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McGee, Daniel Lee Jr. "Applications of neural networks to partial differential equations." Diss., The University of Arizona, 1994. http://hdl.handle.net/10150/186885.
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Efforts to improve hyperthermia treatments of cancer have motivated this research. Three fundamental goals that have have been defined are the identification of tissue parameter values, the prediction of tissue temperature behavior, and the control of tissue temperature behavior during hyperthermia treatments. This dissertation consists of three independent studies plying neural networks to hyperthermia. These studies examine neural network based systems and how these systems apply to the identification, prediction, and control of tissue temperature behavior during a hyperthermia treatment. The first study examines the ability of neural networks to estimate the tissue perfusion values and the minimum temperature associated with numerically calculated steady state hyperthermia temperature fields. This study utilizes a limited number of measured temperatures within this field. We show that a hierarchical system of neural networks consisting of a first layer of pattern recognizing neural networks and a second layer of hypersurface reconstructing neural networks is capable of estimating these variables within a selected error tolerance. Additional results indicate that if the locations of the measured temperatures within the temperature field are selected effectively, the hierarchical system of neural networks can tolerate a moderate level of model mismatch. The second study examines the feasibility of using a system of neural networks to estimate the Laplacian values at the sensor locations in a tissue sample by using the measured temperature data. By combining these neural network Laplacian estimates with the measured data, numerical values for three different components (conduction, advection, and external power) can be obtained at the sensor locations. These thermal terms can then be used in a model of the tissue to predict future temperatures. Using only measured data collected early in the treatment, we show that recursive application of this estimation process can provide accurate predictions of the temperature behavior at the sensor locations of a tissue sample for the duration of a treatment. This system was also found to be robust with respect to the addition of white noise to both the sensor measurements and the amount of power delivered to the sensor locations. The final study explores the ability of a neural network based predictive system to formulate a predictive control strategy. We show that with certain restrictions on the power deposition patterns, desired temperature trajectories within the tissue model can be achieved with a neural network based predictive control system.
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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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Seiler, Werner Markus. "Analysis and application of the formal theory of partial differential equations." Thesis, Lancaster University, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.238979.
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Wright, S. J. "The application of transmission-line modelling implicit and hybrid algorithms to electromagnetic problems." Thesis, University of Nottingham, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384746.
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Mu, Tingshu. "Backward stochastic differential equations and applications : optimal switching, stochastic games, partial differential equations and mean-field." Thesis, Le Mans, 2020. http://www.theses.fr/2020LEMA1023.
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Cette thèse est relative aux Equations Différentielles Stochastique Rétrogrades (EDSRs) réfléchies avec deux obstacles et leurs applications aux jeux de switching de somme nulle, aux systèmes d’équations aux dérivées partielles, aux problèmes de mean-field. Il y a deux parties dans cette thèse. La première partie porte sur le switching optimal stochastique et est composée de deux travaux. Dans le premier travail, nous montrons l’existence de la solution d’un système d’EDSR réfléchies à obstacles bilatéraux interconnectés dans le cadre probabiliste général. Ce problème est lié à un jeu de switching de somme nulle. Ensuite nous abordons la question de l’unicité de la solution. Et enfin nous appliquons les résultats obtenus pour montrer que le système d’EDP associé à une unique solution au sens viscosité, sans la condition de monotonie habituelle. Dans le second travail, nous considérons aussi un système d’EDSRs réfléchies à obstacles bilatéraux interconnectés dans le cadre markovien. La différence avec le premier travail réside dans le fait que le switching ne s’opère pas de la même manière. Cette fois-ci quand le switching est opéré, le système est mis dans l’état suivant importe peu lequel des joueurs décide de switcher. Cette différence est fondamentale et complique singulièrement le problème de l’existence de la solution du système. Néanmoins, dans le cadre markovien nous montrons cette existence et donnons un résultat d’unicité en utilisant principalement la méthode de Perron. Ensuite, le lien avec