We study numerical approximations for the payoff function of the stochastic optimal stopping and control problem. It is known that the payoff function of the optimal stopping and control problem corresponds to the solution of a normalized Bellman PDE. The principal aim of this thesis is to study the rate at which finite difference approximations, derived from the normalized Bellman PDE, converge to the payoff function of the optimal stopping and control problem. We do this by extending results of N.V. Krylov from the Bellman equation to the normalized Bellman equation. To our best knowledge, until recently, no results about the rate of convergence of finite difference approximations to Bellman equations have been known. A major breakthrough has been made by N. V. Krylov. He proved rate of rate of convergence of tau 1/4 + h 1/2 where tau and h are the step sizes in time and space respectively. We will use the known idea of randomized stopping to give a direct proof showing that optimal stopping and control problems can be rewritten as pure optimal control problems by introducing a new control parameter and by allowing the reward and discounting functions to be unbounded in the control parameter. We extend important results of N. V. Krylov on the numerical solutions to the Bellman equations to the normalized Bellman equations associated with the optimal stopping of controlled diffusion processes. We obtain the same rate of convergence of tau1/4 + h1/2. This rate of convergence holds for finite difference schemes defined on a grid on the whole space [0, T]×Rd i.e. on a grid with infinitely many elements. This leads to the study of localization error, which arises when restricting the finite difference approximations to a cylindrical domain. As an application of our results, we consider an optimal stopping problem from mathematical finance: the pricing of American put option on multiple assets. We prove the rate of convergence of tau1/4 + h1/2 for the finite difference approximations.

Multiple myeloma is a plasma cell cancer that affects the bones, immune system, and kidneys. In this thesis, we focus on the impact on the bone, specifically routine bone remodeling. The bone remodeling process is governed by chemical signaling between several cell populations. In multiple myeloma patients, this process is out of balance. Bone destruction outpaces bone replacement, leaving patients with bone lesions. We describe the cell-signaling network that regulates bone remodeling and explain how it is impacted by multiple myeloma. We then present a series of mathematical models describing the bone remodeling process. We lay a thorough mathematical foundation, starting with the derivation of Savageau's power law approximations. Next, we introduce a novel one-dimensional moving-boundary partial differential equation model of this biological system. Our model improves upon models from the literature by including new cell populations, specifically osteoclast precursors, stromal cells, and tumor cells. We also discuss the model's computational results and their significance. We then discuss image processing techniques that can be used on bone marrow biopsies to gather data on the growth of a multiple myeloma tumor. By analyzing these medical images, we can extract tumor cell counts. In particular, we give the results of such an analysis for one patient using color unmixing. Image processing techniques, such as the ones presented here, could be used for validation of the models we present. The long-term goal of this project is the creation of a diagnostic tool that will aid oncologists in selecting the best treatment plan for their patients with multiple myeloma.

Computing highly-accurate approximate solutions to partial differential equations (PDEs) requires both a robust numerical method and a powerful machine. We present a parallel implementation of the discontinuous Galerkin (DG) method on graphics processing units (GPUs). In addition to being flexible and highly accurate, DG methods accommodate parallel architectures well, as their discontinuous nature produces entirely element-local approximations. While GPUs were originally intended to compute and display computer graphics, they have recently become a popular general purpose computing device. These cheap and extremely powerful devices have a massively parallel structure. With the recent addition of double precision floating point number support, GPUs have matured as serious platforms for parallel scientific computing. In this thesis, we present an implementation of the DG method applied to systems of hyperbolic conservation laws in two dimensions on a GPU using NVIDIA’s Compute Unified Device Architecture (CUDA). Numerous computed examples from linear advection to the Euler equations demonstrate the modularity and usefulness of our implementation. Benchmarking our method against a single core, serial implementation of the DG method reveals a speedup of a factor of over fifty times using a USD $500.00 NVIDIA GTX 580.

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.

A new construction of regular solutions to the three dimensional Navier{Stokes equa- tions is introduced and applied to the study of their asymptotic expansions. This construction and other Phragmen-Linderl??of type estimates are used to establish su??- cient conditions for the convergence of those expansions. The construction also de??nes a system of inhomogeneous di??erential equations, called the extended Navier{Stokes equations, which turns out to have global solutions in suitably constructed normed spaces. Moreover, in these spaces, the normal form of the Navier{Stokes equations associated with the terms of the asymptotic expansions is a well-behaved in??nite system of di??erential equations. An application of those asymptotic expansions of regular solutions is the analysis of the helicity for large times. The dichotomy of the helicity's asymptotic behavior is then established. Furthermore, the relations between the helicity and the energy in several cases are described.

