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Dareiotis, Anastasios Constantinos. "Stochastic partial differential and integro-differential equations." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/14186.
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In this work we present some new results concerning stochastic partial differential and integro-differential equations (SPDEs and SPIDEs) that appear in non-linear filtering. We prove existence and uniqueness of solutions of SPIDEs, we give a comparison principle and we suggest an approximation scheme for the non-local integral operators. Regarding SPDEs, we use techniques motivated by the work of De Giorgi, Nash, and Moser, in order to derive global and local supremum estimates, and a weak Harnack inequality.
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Ranner, Thomas. "Computational surface partial differential equations." Thesis, University of Warwick, 2013. http://wrap.warwick.ac.uk/57647/.
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Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties.
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Fedrizzi, Ennio. "Partial differential equations and noise." Paris 7, 2012. http://www.theses.fr/2012PA077176.
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Dans ce travail, nous présentons quelques exemples des effets du bruit sur la solution d'une équation aux dérivées partielles dans trois contextes différents. Nous examinons d'abord deux équations aux dérivées partielles non linéaires dispersives, l'équation de Schrodinger non linéaire et l'équation de Korteweg - de | Vries. Nous analysons les effets d'une condition initiale aléatoire sur certaines solutions spéciales, les ! solitons. Le deuxième cas considéré est une équation aux dérive��es partielles linéaire, l'équation d'onde, avec conditions initiales aléatoires. Nous montrons qu'avec des conditions initiales aléatoires particulières c'est possible de réduire considérablement les coûts de stockage des données et de calcul d'un algorithme pour résoudre un problème inverse basé sur les mesures de la solution de cette équation au bord du domaine. Enfin, le troisième exemple considéré est celui de l'équation de transport linéaire avec un terme de dérive singulière. Nous allons montrer que l'ajout d'un terme de bruit multiplicatif interdit l'explosion | des solutions, et cela sous des hypothèses très faibles pour lesquelles dans le cas déterministe on peut avoir l'explosion de la solution à temps finiIn this work we present examples of the effects of noise on the solution of a partial differential equation in three different settings. We first consider random initial conditions for two nonlinear dispersive partial differential equations, the nonlinear Schrodinger equation and the Korteweg - de Vries equation, and analyze their effects on some special solutions, the soliton solutions. The second case considered is a linear PDE, the wave equation, with random initial conditions. We show that special random initial conditions allow to I substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, where we will show that the addition of a multiplicative noise term forbids the blow up of solutions, under very weak hypothesis for which we have finite-time blow up of solutions in the deterministic case
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Tarkhanov, Nikolai. "Unitary solutions of partial differential equations." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2985/.
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Enstedt, Mattias. "Selected Topics in Partial Differential Equations." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-145763.
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This Ph.D. thesis consists of five papers and an introduction to the main topics of the thesis. In Paper I we give an abstract criteria for existence of multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators. In paper II we establish existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations for Coulomb systems along with properties of the solutions. In Paper III we establish existence of a ground state to the magnetic Hartree-Fock equations. In Paper IV we study the Choquard equation with general potentials (including quasirelativistic and magnetic versions of the equation) and establish existence of multiple solutions. In Paper V we prove that, under some assumptions on its nonmagnetic counterpart, a magnetic Schrödinger operator admits a representation with a positive Lagrange density and we derive consequences of this property.I den tryckta boken har förlag felaktigt angivits som Acta Universitatis Upsaliensis.
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Guo, Yujin. "Partial differential equations of electrostatic MEMS." Thesis, University of British Columbia, 2007. http://hdl.handle.net/2429/31315.