un jeu de switching spécifique est établi dans deux cadres. Dans la seconde partie nous étudions les EDSR réfléchies unidimensionnelles à deux obstacles de type mean-field. Par la méthode du point fixe, nous montrons l’existence et l’unicité de la solution dans deux cadres, en fonction de l’intégrabilité des donnéesThis thesis is related to Doubly Reflected Backward Stochastic Differential Equations (DRBSDEs) with two obstacles and their applications in zero-sum stochastic switching games, systems of partial differential equations, mean-field problems.There are two parts in this thesis. The first part deals with optimal stochastic switching and is composed of two works. In the first work we prove the existence of the solution of a system of DRBSDEs with bilateral interconnected obstacles in a probabilistic framework. This problem is related to a zero-sum switching game. Then we tackle the problem of the uniqueness of the solution. Finally, we apply the obtained results and prove that, without the usual monotonicity condition, the associated PDE system has a unique solution in viscosity sense. In the second work, we also consider a system of DRBSDEs with bilateral interconnected obstacles in the markovian framework. The difference between this work and the first one lies in the fact that switching does not work in the same way. In this second framework, when switching is operated, the system is put in the following state regardless of which player decides to switch. This difference is fundamental and largely complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result by the Perron’s method. Later on, two particular switching games are analyzed.In the second part we study a one-dimensional Reflected BSDE with two obstacles of mean-field type. By the fixed point method, we show the existence and uniqueness of the solution in connection with the integrality of the data
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Chen, Kewang. "Mathematical Analysis of Some Partial Differential Equations with Applications." ScholarWorks @ UVM, 2019. https://scholarworks.uvm.edu/graddis/1053.
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In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2.
In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature.
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Pierson, Mark A. "Theory and Application of a Class of Abstract Differential-Algebraic Equations." Diss., Virginia Tech, 2005. http://hdl.handle.net/10919/27416.
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We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed.Ph. D.
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Guffey, Stephen. "Application of a Numerical Method and Optimal Control Theory to a Partial Differential Equation Model for a Bacterial Infection in a Chronic Wound." TopSCHOLAR®, 2015. https://digitalcommons.wku.edu/theses/1494.
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In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the spatial variable over an N-point mesh, thereby converting each partial differential equation into N ordinary differential equations. These equations can then be solved using numerical routines defined for ordinary differential equations.
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Coros, Corina Alexandra. "Analysis of partial differential equations with time-periodic forcing, applications to Navier-Stokes equations." Thesis, University of Ottawa (Canada), 2006. http://hdl.handle.net/10393/29346.
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Flows with time-periodic forcing can be found in various applications, such as the circulatory and respiratory systems, or industrial mixers. In this thesis, we address few questions in relation with the time-periodic forcing of flows and related partial differential equations (PDE), including the linear Advection-Diffusion equation.
In Chapter 2, we first study linear PDE's with non-symmetric operators subject to time-periodic forcing. We prove that they have a unique time-periodic solution which is stable and attracts any initial solution if the bilinear form associated to the operator is coercive, and we obtain an error estimate for finite element method with a backward Euler time-stepping scheme. That general theory is applied to the Advection-Diffusion equation and the Stokes problem. The first equation has a non-symmetric operator, while the second has a symmetric operator but two unknowns, the velocity and pressure. To apply the general theory, we prove an error estimate for a Riesz projection operator, using a special Aubin-Nistche argument for the Advection-Diffusion equation with a tune-dependent advective velocity. A spectral analysis for the 1-D Advection-Diffusion equation, relevant parameters that control the speed of convergence of any initial solution to the time-periodic solution are identified.
In Chapter 3, we extend a theorem of J.L. Lions about the existence of time-periodic solutions of Navier-Stokes equations under periodic distributed forcing with homogeneous Dirichlet boundary conditions to the case of non-homogeneous time-periodic Dirichlet boundary conditions. Our theorem predicts the existence of a time-periodic solution for Navier-Stokes equations subject to time-periodic forcing but the stability of these time-periodic solutions is not known.