Computing highly-accurate approximate solutions to partial differential equations (PDEs) requires both a robust numerical method and a powerful machine. We present a parallel implementation of the discontinuous Galerkin (DG) method on graphics processing units (GPUs). In addition to being flexible and highly accurate, DG methods accommodate parallel architectures well, as their discontinuous nature produces entirely element-local approximations. While GPUs were originally intended to compute and display computer graphics, they have recently become a popular general purpose computing device. These cheap and extremely powerful devices have a massively parallel structure. With the recent addition of double precision floating point number support, GPUs have matured as serious platforms for parallel scientific computing. In this thesis, we present an implementation of the DG method applied to systems of hyperbolic conservation laws in two dimensions on a GPU using NVIDIA’s Compute Unified Device Architecture (CUDA). Numerous computed examples from linear advection to the Euler equations demonstrate the modularity and usefulness of our implementation. Benchmarking our method against a single core, serial implementation of the DG method reveals a speedup of a factor of over fifty times using a USD $500.00 NVIDIA GTX 580.

Made available in DSpace on 2014-06-12T17:20:21Z (GMT). No. of bitstreams: 2 arquivo932_1.pdf: 2343258 bytes, checksum: 9228fa008881d546951e2dd04251ce75 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2009 / Esta tese buscou analisar a política do PDE na Zona da Mata Norte do Estado de Pernambuco, no período de 1999 a 2007. Partiu-se do pressuposto de que o PDE é um marco na política educacional originado no contexto de políticas públicas impulsionadas pelo BM, a partir da década de 1990, e que, como processo de política passa por um modo dinâmico de regulação em todas as suas dimensões. O caminho teórico metodológico apoiou-se na abordagem cognitiva de política pública defendida por Muller & Surel (2002), combinado com a teoria de discurso e de análise de discurso em Norman Fairclough (2001). Considerou-se nesse sentido o discurso em uma perspectiva tridimensional, como interdiscursividade, intertextualidade e como prática social. Observou-se que em Pernambuco o PDE encontrou um discurso de resistência a sua concepção de política educacional. O sentido dos termos descentralização, autonomia, qualidade da educação e participação foram objeto de disputa, envolvendo o Fundescola/BM, setores organizados da sociedade civil, organizações acadêmicas, Secretaria Estadual de Educação, Sindicato de professores e membros da escola. Na Região investigada, o crescimento do PDE esteve vinculado ao financiamento do PDDE. A participação proposta restringiu-se a funções burocráticas e contribuiu para negar o seu valor pedagógico e político, uma vez que diminuiu a capacidade de autonomia das escolas e ampliou os mecanismos de regulação, de controle e de avaliação externa. Observou-se ainda que o PDE ocupou um espaço deixado pela ausência de uma cultura de planejamento no âmbito da escola e que, em alguns casos, sobrepôs o processo de construção do PPP.

In this dissertation we applied geometric methods to study underdetermined second order scalar ordinary differential equations (called general Monge equations), nonlinear involutive systems of two scalar partial differential equations in two independent variables and one unknown and non-Monge-Ampere Goursat parabolic scalar PDE in the plane. These particular kinds of differential equations are related to general rank-3 Pfaffian systems in five variables. Cartan studied these objects in his 1910 paper. In this work Cartan provided normal forms only for some general rank-3 Pfaffian systems with 14-, 7-, and 6-dimensional symmetry algebra. We applied our normal forms to [i] sharpen Cartan's integration method of nonlinear involutive systems, [ii] classify all general Monge equations with a freely acting transverse 3-dimensional symmetry algebra, of which many new examples are presented, and [iii] provide a broad classification of non-Monge-Ampere Darboux integrable hyperbolic PDE in the plane. We developed a computer software, called FiveVariables, that classifies general rank-3 Pfaffian systems. FiveVariables runs in the environment DifferentialGeometry of Maple, version 11 and later.

We study a two-dimensional Fermi-Pasta-Ulam lattice in the long-amplitude, small-wavelength limit. The one-dimensional lattice has been thoroughly studied in this limit, where it has been established that the dynamics of the lattice is well-approximated by the Korteweg–De Vries (KdV) equation for timescales of the order ε^−3. Further it has been shown that solitary wave solutions of the FPU lattice in the one dimensional case are well approximated by solitary wave solutions of the KdV equation. A two-dimensional analogue of the KdV equation, the Kadomtsev–Petviashvili (KP-II) equation, is known to be a good approximation of certain two-dimensional FPU lattices for similar timescales, although no proof exists. In this thesis we present a rigorous justification that the KP-II equation is the long-amplitude, small-wavelength limit of a two-dimensional FPU model we introduce, analogous to the one-dimensional FPU system with quadratic nonlinearity. We also prove that the cubic KP-II equation is the limit of a model analogous to a one-dimensional FPU system with cubic nonlinearity. Further we study whether stability of line solitons in the KP-II equation extends to stability of one-dimensional FPU solitary waves in the two-dimensional lattices. / Thesis / Doctor of Philosophy (PhD)

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.