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Micro-Electromechanical Systems (MEMS) combine electronics with micro-size mechanical devices in the process of designing various types of microscopic machinery, especially those involved in conceiving and building modern sensors. Since their initial development in the 1980s, MEMS has revolutionized numerous branches of science and industry. Indeed, MEMS-based devices are now essential components of modern designs in a variety of areas, such as in commercial systems, the biomedical industry, space exploration, telecommunications, and other fields of applications. As it is often the case in science and technology, the quest for optimizing the attributes of MEMS devices according to their various uses, led to the development of mathematical models that try to capture the importance and the impact of the multitude of parameters involved in their design and production. This thesis is concerned with one of the simplest mathematical models for an idealized electrostatic MEMS, which was recently developed and popularized in a relatively recent monograph by J. Pelesko and D. Bernstein. These models turned out to be an incredibly rich source of interesting mathematical phenomena. The subject of this thesis is the mathematical analysis combined with numerical simulations of a nonlinear parabolic problem u[sub t] = Δu - [See Thesis for Equation] on a bounded domain of R[sup N] with Dirichlet boundary conditions. This equation models the dynamic deflection of a simple idealized electrostatic MEMS device, which consists of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage -represented here by λ- is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) may occur when it exceeds a certain critical value λ* (pull-in voltage). This creates a so-called pull-in instability which greatly affects the design of many devices. In order to achieve better MEMS design, the elastic membrane is fabricated with a spatially varying dielectric permittivity profile f (x). The first part of this thesis is focussed on the pull-in voltage λ* and the quantitative and qualitative description of the steady states of the equation. Applying analytical and numerical techniques, the existence of λ* is established together with rigorous bounds. We show the existence of at least one steady state when λ < λ* (and when λ = λ* in dimension N < 8), while none is possible for λ > λ*. More refined properties of steady states--such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results--are shown to depend on the dimension of the ambient space and on the permittivity profile. The second part of this thesis is devoted to the dynamic aspect of the parabolic equation. We prove that the membrane globally converges to its unique maximal negative steady-state when λ ≤ λ*, with a possibility of touchdown at infinite time when λ = λ* and N ≥ 8. On the other hand, if λ > λ* the membrane must touchdown at finite time T , which cannot take place at the location where the permittivity profile f ( x ) vanishes. Both larger pull-in distance and larger pull-in voltage can be achieved by properly tailoring the permittivity profile. We analyze and compare finite touchdown times by using both analytical and numerical techniques. When λ > λ*, some a priori estimates of touchdown behavior are established, based on which, we can give a refined description of touchdown profiles by adapting recently developed self-similarity methods as well as center manifold analysis. Applying various analytical and numerical methods, some properties of the touchdown set - such as compactness, location and shape - are also discussed for different classes of varying permittivity profiles f (x).Science, Faculty of
Mathematics, Department of
Graduate
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Keane, Therese Alison Mathematics & Statistics Faculty of Science UNSW. "Combat modelling with partial differential equations." Awarded By:University of New South Wales. Mathematics & Statistics, 2009. http://handle.unsw.edu.au/1959.4/43086.
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In Part I of this thesis we extend the Lanchester Ordinary Differential Equations and construct a new physically meaningful set of partial differential equations with the aim of more realistically representing soldier dynamics in order to enable a deeper understanding of the nature of conflict. Spatial force movement and troop interaction components are represented with both local and non-local terms, using techniques developed in biological aggregation modelling. A highly accurate flux limiter numerical method ensuring positivity and mass conservation is used, addressing the difficulties of inadequate methods used in previous research. We are able to reproduce crucial behaviour such as the emergence of cohesive density profiles and troop regrouping after suffering losses in both one and two dimensions which has not been previously achieved in continuous combat modelling. In Part II, we reproduce for the first time apparently complex cellular automaton behaviour with simple partial differential equations, providing an alternate mechanism through which to analyse this behaviour. Our PDE model easily explains behaviour observed in selected scenarios of the cellular automaton wargame ISAAC without resorting to anthropomorphisation of autonomous 'agents'. The insinuation that agents have a reasoning and planning ability is replaced with a deterministic numerical approximation which encapsulates basic motivational factors and demonstrates a variety of spatial behaviours approximating the mean behaviour of the ISAAC scenarios. All scenarios presented here highlight the dangers associated with attributing intelligent reasoning to behaviour shown, when this can be explained quite simply through the effects of the terms in our equations. A continuum of forces is able to behave in a manner similar to a collection of individual autonomous agents, and shows decentralised self-organisation and adaptation of tactics to suit a variety of combat situations. We illustrate the ability of our model to incorporate new tactics through the example of introducing a density tactic, and suggest areas for further research.
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Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.
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In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Lloyd, David J. B. "Localised solutions of partial differential equations." Thesis, University of Bristol, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.434765.
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Lorz, Alexander Stephan Richard. "Partial differential equations modelling biophysical phenomena." Thesis, University of Cambridge, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.609381.
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Markall, Graham. "Multilayered abstractions for partial differential equations." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/22175.
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How do we build maintainable, robust, and performance-portable scientific applications? This thesis argues that the answer to this software engineering question in the context of the finite element method is through the use of layers of Domain-Specific Languages (DSLs) to separate the various concerns in the engineering of such codes. Performance-portable software achieves high performance on multiple diverse hardware platforms without source code changes. We demonstrate that finite element solvers written in a low-level language are not performance-portable, and therefore code must be specialised to the target architecture by a code generation framework. A prototype compiler for finite element variational forms that generates CUDA code is presented, and is used to explore how good performance on many-core platforms in automatically-generated finite element applications can be achieved. The differing code generation requirements for multi- and many-core platforms motivates the design of an additional abstraction, called PyOP2, that enables unstructured mesh applications to be performance-portable. We present a runtime code generation framework comprised of the Unified Form Language (UFL), the FEniCS Form Compiler, and PyOP2. This toolchain separates the succinct expression of a numerical method from the selection and generation of efficient code for local assembly. This is further decoupled from the selection of data formats and algorithms for efficient parallel implementation on a specific target architecture. We establish the successful separation of these concerns by demonstrating the performance-portability of code generated from a single high-level source code written in UFL across sequential C, CUDA, MPI and OpenMP targets. The performance of the generated code exceeds the performance of comparable alternative toolchains on multi-core architectures.
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Hu, Jiaxin. "Nonlinear partial differential equations on fractals." Thesis, University of St Andrews, 2001. http://hdl.handle.net/10023/15180.