In Chapter 4, we investigate the stability of these time-periodic solutions, through numerical simulations with test cases in a 2-D time-periodic lid driven cavity and a 2-D constricted channel with a time-periodic inflow. From our numerical simulations, it seems that a bifurcation occurs in the range 3000--8000 in the periodically driven cavity, and the range 400--1200 in the periodically driven channel.
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Czapinski, Michal. "Solvers on Advanced Parallel Architectures with Application to Partial Differential Equations and Discrete Optimisation." Thesis, Cranfield University, 2014. http://dspace.lib.cranfield.ac.uk/handle/1826/9315.
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This thesis investigates techniques for the solution of partial differential equations
(PDE) on advanced parallel architectures comprising central processing units (CPU)
and graphics processing units (GPU). Many physical phenomena studied by scientists
and engineers are modelled with PDEs, and these are often computationally expensive
to solve. This is one of the main drivers of large-scale computing development.
There are many well-established PDE solvers, however they are often inherently
sequential. In consequence, there is a need to redesign the existing algorithms, and
to develop new methods optimised for advanced parallel architectures. This task is
challenging due to the need to identify and exploit opportunities for parallelism, and
to deal with communication overheads. Moreover, a wide range of parallel platforms
are available — interoperability issues arise if these are employed to work together.
This thesis offers several contributions. First, performance characteristics of hybrid
CPU-GPU platforms are analysed in detail in three case studies. Secondly, an
optimised GPU implementation of the Preconditioned Conjugate Gradients (PCG)
solver is presented. Thirdly, a multi-GPU iterative solver was developed — the
Distributed Block Direct Solver (DBDS). Finally, and perhaps the most significant
contribution, is the innovative streaming processing for FFT-based Poisson solvers.
Each of these contributions offers significant insight into the application of advanced
parallel systems in scientific computing. The techniques introduced in the case
studies allow us to hide most of the communication overhead on hybrid CPU-GPU
platforms. The proposed PCG implementation achieves 50–68% of the theoretical
GPU peak performance, and it is more than 50% faster than the state-of-the-art
solution (CUSP library). DBDS follows the Block Relaxation scheme to find the
solution of linear systems on hybrid CPU-GPU platforms. The convergence of DBDS
has been analysed and a procedure to compute a high-quality upper bound is derived.
Thanks to the novel streaming processing technique, our FFT-based Poisson solvers
are the first to handle problems larger than the GPU memory, and to enable multi-
GPU processing with a linear speed-up. This is a significant improvement over the
existing methods, which are designed to run on a single GPU, and are limited by
the device memory size. Our algorithm needs only 6.9 seconds to solve a 2D Poisson
problem with 2.4 billion variables (9 GB) on two Tesla C2050 GPUs (3 GB memory).
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Chen, Linan Ph D. Massachusetts Institute of Technology. "Applications of probability to partial differential equations and infinite dimensional analysis." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/67787.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 79-80).
This thesis consists of two parts. The first part applies a probabilistic approach to the study of the Wright-Fisher equation, an equation which is used to model demographic evolution in the presence of diffusion. The fundamental solution to the Wright-Fisher equation is carefully analyzed by relating it to the fundamental solution to a model equation which has the same degeneracy at one boundary. Estimates are given for short time behavior of the fundamental solution as well as its derivatives near the boundary. The second part studies the probabilistic extensions of the classical Cauchy functional equation for additive functions both in finite and infinite dimensions. The connection between additivity and linearity is explored under different circumstances, and the techniques developed in the process lead to results about the structure of abstract Wiener spaces. Both parts are joint work with Daniel W. Stroock.
by Linan Chen.
Ph.D.
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Singler, John. "Sensitivity Analysis of Partial Differential Equations With Applications to Fluid Flow." Diss., Virginia Tech, 2005. http://hdl.handle.net/10919/28051.
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For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the non-normality of the linearized operator. These new â mostly linearâ theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored. The main goal of this work is to begin to study the role of nonlinearity in transition. We use model problems to illustrate that small unmodeled disturbances can cause transition through movement or bifurcation of equilibria. We also demonstrate that small wall roughness can lead to transition by causing the linearized system to become unstable. Sensitivity methods are used to obtain important information about the disturbed problem and to illustrate that it is possible to have a precursor to predict transition. Finally, we apply linear feedback control to the model problems to illustrate the power of feedback to delay transition and even relaminarize fully developed chaotic flows.Ph. D.
