by using a fast iterative method. The second operation is performed efficiently by computing a sequence of closed-form projectors. The proposed methods have been compared with the leading state-of-the-art signal-recovery methods, namely, the gradient-projection method of Figueiredo, Nowak, and Wright, the L1-LS method of Kim, Koh, Lustig, Boyd, and Gorinevsky, the L1-Magic method of Candes and Romberg, the spectral projected-gradient L1-norm method of Berg and Friedlander, the iteratively reweighted least squares method of Chartrand and Yin, the difference-of-two-convex-functions method of Gasso, Rakotomamonjy, and Canu, and the NESTA method of Becker, Bobin, and Candes. The comparisons concerned the capability of the proposed and competing algorithms in recovering signals in a wide range of test problems and also the computational efficiency of the various algorithms. Simulation results demonstrate that improved reconstruction performance, measurement consistency, and comparable computational cost are achieved with the proposed methods relative to the competing methods. The proposed methods are robust, are supported by known convergence theorems, and lead to fast convergence. They are, as a consequence, particularly suitable for the solution of hard recovery problems of large size that entail large dynamic range and, are, in effect, strong candidates for use in many real-world applications. / Graduate / 0544 / eng.flavio.teixeira@gmail.com

This thesis presents a new preconditioner for elliptic PDE problems on unstructured meshes. Using ideas from second generation wavelets, a multi-resolution basis is constructed to effectively compress the inverse of the matrix, resolving the sparsity vs. quality problem of standard approximate inverses. This finally allows the approximate inverse approach to scale well, giving fast convergence for Krylov subspace accelerators on a wide variety of large unstructured problems. Implementation details are discussed, including ordering and construction of factored approximate inverses, discretization and basis construction in one and two dimensions, and possibilities for parallelism. The numerical experiments in one and two dimensions confirm the capabilities of the scheme. Along the way I highlight many new avenues for research, including the connections to multigrid and other multi-resolution schemes.

This thesis is concerned with changes in the behaviour of solutions to parameter-dependent stochastic delay differential equations.