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The study of nonlinear partial differential equations on fractals is a burgeoning inter-disciplinary topic, allowing dynamic properties on fractals to be investigated. In this thesis we will investigate nonlinear PDEs of three basic types on bounded and unbounded fractals. We first review the definition of post-critically finite (p.c.f.) self-similar fractals with regular harmonic structure. A Dirichlet form exists on such a fractal; thus we may define a weak version of the Laplacian. The Sobolev-type inequality, established on p.c.f. self-similar fractals satisfying the separation condition, plays a crucial role in the analysis of PDEs on p.c.f. self-similar fractals. We use the classical approach to study the linear eigenvalue problem on p.c.f. self-similar fractals, which depends on the Sobolev-type inequality. Fundamental solutions such as Green's function, wave propagator and heat kernel are then explicitly expressed in terms of eigenvalues and eigenfunctions. The main aim of the thesis is to study nonlinear PDEs on fractals. We begin with nonlinear elliptic equations on p.c.f. self-similar fractals. We prove the existence of non-trivial solutions to elliptic equations with zero Dirichlet boundary conditions using the mountain pass theorem and the saddle point theorem. For nonlinear wave equations on p.c.f. self-similar fractals, we show the existence of global solutions for appropriate initial and boundary data. We also examine blow up at finite time which may occur for certain initial data. Finally, we consider nonlinear diffusion equations on p.c.f. self-similar fractals and unbounded fractals. Using the upper-lower solution technique, we prove the global existence of solutions of the nonlinear diffusion equation with initial value and boundary conditions on p.c.f. self-similar fractals. For unbounded fractals, starting with a heat kernel satisfying certain assumptions, we prove that the diffusion equation with a nonlinear term of the form up possesses a global solution if the initial data is small and p > 1 + ds/2, while solutions blow up if p ≤ 1 + ds/2 even for small initial data, where dg is the spectral dimension of the fractal. We investigate smoothness and Holder continuity of solutions when they exist.
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Liu, Yichen. "Optimization problems in partial differential equations." Thesis, University of Liverpool, 2015. http://livrepository.liverpool.ac.uk/2015545/.
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The primary objective of this research is to investigate various optimization problems connected with partial differential equations (PDE). In chapter 2, we utilize the tool of tangent cones from convex analysis to prove the existence and uniqueness of a minimization problem. Since the admissible set considered in chapter 2 is a suitable convex set in $L^\infty(D)$, we can make use of tangent cones to derive the optimality condition for the problem. However, if we let the admissible set to be a rearrangement class generated by a general function (not a characteristic function), the method of tangent cones may not be applied. The central part of this research is Chapter 3, and it is conducted based on the foundation work mainly clarified by Geoffrey R. Burton with his collaborators near 90s, see [7, 8, 9, 10]. Usually, we consider a rearrangement class (a set comprising all rearrangements of a prescribed function) and then optimize some energy functional related to partial differential equations on this class or part of it. So, we call it rearrangement optimization problem (ROP). In recent years this area of research has become increasingly popular amongst mathematicians for several reasons. One reason is that many physical phenomena can be naturally formulated as ROPs. Another reason is that ROPs have natural links with other branches of mathematics such as geometry, free boundary problems, convex analysis, differential equations, and more. Lastly, such optimization problems also offer very challenging questions that are fascinating for researchers, see for example [2]. More specifically, Chapter 2 and Chapter 3 are prepared based on four papers [24, 40, 41, 42], mainly in collaboration with Behrouz Emamizadeh. Chapter 4 is inspired by [5]. In [5], the existence and uniqueness of solutions of various PDEs involving Radon measures are presented. In order to establish a connection between rearrangements and PDEs involving Radon measures, the author try to investigate a way to extend the notion of rearrangement of functions to rearrangement of Radon measures in Chapter 4.
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Athreya, Siva. "Probability and semilinear partial differential equations /." Thesis, Connect to this title online; UW restricted, 1998. http://hdl.handle.net/1773/5799.
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Morian, Christina. "Partial differential equations on time scales /." free to MU campus, to others for purchase, 2000. http://wwwlib.umi.com/cr/mo/fullcit?p9974665.
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Ugail, Hassan, and S. Kirmani. "Shape reconstruction using partial differential equations." World Scientific and Engineering Academy and Society (WSEAS), 2006. http://hdl.handle.net/10454/2645.
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We present an efficient method for reconstructing complex geometry using an elliptic Partial Differential Equation (PDE) formulation. The integral part of this work is the use of three-dimensional curves within the physical space which act as boundary conditions to solve the PDE. The chosen PDE is solved explicitly for a given general set of curves representing the original shape and thus making the method very efficient. In order to improve the quality of results for shape representation we utilize an automatic parameterization scheme on the chosen curves. With this formulation we discuss our methodology for shape representation using a series of practical examples.
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Thompson, Jeremy R. (Jeremy Ray). "Physical Motivation and Methods of Solution of Classical Partial Differential Equations." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277898/.
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We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
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Nguyen, Thieu-Huy. "Functional partial differential equations and evolution semigroups." [S.l.] : [s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=973911344.