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von, Schwerin Erik. "Adaptivity for Stochastic and Partial Differential Equations with Applications to Phase Transformations." Doctoral thesis, KTH, Numerisk Analys och Datalogi, NADA, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4477.
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his work is concentrated on efforts to efficiently compute properties of systems, modelled by differential equations, involving multiple scales. Goal oriented adaptivity is the common approach to all the treated problems. Here the goal of a numerical computation is to approximate a functional of the solution to the differential equation and the numerical method is adapted to this task. The thesis consists of four papers. The first three papers concern the convergence of adaptive algorithms for numerical solution of differential equations; based on a posteriori expansions of global errors in the sought functional, the discretisations used in a numerical solution of the differential equiation are adaptively refined. The fourth paper uses expansion of the adaptive modelling error to compute a stochastic differential equation for a phase-field by coarse-graining molecular dynamics. An adaptive algorithm aims to minimise the number of degrees of freedom to make the error in the functional less than a given tolerance. The number of degrees of freedom provides the convergence rate of the adaptive algorithm as the tolerance tends to zero. Provided that the computational work is proportional to the degrees of freedom this gives an estimate of the efficiency of the algorithm. The first paper treats approximation of functionals of solutions to second order elliptic partial differential equations in bounded domains of ℝd, using isoparametric $d$-linear quadrilateral finite elements. For an adaptive algorithm, an error expansion with computable leading order term is derived %. and used in a computable error density, which is proved to converge uniformly as the mesh size tends to zero. For each element an error indicator is defined by the computed error density multiplying the local mesh size to the power of 2+d. The adaptive algorithm is based on successive subdivisions of elements, where it uses the error indicators. It is proved, using the uniform convergence of the error density, that the algorithm either reduces the maximal error indicator with a factor or stops; if it stops, then the error is asymptotically bounded by the tolerance using the optimal number of elements for an adaptive isotropic mesh, up to a problem independent factor. Here the optimal number of elements is proportional to the d/2 power of the Ldd+2 quasi-norm of the error density, whereas a uniform mesh requires a number of elements proportional to the d/2 power of the larger L1 norm of the same error density to obtain the same accuracy. For problems with multiple scales, in particular, these convergence rates may differ much, even though the convergence order may be the same. The second paper presents an adaptive algorithm for Monte Carlo Euler approximation of the expected value E[g(X(τ),\τ)] of a given function g depending on the solution X of an \Ito\ stochastic differential equation and on the first exit time τ from a given domain. An error expansion with computable leading order term for the approximation of E[g(X(T))] with a fixed final time T>0 was given in~[Szepessy, Tempone, and Zouraris, Comm. Pure and Appl. Math., 54, 1169-1214, 2001]. This error expansion is now extended to the case with stopped diffusion. In the extension conditional probabilities are used to estimate the first exit time error, and difference quotients are used to approximate the initial data of the dual solutions. For the stopped diffusion problem the time discretisation error is of order N-1/2 for a method with N uniform time steps. Numerical results show that the adaptive algorithm improves the time discretisation error to the order N-1, with N adaptive time steps. The third paper gives an overview of the application of the adaptive algorithm in the first two papers to ordinary, stochastic, and partial differential equation. The fourth paper investigates the possibility of computing some of the model functions in an Allen--Cahn type phase-field equation from a microscale model, where the material is described by stochastic, Smoluchowski, molecular dynamics. A local average of contributions to the potential energy in the micro model is used to determine the local phase, and a stochastic phase-field model is computed by coarse-graining the molecular dynamics. Molecular dynamics simulations on a two phase system at the melting point are used to compute a double-well reaction term in the Allen--Cahn equation and a diffusion matrix describing the noise in the coarse-grained phase-field.QC 20100823
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Schwerin, Erik von. "Adaptivity for stochastic and partial differential equations with applications to phase transformations /." Stockholm : Numerisk analys och datalogi, Kungliga Tekniska högskolan, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4477.
