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Fast Marching Methods - Parallel Implementation And Analysis, Maria Cristina Tugurlan
Fast Marching Methods - Parallel Implementation And Analysis, Maria Cristina Tugurlan
LSU Doctoral Dissertations
Fast Marching represents a very efficient technique for solving front propagation problems, which can be formulated as partial differential equations with Dirichlet boundary conditions, called Eikonal equation: $F(x)|\nabla T(x)|=1$, for $x \in \Omega$ and $T(x)=0$ for $x \in \Gamma$, where $\Omega$ is a domain in $\mathbb{R}^n$, $\Gamma$ is the initial position of a curve evolving with normal velocity F>0. Fast Marching Methods are a necessary step in Level Set Methods, which are widely used today in scientific computing. The classical Fast Marching Methods, based on finite differences, are typically sequential. Parallelizing Fast Marching Methods is a step forward for …
Stochastic And Copula Models For Credit Derivatives, Chao Meng
Stochastic And Copula Models For Credit Derivatives, Chao Meng
LSU Doctoral Dissertations
We prove results relating to the exit time of a stochastic process from a region in N-dimensional space. We compute certain stochastic integrals involving the exit time. Taking a Gaussian copula model for the hitting time behavior, we prove several results on the sensitivity of quantities connected with the hitting times to parameters of the model, as well as the large-N behavior. We discuss the relationship of these results to certain credit derivative instruments. Relevant simulations are presented.
Laplace Transform Inversion And Time-Discretization Methods For Evolution Equations, Koray Ozer
Laplace Transform Inversion And Time-Discretization Methods For Evolution Equations, Koray Ozer
LSU Doctoral Dissertations
In this dissertation, we introduce Post-Widder-type inversion methods for the Laplace transform based on A-stable rational approximations of the exponential function. Since the results hold for Banach-space-valued functions, they yield efficient time-discretization methods for evolution equations of convolution type; e.g., linear first and higher order abstract Cauchy problems, inhomogeneous Cauchy problems, delay equations, Volterra and integro-differential equations, and problems that can be re-written as an abstract Cauchy problem on an appropriate state space.
Differential Geometry In Cartesian Closed Categories Of Smooth Spaces, Martin Laubinger
Differential Geometry In Cartesian Closed Categories Of Smooth Spaces, Martin Laubinger
LSU Doctoral Dissertations
The main categories of study in this thesis are the categories of diffeological and Fr\"olicher spaces. They form concrete cartesian closed categories. In Chapter 1 we provide relevant background from category theory and differentiation theory in locally convex spaces. In Chapter 2 we define a class of categories whose objects are sets with a structure determined by functions into the set. Fr\"olicher's $M$-spaces, Chen's differentiable spaces and Souriau's diffeological spaces fall into this class of categories. We prove cartesian closedness of the two main categories, and show that they have all limits and colimits. We exhibit an adjunction between the …
Rational Approximation Schemes For Solutions Of Abstract Cauchy Problems And Evolution Equations, Patricio Gabriel Jara
Rational Approximation Schemes For Solutions Of Abstract Cauchy Problems And Evolution Equations, Patricio Gabriel Jara
LSU Doctoral Dissertations
In this dissertation we study time and space discretization methods for approximating solutions of abstract Cauchy problems and evolution equations in a Banach space setting. Two extensions of the Hille-Phillips functional calculus are developed. The first result is the Hille-Phillips functional calculus for generators of bi-continuous semigroups, and the second is a C-regularized version of the Hille-Phillips functional calculus for generators of C-regularized semigroups. These results are used in order to study time discretization schemes for abstract Cauchy problems associated with generators of bi-continuous semigroups as well as C-regularized semigoups. Stability, convergence results, and error estimates for rational approximation schemes …
Multiscale Analysis Of Heterogeneous Media For Local And Nonlocal Continuum Theories, Bacim Alali
Multiscale Analysis Of Heterogeneous Media For Local And Nonlocal Continuum Theories, Bacim Alali
LSU Doctoral Dissertations
The dissertation provides new multiscale methods for the analysis of heterogeneous media. The first part of the dissertation treats heterogeneous media using the theory of linear elasticity. In this context, a methodology is presented for bounding the higher order moments of the local stress and strain fields inside random elastic media. Optimal lower bounds that are given in terms of the applied loading and the volume (area) fractions for random two-phase composites are presented. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. The second part of …
Surgery Description Of Colored Knots, Steven Daniel Wallace
Surgery Description Of Colored Knots, Steven Daniel Wallace
LSU Doctoral Dissertations
By a knot, or link, we mean a circle, or a collection of circles, embedded in the three-sphere S3. The study of knots is a very rich subject and plays a key role in the area of low-dimensional topology. In fact, a theorem of W.B.R. Lickorish and A.D. Wallace states that any three-dimensional manifold may be described by Dehn surgery along a link which is the process of removing the link from S3 and then gluing it back in a way that possibly changes the resulting manifold. In this dissertation, we will be interested in the pair (K, ρ) consisting …
Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith
Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith
LSU Doctoral Dissertations
This dissertation is concerned with providing a description of certain symmetric bilinear forms, called trace forms, associated with finite normal extensions N/K of an algebraic number field K, with abelian Galois group Gal(N/K). These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). Complete descriptions are obtained when the base field K has exactly one dyadic prime and either no real embeddings or one real embedding. For these fields K, the set of abelian trace forms is closed under multiplication in the Witt ring W(K).