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Aksoy, Umit. "Schwarz Problem For Complex Partial Differential Equations." Phd thesis, METU, 2006. http://etd.lib.metu.edu.tr/upload/3/12607977/index.pdf.
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This study consists of four chapters. In the first chapter we
give some historical background of the problem, basic definitions
and properties. Basic integral operators of complex analysis and
and Schwarz problem for model equations are presented in Chapter
2. Chapter 3 is devoted to the investigation of the properties of
a class of strongly singular integral operators. In the last
chapter we consider the Schwarz boundary value problem for the
general partial complex differential equations of higher order.
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Keras, Sigitas. "Numerical methods for parabolic partial differential equations." Thesis, University of Cambridge, 1997. https://www.repository.cam.ac.uk/handle/1810/251611.
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Matetski, Kanstantsin. "Discretisations of rough stochastic partial differential equations." Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/81460/.
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This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use the theory of (controlled) rough paths to define a solution for one-dimensional stochastic PDEs of Burgers type driven by an additive space-time white noise. We prove that natural numerical approximations of these equations converge to the solution of a corrected continuous equation and that their optimal convergence rate in the uniform topology (in probability) is arbitrarily close to 1/2 . In the second part of the thesis we develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical �43 model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the �43 measure is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.
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Wan, Andy Tak Shik. "Finding conservation laws for partial differential equations." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/28135.
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In this thesis, we discuss systematic methods of finding conservation laws for systems of partial differential equations (PDEs). We first review the direct method of finding conservation laws. In order to use the direct method, one first seeks a set of conservation law multipliers so that a linear combination of the PDEs with the multipliers will yield a divergence expression. Once a set of conservation law multipliers is determined, one proceeds to find the fluxes of the conservation law.
As the solution to the problem of finding conservation law multipliers is well-understood, in this thesis we focus on constructing the fluxes assuming the knowledge of a set of conservation law multipliers. First, we derive a new method called the flux equation method and show that, in general, fluxes can be found by at most computing a line integral. We show that the homotopy integral formula is a special case of the line integral formula obtained from the flux equations. We also show how the line integral formula can be simplified in the presence of a point symmetry of the PDE system and of the set of conservation law multipliers. By examples, we illustrate that the flux equation method can derive fluxes which would be otherwise difficult to find. We also review existing known methods of finding fluxes and make comparison with the flux equation method.
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Bratsos, A. G. "Numerical solutions of nonlinear partial differential equations." Thesis, Brunel University, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332806.
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Brackenridge, Kenneth. "Multigrid solution of elliptic partial differential equations." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260756.
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Elton, Daniel M. "Hyperbolic partial differential equations with singular coefficients." Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389210.
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Sharp, Benjamin G. "Compensation phenomena in geometric partial differential equations." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/50026/.
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In this thesis we present optimal and improved estimates for systems of critical elliptic PDE which arise as generalisations of natural geometric problems. We provide optimal regularity and compactness results for ‘Rivière’s equation’ for two dimensional domains via a new decay estimate, andwe exhibit examples to showthat the results are sharp. These results are presented in chapters 4 and 5. Such estimates generalise and improve known results in the classical setting. In chapter 6 we improve the known regularity for the higher dimensional theory introduced by Rivière-Struwe leading to better estimates for solutions in this case. Such estimates in particular lead to an easy proof for the regularity for stationary harmonic maps. We also present (in chapter 7) sharp results for a complex system of PDE, a consequence of which is a short proof of the full regularity for weakly harmonic maps from a Riemann surface into a closed Riemannian manifold.
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Nguyen, Thi Nhu Thuy. "Uniform controllability of discrete partial differential equations." Phd thesis, Université d'Orléans, 2012. http://tel.archives-ouvertes.fr/tel-00919255.
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In this thesis, we study uniform controllability properties of semi-discrete approximations for parabolic systems. In a first part, we address the minimization of the Lq-norm (q > 2) of semidiscrete controls for parabolic equation. Our goal is to overcome the limitation of [LT06] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform observability property also holds in Lq (q > 2) even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls is provided. The study of Lq optimality in the first part is in a general context. However, the discrete observability inequalities that are obtained are not so precise than the ones derived then with Carleman estimates. In a second part, in the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi discrete version of the parabolic operator @t − @x(c@x) which allows one to derive observability inequalities that are far more precise. Here we consider in case that the diffusion coefficient has a jump which yields a transmission problem formulation. Consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of linear and semi-linear parabolic equations.
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Wood, J. "Some problems associated with partial differential equations." Thesis, Bucks New University, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.356214.
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Sheng, Qin. "Solving partial differential equations by exponential splitting." Thesis, University of Cambridge, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.317937.
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Spantini, Alessio. "Preconditioning techniques for stochastic partial differential equations." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/82507.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2013.This thesis was scanned as part of an electronic thesis pilot project.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 149-155).