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Arjmand, Doghonay. "Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations." Doctoral thesis, KTH, Numerisk analys, NA, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-160122.
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This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.QC 20150216
Multiscale methods for wave propagation
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Jayes, Mohd Idris. "Numerical solution of ordinary and partial differential equations occurring in scientific applications." Thesis, Loughborough University, 1992. https://dspace.lboro.ac.uk/2134/32103.
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Deb, Manas Kumar. "Solution of stochastic partial differential equations (SPDEs) using Galerkin method : theory and applications /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.
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Bagri, Gurjeet S. "Existence theorems for periodic solutions to partial differential equations with applications in hydrodynamics." Thesis, Loughborough University, 2010. https://dspace.lboro.ac.uk/2134/6724.
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The thesis looks at a number of existence theorems that prove the existence of small-amplitude periodic solutions to systems of partial differential equations. The existence theorems we consider are the Hopf bifurcation theorem, the Lyapunov centre theorem, the Weinstein-Moser theorem, and extensions of these theorems; the Hopf-Iooss bifurcation theorem, the Lyapunov-Iooss centre theorem and the Weinstein-Moser-Iooss theorem, respectively. The theorems have been derived so that they are applicable to functional analytical problems, and have been represented in a coherent and uniform manner in order to bridge the fundamental structure common to them all. Applications of these theorems, in this standardised form, are then applied in a systematic way to two particular hydrodynamical problems; the water wave problem and the Navier-Stokes equations. The classic water wave problem concerns the irrotational flow of a perfect fluid of unit density, subject to the forces of gravity and surface tension. We apply the Lyapunov-Iooss centre theorem to prove the existence of doubly-periodic waves; a doubly-periodic wave is a travelling wave that possess spatially periodic profiles in two different horizontal directions. Fundamental to our approach is the spatial dynamics formulation. The spatial dynamics formulation involves formulating a system of partial differential equations, defined on some spatial domain, as a dynamical system where one of the unbounded spatial variables plays the role of time. We catalogue a variety of parameter values for which it is possible to obtain doubly periodic waves, and we conclude with an existence result for doubly periodic waves under specific parameter restrictions. The Navier-Stokes equations in an exterior domain models the flow of an incompressible, viscous fluid past an obstacle. We apply the Hopf-Iooss bifurcation theorem to the defining equations to determine the existence of time-periodic waves. Our approach involves a careful examination of the Oseen problem to which we apply a 'cut-off' technique. This technique is used to constructs a solution to the Oseen problem using the respective solutions to the Oseen problem on a bounded domain and free space (the existence of which are well established). Time-periodic solutions are established using the Hopf-Iooss bifurcation theorem provided certain spectral conditions are met. The verification of the conditions may only be possible numerically, and so beyond the scope of our investigation.
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Jin, Xiao-qing. "Circulant preconditioners for Toeplitz matrices and their applications in solving partial differential equations /." [Hong Kong : University of Hong Kong], 1992. http://sunzi.lib.hku.hk/hkuto/record.jsp?B13391756.
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Yousept, Irwin. "Optimal control of partial differential equations involving pointwise state constraints: regularization and applications." Göttingen Cuvillier, 2008. http://d-nb.info/990426513/04.
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Seadler, Bradley T. "Signed-Measure Valued Stochastic Partial Differential Equations with Applications in 2D Fluid Dynamics." Case Western Reserve University School of Graduate Studies / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=case1333062148.
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Manay, Siddharth. "Applications of anti-geometric diffusion of computer vision : thresholding, segmentation, and distance functions." Diss., Georgia Institute of Technology, 2003. http://hdl.handle.net/1853/33626.
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Gahlawat, Aditya. "Analysis and control of parabolic partial differential equations with application to tokamaks using sum-of-squares polynomials." Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAT111/document.
