We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method relies on a numerical resolution of the corresponding Monge-Ampère equation. We use an existing Newton-like algorithm that we generalize to the case of a non uniform final density. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we discretize and solve at each iteration, in two different ways: a second order finite difference scheme and a Fourier transform (FT) method. The FT method, made possible thanks to a preconditioning step based on the coefficient-averaged equation, results in an overall O(P LogP )-operations algorithm, where P is the number of discretization points. We prove that the generalized algorithm converges to the solution of the optimal transport problem, under suitable conditions on the initial and final densities. Numerical experiments demonstrating the robustness and efficiency of the method on several examples of image processing, including an application to multiple sclerosis disease detection, are shown. We also demonstrate by numerical tests that the method is competitive against some other methods available.

An n × n sign pattern A is potentially stable (PS) if there exists a real matrix A having the sign pattern A and with all its eigenvalues having negative real parts. The identification of non-trivial necessary and sufficient conditions for potential stability remains a long standing open problem. Here we review some of the previous results and give simplified proofs for some of these results. Three techniques are given for the construction of larger order PS sign patterns from given PS sign patterns. These techniques are: construction of a sign pattern that allows a nested sequence of properly signed principal minors (a nest), bordering of a PS sign pattern with additional rows and columns, and use of a similarity transformation of a matrix that is reducible with two diagonal blocks (one of which is a stable matrix and the other a negative scalar). The minimum number of nonzero entries in an irreducible minimally PS sign pattern is determined for n = 2, . . . , 6 and for an arbitrary sign pattern that allows a nest. We also determine lower bounds for the number of nonzero entries in irreducible minimally PS sign patterns having certain sign patterns for their diagonal entries. For irreducible PS sign patterns of order at least four, a bordering construction leads to a new upper bound for the minimum number of nonzero entries.

In this study, we consider two bi-objective bin packing problems that assign a number of weighted items to bins having identical capacities. Firstly, we aim to minimize total deviation over bin capacity and minimize number of bins. We show that these two objectives are conflicting. Secondly, we study the problem of minimizing maximum overdeviation and minimizing the number of bins. We show the similarities of these two problems to parallel machine scheduling problems and benefit from the results while developing our solution approaches. For both problems, we propose exact procedures that generate efficient solutions relative to two objectives. To increase the efficiency of the solutions, we propose some lower and upper bounding procedures. The results of our experiments show that total overdeviation problem is easier to solve compared to maximum overdeviation problem and the bin capacity, the weight of items and the number of items are important factors that effect the solution time and quality. Our procedures can solve the problems with up to 100 items in reasonable solution times.

Contact is a phenomenon faced in every day life, which is actually a complex problem to tackle for engineers. Most of the times, may be impossible to get analytic or exact results for the interaction of bodies in contact. In this thesis work, solution of the frictionless contact of an elastic body, touching to a rigid planar surface for two-dimensional elasticity / namely plane stress, plane strain and axi-symmetric formulations is aimed. The problem is solved numerically, with Finite Element Method, and an Object Oriented computer program in C++ for this purpose is written, and the results are verified with some basic analytic solutions and ABAQUS package program. It is not aimed in this thesis work to give a new solution in the area of solution of contact problems, but instead, it is aimed to form a strong basis, and computational library, which is extendible for further development of the subject to include friction, plasticity, and different material modeling in this advanced field of mechanics.

In this study, a long annulus bonded to a rigid cylinder containing an axisymmetric circumferential crack of width (d-c) at the midplane is considered. The material of the annulus is assumed to be linearly elastic and isotropic. The external surface of the annulus is free of stress. Surfaces of the crack are subject to distributed compressive loads. The Fourier and Hankel transform techniques are used to solve the governing equations which are reduced to a singular integral equation for crack surface displacement derivative. This integral equation is converted to a system of linear algebraic equations which are solved numerically by using Gauss-Lobatto and Gauss-Jacobi quadrature formulas. Then, the stress intensity factors at the edges of the crack are calculated. Results are presented in graphical form.

Thesis (M. App. Sc.)--Carleton University, 2001. / Includes bibliographical references (p. 159-164). Also available in electronic format on the Internet.

Thesis (M. S.)--Public Policy, Georgia Institute of Technology, 2006. / Dr. Monica Gaughan, Committee Member ; Dr. Donna Llewellyn, Committee Member ; Dr. Gordon Kingsley, Committee Chair. Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (leaves 109-124).

Thesis (M.Eng.)(Mechanical)--University of Pretoria, 2006. / Includes summary. Includes bibliographical references. Available on the Internet via the World Wide Web.

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