This thesis is about preconditioning techniques for time dependent stochastic Partial Differential Equations arising in the broader context of Uncertainty Quantification. State-of-the-art methods for an efficient integration of stochastic PDEs require the solution field to lie on a low dimensional linear manifold. In cases when there is not such an intrinsic low rank structure we must resort on expensive and time consuming simulations. We provide a preconditioning technique based on local time stretching capable to either push or keep the solution field on a low rank manifold with substantial reduction in the storage and the computational burden. As a by-product we end up addressing also classical issues related to long time integration of stochastic PDEs.
by Alessio Spantini.
S.M.
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Bagherinakhjavanlo, Bashir. "Partial differential equations for medical image segmentation." Thesis, Kingston University, 2014. http://eprints.kingston.ac.uk/29993/.
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This study is concerned with image segmentation techniques using mathematical models based on elastic curves or surfaces defined within an image domain that can move under the influence of a defined energy. These active contour models use internal and external forces generated from curves or surfaces in 2D and 3D image data. The algorithms that measure these energies must cope with non-homogeneous objects and regions, low contrast boundaries and image noise. It investigates level sets, which employ an energy formulation defined by partial differential equations (PDEs), that are sensitive to weak boundaries yet are robust to noise whilst maintaining computational stability. The methodology is evaluated using medical imagery, which commonly suffer from high levels of noise, blur and exhibit weak boundaries between different types of adjacent tissue. An energy based on PDEs has been used to evolve an image contour from an initial guess using image forces derived from region properties to drive the search to locate the boundaries of the desired objects that includes the maximum and minimum curvature function to enable length shortening in the curve evolution. It is applied to both 2D and 3D CTA datasets for the segmentation of abdominal and thoracic aortic aneurysm (AAA&TAA). For some image data the methodology can be initialised automatically using a contour detected after intensity thresholding. Non-homogeneous regions require a manual initialisation that crosses the boundary between the aorta and thrombus. Sussman’s re-initialization has been used in the 3D algorithm to maintain stability in the evolving boundary, as a consequence of the re-formulation from the continuous to the discrete domain. A hybrid method is developed that combines a novel approach using region information (i.e. intensities inside and outside the object) and edge information, computed using a diffusion-based approach integrated into a level set formulation, to guide the initial curve to the object boundary by finding strong edges with local minima. Boundary information supports finding a local minimum length curve on evaluation and only examines data on the contour. Using Green’s theorem, region information is be used to address the boundary leakage problem, as it minimizes the energy related to the whole image data and the moving curve is stopped by strong gradients on the borders of objects. Finally, a Gabor filter has been integrated into the hybrid algorithm to enhance the image and support the detection of textured regions of interest. The method is evaluated on both synthetic and real image data and compared with the region-based methods of Chan-Vese and Li et al.
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Prerapa, Surya Mohan. "Projection schemes for stochastic partial differential equations." Thesis, University of Southampton, 2009. https://eprints.soton.ac.uk/342800/.
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The focus of the present work is to develop stochastic reduced basis methods (SRBMs) for solving partial differential equations (PDEs) defined on random domains and nonlinear stochastic PDEs (SPDEs). SRBMs have been extended in the following directions: Firstly, an h-refinement strategy referred to as Multi-Element-SRBMs (ME-SRBMs) is developed for local refinement of the solution process. The random space is decomposed into subdomains where SRBMs are employed in each subdomain resulting in local response statistics. These local statistics are subsequently assimilated to compute the global statistics. Two types of preconditioning strategies namely global and local preconditioning strategies are discussed due to their merits such as degree of parallelizability and better convergence trends. The improved accuracy and convergence trends of ME-SRBMs are demonstrated by numerical investigation of stochastic steady state elasticity and stochastic heat transfer applications. The second extension involves the development of a computational approach employing SRBMs for solving linear elliptic PDEs defined on random domains. The key idea is to carry out spatial discretization of the governing equations using finite element (FE) methods and mesh deformation strategies. This results in a linear random algebraic system of equations whose coefficients of expansion can be computed nonintrusively either at the element or the global level. SRBMs are subsequently applied to the linear random algebraic system of equations to obtain the response statistics. We establish conditions that the input uncertainty model must satisfy to ensure the well-posedness of the problem. The proposed formulation is demonstrated on two and three dimensional model problems with uncertain boundaries undergoing steady state heat transfer. A large scale study involving a three-dimensional gas turbine model with uncertain boundary, has been presented in this context. Finally, a numerical scheme that combines SRBMs with the Picard iteration scheme is proposed for solving nonlinear SPDEs. The governing equations are linearized using the response process from the previous iteration and spatially discretized. The resulting linear random algebraic system of equations are solved to obtain the new response process which acts as a guess for the next iteration. These steps of linearization, spatial discretization, solving the system of equations and updating the current guess are repeated until the desired accuracy is achieved. The effectiveness and the limitations of the formulation are demonstrated employing numerical studies in nonlinear heat transfer and the one-dimensional Burger’s equation.
33
Williamson, Rosemary Anne. "Numerical solution of hyperbolic partial differential equations." Thesis, University of Cambridge, 1985. https://www.repository.cam.ac.uk/handle/1810/278503.