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Dans ce travail, nous abordons les problèmes de l'analyse de la stabilité et de la synthèse de contrôleur pour une Equation aux Dérivées Partielles (EDP) parabolique linéaire de dimension 1. Ces problèmes sont résolus avec des méthodologies analogues au cadre des inégalités matricielles linéaires (LMI) pour les équations différentielles ordinaires (EDO). Nous développons une méthode pour EDP paraboliques dans laquelle nous testons la faisabilité de certaines LMIs utilisant la programmation semi-définie (SDP) pour construire des fonctions de Lyapunov quadratiques et des contrôleurs. Le cœur de notre démarche est la construction de fonctions de Lyapunov quadratiques paramétrées par les opérateurs définis positifs sur les espaces de Hilbert de dimension infinie. Contrairement aux matrices positives, il n'y a pas de méthode unique paramétrisant l'ensemble des opérateurs positifs sur un espace de Hilbert. Bien sûr, nous pouvons toujours paramétrer un sous-ensemble des opérateurs positifs en utilisant, par exemple, des scalaires positifs. Cependant, nous devons nous assurer que le paramétrage des opérateurs positifs ne doit pas être conservatif. Notre contribution est de construire une paramétrisation qui a seulement une petite quantité de conservatisme comme indiqué par nos résultats numériques. Nous utilisons des polynômes en somme des carrés (SOS) pour paramétrer l'ensemble des opérateurs positifs, linéaire et bornés sur les espaces de Hilbert. Comme son nom l'indique, un polynôme SOS est celui qui peut être représenté comme une somme de polynômes carrés. La propriété la plus importante d'un polynôme SOS est qu'il peut être représenté au moyen d'une matrice (semi-)définie positive. Cela implique que, même si le problème de polynôme (semi-)positif est NP-difficile, le problème de vérifier si polynôme est SOS (et donc (semi-)positif) peut être résolu en utilisant la SDP. Par conséquent, nous nous efforçons de construire des fonctions de Lyapunov quadratiques paramétrées par les opérateurs positifs. Ces opérateurs positifs sont à leur tour paramétrés par des polynômes SOS. Cette paramétrisation SOS nous permet de formuler le problème de faisabilité pour l'existence d'une fonction de Lyapunov quadratique comme un problème de faisabilité LMI. Le problème de la faisabilité LMI peut alors être adressé à l'aide de SDP. Dans la première partie de la thèse nous considérons analyse de stabilité et la synthèse de contrôleur aux frontières pour une large classe d'EDP paraboliques. Les EDP ont des coefficients de transport distribués spatialement. Ces EDP sont utilisés pour modéliser les processus de diffusion, de convection et de réaction de quantités physiques dans les milieux anisotropes. Nous considérons la synthèse de contrôleurs limite à la fois pour le cas de retour d'état et le cas de retour de sortie (à l'aide d'un observateur). Dans la deuxième partie de la thèse, nous concevons un contrôleur distribué pour la régulation du flux magnétique poloïdal dans un tokamak (procédé de fusion thermonucléaire par confinement magnétique). Tout d'abord, nous concevons un contrôleur régulant la pente des lignes de champ magnétique (le facteur de sécurité). La régulation du profil du facteur de sécurité est importante pour supprimer les instabilités MHD dans un tokamak. Ensuite, nous concevons un contrôleur maximisant la densité de courant bootstrap généré en interne. Une proportion accrue du courant bootstrap conduirait à une réduction des besoins énergétiques exogènes pour l'exploitation d'un tokamakIn this work we address the problems of stability analysis and controller synthesis for one dimensional linear parabolic Partial Differential Equations (PDEs). To achieve the tasks of stability analysis and controller synthesis we develop methodologies akin to the Linear Matrix Inequality (LMI) framework for Ordinary Differential Equations (ODEs). We develop a method for parabolic PDEs wherein we test the feasibility of certain LMIs using SDP to construct quadratic Lyapunov functions and controllers. The core of our approach is the construction of quadratic Lyapunov functions parametrized by positive definite operators on infinite dimensional Hilbert spaces. Unlike positive matrices, there is no single method of parametrizing the set of all positive operators on a Hilbert space. Of course, we can always parametrize a subset of positive operators, using, for example, positive scalars. However, we must ensure that the parametrization of positive operators should not be conservative. Our contribution is constructing a parametrization which has only a small amount of conservatism as indicated by our numerical results. We use Sum-of-Squares (SOS) polynomials to parametrize the set of positive, linear and bounded operators on Hilbert spaces. As the name indicates, an SOS polynomial is one which can be represented as a sum of squared polynomials. The most important property of an SOS polynomial is that it can be represented using a positive (semi)-definite matrix. This implies that even though the problem of polynomial (semi)-positivity is NP-hard, the problem of checking if polynomial is SOS (and hence (semi)-positive) can be solved using SDP. Therefore, we aim to construct quadratic Lyapunov functions parametrized by positive operators. These positive operators are in turn parametrized by SOS