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34
Osman, Abdusslam. "3D modelling using partial differential equations (PDEs)." Thesis, Sheffield Hallam University, 2014. http://shura.shu.ac.uk/20153/.
Full textAbstract:
Partial differential equations (PDEs) are used in a wide variety of contexts in computer science ranging from object geometric modelling to simulation of natural phenomena such as solar flares, and generation of realistic dynamic behaviour in virtual environments including variables such as motion, velocity and acceleration. A major challenge that has occupied many players in geometric modelling and computer graphics is the accurate representation of human facial geometry in 3D. The acquisition, representation and reconstruction of such geometries are crucial for an extensive range of uses, such as in 3D face recognition, virtual realism presentations, facial appearance simulations and computer-based plastic surgery applications among others. The principle aim of this thesis should be to tackle methods for the representation and reconstruction of 3D geometry of human faces depending on the use of partial differential equations and to enable the compression of such 3D data for faster transmission over the Internet. The actual suggested techniques are based on sampling surface points at the intersection of horizontal and vertical mesh cutting planes. The set of sampled points contains the explicit structure of the cutting planes with three important consequences: 1) points in the plane can be defined as a one dimensional signal and are thus, subject to a number of compression techniques; 2) any two mesh cutting planes can be used as PDE boundary conditions in a rectangular domain; and 3) no connectivity information needs to be coded as the explicit structure of the vertices in 3D renders surface triangulation a straightforward task. This dissertation proposes and demonstrates novel algorithms for compression and uncompression of 3D meshes using a variety of techniques namely polynomial interpolation, Discrete Cosine Transform, Discrete Fourier Transform, and Discrete Wavelet Transform in connection with partial differential equations. In particular, the effectiveness of the partial differential equations based method for 3D surface reconstruction is shown to reduce the mesh over 98.2% making it an appropriate technique to represent complex geometries for transmission over the network.
35
Dowie, Ellen. "Rational solutions of nonlinear partial differential equations." Thesis, University of Kent, 2018. https://kar.kent.ac.uk/66565/.
Full textAbstract:
The work in this thesis considers rational solutions of nonlinear partial differential equations formed from polynomials. The main work will be on the Boussinesq equation and the Kadomtsev-Petviashvili-I (KP-I) equation, the nonlinear Schroedinger equation will also be included for completeness. Rational solutions of the Boussinesq equation model rogue wave behaviour. These solutions are shown to be highly structured which, it is hypothesised, is due to the inherent structure and form of integrable differential equations. Rogue wave solutions have been observed in equations such as the nonlinear Schr\"odinger equation, KP equation and the Boussinesq equation, to name but a few. By examining the form of these solutions and considering the behaviour of the roots, the aim is to establish the behaviour of this family of solutions. All solutions are bounded and real. Additionally, since a generating function for the KP equation solutions already exists, a characterisation of the solutions will be made along with an attempt at understanding the current generating function in order to improve its adaptability. Links between solutions of the three equations will be shown as well as a function that can solve all three equations subject to certain criteria on the parameters.
36
Ugail, Hassan. "Generalized partial differential equations for interactive design." World Scientific Publishing Company, 2007. http://hdl.handle.net/10454/2642.
Full textAbstract:
This paper presents a method for interactive design by means of extending the PDE
based approach for surface generation. The governing partial differential equation is
generalized to arbitrary order allowing complex shapes to be designed as single patch
PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating
the geometry of shape that satisfying an arbitrary set of boundary conditions.
Both the boundary conditions which are defined as curves in 3-space and the spine of the
corresponding PDE are utilized as interactive design tools for creating and manipulating
geometry intuitively. In order to facilitate interactive design in real time, a compact
analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme
even in the case of general boundary conditions satisfies exactly the boundary conditions
where the resulting surface has an closed form representation allowing real time
shape manipulation. In order to enable users to appreciate the powerful shape design
and manipulation capability of the method, we present a set of practical examples.
37
McKay, Steven M. "Brownian Motion Applied to Partial Differential Equations." DigitalCommons@USU, 1985. https://digitalcommons.usu.edu/etd/6992.
Full textAbstract:
This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.
38
Zhang, Qi. "Stationary solutions of stochastic partial differential equations and infinite horizon backward doubly stochastic differential equations." Thesis, Loughborough University, 2008. https://dspace.lboro.ac.uk/2134/34040.
Full textAbstract:
In this thesis we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. For this, we prove the existence and uniqueness of the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued solutions of BDSDEs with Lipschitz nonlinear term on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary weak solutions (independent of any initial value) of SPDEs. Also the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued BDSDE with non-Lipschitz term is considered. Moreover, we verify the time and space continuity of solutions of real-valued BDSDEs, so obtain the stationary stochastic viscosity solutions of real-valued SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.
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Athanasopoulos, Michael. "Modelling and Animation using Partial Differential Equations. Geometric modelling and computer animation of virtual characters using elliptic partial differential equations." Thesis, University of Bradford, 2011. http://hdl.handle.net/10454/5437.
Full textAbstract:
This work addresses various applications pertaining to the design, modelling and animation of parametric surfaces using elliptic Partial Differential Equations (PDE) which are produced via the PDE method. Compared with traditional surface generation techniques, the PDE method is an effective technique that can represent complex three-dimensional (3D) geometries in terms of a relatively small set of parameters. A PDE-based surface can be produced from a set of pre-configured curves that are used as the boundary conditions to solve a number of PDE. An important advantage of using this method is that most of the information required to define a surface is contained at its boundary. Thus, complex surfaces can be computed using only a small set of design parameters. In order to exploit the advantages of this methodology various applications were developed that vary from the interactive design of aircraft configurations to the animation of facial expressions in a computer-human interaction system that utilizes an artificial intelligence (AI) bot for real time conversation. Additional applications of generating cyclic motions for PDE based human character integrated in a Computer-Aided Design (CAD) package as well as developing techniques to describe a given mesh geometry by a set of boundary conditions, required to evaluate the PDE method, are presented. Each methodology presents a novel approach for interacting with parametric surfaces obtained by the PDE method. This is due to the several advantages this surface generation technique has to offer. Additionally, each application developed in this thesis focuses on a specific target that delivers efficiently various operations in the design, modelling and animation of such surfaces.
40
Kadamani, Sami M. "USFKAD: An Expert System For Partial Differential Equations." [Tampa, Fla.] : University of South Florida, 2005. http://purl.fcla.edu/fcla/etd/SFE0001144.
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Ozmen, Neslihan. "Image Segmentation And Smoothing Via Partial Differential Equations." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/12610395/index.pdf.
Full textAbstract:
In image processing, partial differential equation (PDE) based approaches have been extensively used in segmentation and smoothing applications. The Perona-Malik nonlinear diffusion
model is the first PDE based method used in the image smoothing tasks. Afterwards the classical Mumford-Shah model was developed to solve both image segmentation and smoothing problems and it is based on the minimization of an energy functional. It has numerous application areas such as edge detection, motion analysis, medical imagery, object tracking etc. The model is a way of finding a partition of an image by using a piecewise smooth representation of the image. Unfortunately numerical procedures for minimizing the Mumford-Shah functional have some difficulties because the problem is non convex and it has numerous local minima, so approximate approaches have been proposed. Two such methods are the Ambrosio-Tortorelli approximation and the Chan-Vese active contour method. Ambrosio and Tortorelli have developed a practical numerical implementation of the Mumford-Shah model which based on an elliptic approximation of the original functional. The Chan-Vese model is a piecewise constant generalization of the Mumford-Shah functional and it is based on level set formulation. Another widely used image segmentation technique is the &ldquoActive Contours (Snakes)&rdquo
model and it is correlated with the Chan-Vese model. In this study, all these approaches have been examined in detail. Mathematical and numerical analysis of these models are studied and some experiments are performed to compare their performance.
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Le, Gia Quoc Thong. "Approximation of linear partial differential equations on spheres." Texas A&M University, 2003. http://hdl.handle.net/1969.1/22.
Full textAbstract:
The theory of interpolation and approximation of solutions to
differential and integral equations on spheres has attracted
considerable interest in recent years; it has also been applied
fruitfully in fields such as physical geodesy, potential theory,
oceanography, and meteorology.
In this dissertation we study the approximation of linear
partial differential equations on spheres, namely a class of
elliptic partial differential equations
and the heat equation on the unit sphere.
The shifts of a spherical basis
function are used to construct the approximate solution. In the
elliptic case, both the finite element method and the collocation method
are discussed. In the heat equation, only the collocation method is
considered. Error estimates in the supremum norms and the Sobolev norms
are obtained when certain regularity conditions are imposed on
the spherical basis functions.
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Eftang, Jens Lohne. "Reduced basis methods for parametrized partial differential equations." Doctoral thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-12550.
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Chow, Tanya L. M., of Western Sydney Macarthur University, and Faculty of Business and Technology. "Systems of partial differential equations and group methods." THESIS_FBT_XXX_Chow_T.xml, 1996. http://handle.uws.edu.au:8081/1959.7/43.
Full textAbstract:
This thesis is concerned with the derivation of similarity solutions for one-dimensional coupled systems of reaction - diffusion equations, a semi-linear system and a one-dimensional tripled system. The first area of research in this thesis involves a coupled system of diffusion equations for the existence of two distinct families of diffusion paths. Constructing one-parameter transformation groups preserving the invariance of this system of equations enables similarity solutions for this coupled system to be derived via the classical and non-classical procedures. This system of equation is the uncoupled in the hope of recovering further similarity solutions for the system. Once again, one-parameter groups leaving the uncoupled system invariant are obtained, enabling similarity solutions for the system to be elicited. A one-dimensional pattern formation in a model of burning forms the next component of this thesis. The primary focus of this area is the determination of similarity solutions for this reaction - diffusion system by means of one-parameter transformation group methods. Consequently, similarity solutions which are a generalisation of the solutions of the one-dimensional steady equations derived by Forbes are deduced. Attention in this thesis is then directed toward a semi-linear coupled system representing a predator - prey relationship. Two approaches to solving this system are made using the classical procedure, leading to one-parameter transformation groups which are instrumental in elicting the general similarity solution for this system. A triple system of equations representing a one-dimensional case of diffusion in the presence of three diffusion paths constitutes the next theme of this thesis. In association with the classical and non-classical procedures, the derivation of one-parameter transformation groups leaving this system invariant enables similarity solutions for this system to be deduced. The final strand of this thesis involves a one- dimensional case of the general linear system of coupled diffusion equations with cross-effects for which one-parameter transformation group methods are once more employed. The one-parameter groups constructed for this system prove instrumental in enabling the attainment of similarity solutions for this system to be accomplishedFaculty of Business and Technology
45
Chulkov, Sergei. "Topics in analytic theory of partial differential equations /." Stockholm : Dept. of mathematics, Stockholm university, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-782.