polynomials. This parametrization using SOS allows us to cast the feasibility problem for the existence of a quadratic Lyapunov function as the feasibility problem of LMIs. The feasibility problem of LMIs can then be addressed using SDP. In the first part of the thesis we consider stability analysis and boundary controller synthesis for a large class of parabolic PDEs. The PDEs have spatially distributed coefficients. Such PDEs are used to model processes of diffusion, convection and reaction of physical quantities in anisotropic media. We consider boundary controller synthesis for both the state feedback case and the output feedback case (using and observer design). IN the second part of thesis we design distributed controllers for the regulation of poloidal magnetic flux in a tokamak (a thermonuclear fusion devise). First, we design the controllers to regulate the magnetic field line pitch (the safety factor). The regulation of the safety factor profile is important to suppress the magnetohydrodynamic instabilities in a tokamak. Then, we design controllers to maximize the internally generated bootstrap current density. An increased proportion of bootstrap current would lead to a reduction in the external energy requirements for the operation of a tokamak
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Dong, Bin. "Applications of variational models and partial differential equations in medical image and surface processing." Diss., Restricted to subscribing institutions, 2009. http://proquest.umi.com/pqdweb?did=1872060431&sid=3&Fmt=2&clientId=1564&RQT=309&VName=PQD.
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Jiang, Lijian. "Multiscale numerical methods for partial differential equations using limited global information and their applications." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-2991.
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Brito, Loeza Carlos Francisco. "Fast numerical algorithms for high order partial differential equations with applications to image restoration techniques." Thesis, University of Liverpool, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526786.
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Shek, Cheuk-man Edmond. "The continuous and discrete extended Korteweg-de Vries equations and their applications in hydrodynamics and lattice dynamics." Click to view the E-thesis via HKUTO, 2006. http://sunzi.lib.hku.hk/hkuto/record/B36925585.
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Kelome, Djivèdé Armel. "Viscosity solutions of second order equations in a separable Hilbert space and applications to stochastic optimal control." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29159.
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Arjmand, Doghonay. "Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations." Licentiate thesis, KTH, Numerisk analys, NA, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-129237.
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This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers. The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large. In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities.QC 20130926
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Bradley, Aoibhinn Maire. "Analysis of nonlinear spatio-temporal partial differential equations : applications to host-parasite systems and bubble growth." Thesis, University of Strathclyde, 2014. http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=24405.
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The mountain hare population currently appears to be under threat in Scotland. The natural population cycles exhibited by this species are thought to be, at least in part, due to its infestation by a parasitic worm. We seek to gain an understanding of these population dynamics through a mathematical model of this system and so determine whether low population levels observed in the field are a natural trough associated with this cycling, or whether they point to a more serious decline in overall population densities. A generic result, that can be used to predict the presence of periodic travelling waves (PTWs) in a spatially heterogeneous system, is reported. This result is applicable to any two population host-parasite system with a supercritical Hopf bifurcation in the reaction kinetics. Application of this result to two examples of well studied host-parasite systems, namely the mountain hare and the red grouse systems, predicts and illustrates, for the first time, the existence of PTWs as solutions for these reaction advection diffusion schemes. One method for designing bone scaffolds involves the acoustic irradiation of a reacting polymer foam resulting in a final sample with graded porosity. The work in this thesis represents the first attempt to derive a mathematical model, for this empirical method, in order to inform the experimental design and tailor the porosity profile of samples. We isolate and study the direct effect of the acoustic pressure amplitude as well as its indirect effect on the reaction rate. We demonstrate that the direct effect of the acoustic pressure amplitude is negligible due to a high degree of attenuation by the sample. The indirect effect, on reaction rate, is significant and the standing wave is shown to produce a heterogeneous bubble size distribution. Several suggestions for further work are made.