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Rangelova, Marina. "Error estimation for fourth order partial differential equations." Ann Arbor, Mich. : ProQuest, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3258675.
Full textAbstract:
Thesis (Ph.D. in Computational and Applied Mathematics)--S.M.U., 2007.Title from PDF title page (viewed Mar. 18, 2008). Source: Dissertation Abstracts International, Volume: 68-03, Section: B, page: 1675. Adviser: Peter Moore. Includes bibliographical references.
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Becker, Dulcenéia. "Parallel unstructured solvers for linear partial differential equations." Thesis, Cranfield University, 2006. http://hdl.handle.net/1826/4140.
Full textAbstract:
This thesis presents the development of a parallel algorithm to solve symmetric
systems of linear equations and the computational implementation of a parallel
partial differential equations solver for unstructured meshes. The proposed
method, called distributive conjugate gradient - DCG, is based on a single-level
domain decomposition method and the conjugate gradient method to obtain a
highly scalable parallel algorithm.
An overview on methods for the discretization of domains and partial differential
equations is given. The partition and refinement of meshes is discussed and
the formulation of the weighted residual method for two- and three-dimensions
presented. Some of the methods to solve systems of linear equations are introduced,
highlighting the conjugate gradient method and domain decomposition
methods. A parallel unstructured PDE solver is proposed and its actual implementation
presented. Emphasis is given to the data partition adopted and the
scheme used for communication among adjacent subdomains is explained. A series
of experiments in processor scalability is also reported.
The derivation and parallelization of DCG are presented and the method validated
throughout numerical experiments. The method capabilities and limitations
were investigated by the solution of the Poisson equation with various source
terms. The experimental results obtained using the parallel solver developed as
part of this work show that the algorithm presented is accurate and highly scalable,
achieving roughly linear parallel speed-up in many of the cases tested.
48
Wu, Xiaoming. "Partial differential equations with applications to wave propagation." Thesis, University of Newcastle Upon Tyne, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239573.
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Davidson, Bryan Duncan. "Recursive projection for semi-linear partial differential equations." Thesis, University of Bristol, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.294932.
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Hu, Jing. "Solution of partial differential equations using reconfigurable computing." Thesis, University of Birmingham, 2011. http://etheses.bham.ac.uk//id/eprint/1655/.
Full textAbstract:
This research undergone is an inter-disciplinary project with the Civil Engineering Department, which focuses on acceleration of the numerical solutions of Partial differential equations (PDEs) describing continuous solid bodies (e.g. a dam or an aircraft wing). Numerical techniques for solutions to PDEs are generally computationally demanding and data intensive. One approach to acceleration of their numerical solutions is to use FPGA based reconfigurable computing boards. The aim of this research is to investigate the features of various algorithms for the numerical solution of Laplace’s equation (the targeted PDE problem) in order to establish how well they can be mapped onto reconfigurable hardware accelerators. Finite difference methods and finite element methods are used to solve the PDE and they are characterized in terms of their operation count, sequential and parallel content, communication requirements and amenability to domain decomposition. These are then matched to abstract models of the capabilities of FPGA-based reconfigurable computing platforms. The performance of different algorithms is compared and discussed. The resulting hardware design will be suitable for platforms ranging from single board add-ins for general PCs to reconfigurable supercomputers such as the Cray XD1. However, the principal aim in this research has been to seek methods that perform well on low-cost platforms. In this thesis, several algorithms of solving the PDE are implemented on FPGA-based reconfigurable computing systems. Domain decomposition is used to take advantage of the embedded memory within the FPGA, which is used as a cache to store the data for the current sub-domain in order to eliminate communication and synchronization delays between the sub-domains and to support a very large number of parallel pipelines. Using Fourier decomposition, the 32bit floating-point hardware/software design can achieve a speed-up of 38 for 3-D 256x256x256 finite difference method on a single FPGA board (based on a Virtex2V6000 FPGA) compared to a software solution implemented in the same algorithm on a 2.4 GHz Pentium 4 PC which supports SSE2. The 32 bit floating-point hardware-software coprocessor for the 3D tetrahedral finite element problem with 48,000 elements using the preconditioned conjugate gradient method can achieve a speed-up of 40 for a single FPGA board (based on a Virtex4VLX160 FPGA) compared to a software solution.