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Manukian, Vahagn Emil. "Existence and stability of multi-pulses with applications to nonlinear optics." Connect to resource Connect to this title online, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1117638160.
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Thesis (Ph. D.)--Ohio State University, 2005.Title from first page of PDF file. Document formatted into pages; contains ix, 134 p.; also includes graphics. Includes bibliographical references (p. 130-134). Available online via OhioLINK's ETD Center
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Eneyew, Eyaya Birara. "Efficient computation of shifted linear systems of equations with application to PDEs." Thesis, Stellenbosch : Stellenbosch University, 2011. http://hdl.handle.net/10019.1/17827.
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Thesis (MSc)--Stellenbosch University, 2011.ENGLISH ABSTRACT: In several numerical approaches to PDEs shifted linear systems of the form (zI - A)x = b, need to be solved for several values of the complex scalar z. Often, these linear systems are large and sparse. This thesis investigates efficient numerical methods for these systems that arise from a contour integral approximation to PDEs and compares these methods with direct solvers. In the first part, we present three model PDEs and discuss numerical approaches to solve them. We use the first problem to demonstrate computations with a dense matrix, the second problem to demonstrate computations with a sparse symmetric matrix and the third problem for a sparse but nonsymmetric matrix. To solve the model PDEs numerically we apply two space discrerization methods, namely the finite difference method and the Chebyshev collocation method. The contour integral method mentioned above is used to integrate with respect to the time variable. In the second part, we study a Hessenberg reduction method for solving shifted linear systems with a dense matrix and present numerical comparison of it with the built-in direct linear system solver in SciPy. Since both are direct methods, in the absence of roundoff errors, they give the same result. However, we find that the Hessenberg reduction method is more efficient in CPU-time than the direct solver. As application we solve a one-dimensional version of the heat equation. In the third part, we present efficient techniques for solving shifted systems with a sparse matrix by Krylov subspace methods. Because of their shift-invariance property, the Krylov methods allow one to obtain approximate solutions for all values of the parameter, by generating a single approximation space. Krylov methods applied to the linear systems are generally slowly convergent and hence preconditioning is necessary to improve the convergence. The use of shift-invert preconditioning is discussed and numerical comparisons with a direct sparse solver are presented. As an application we solve a two-dimensional version of the heat equation with and without a convection term. Our numerical experiments show that the preconditioned Krylov methods are efficient in both computational time and memory space as compared to the direct sparse solver.
AFRIKAANSE OPSOMMING: In verskeie numeriese metodes vir PDVs moet geskuifde lineêre stelsels van die vorm (zI − A)x = b, opgelos word vir verskeie waardes van die komplekse skalaar z. Hierdie stelsels is dikwels groot en yl. Hierdie tesis ondersoek numeriese metodes vir sulke stelsels wat voorkom in kontoerintegraalbenaderings vir PDVs en vergelyk hierdie metodes met direkte metodes vir oplossing. In die eerste gedeelte beskou ons drie model PDVs en bespreek numeriese benaderings om hulle op te los. Die eerste probleem word gebruik om berekenings met ’n vol matriks te demonstreer, die tweede probleem word gebruik om berekenings met yl, simmetriese matrikse te demonstreer en die derde probleem vir yl, onsimmetriese matrikse. Om die model PDVs numeries op te los beskou ons twee ruimte-diskretisasie metodes, naamlik die eindige-verskilmetode en die Chebyshev kollokasie-metode. Die kontoerintegraalmetode waarna hierbo verwys is word gebruik om met betrekking tot die tydveranderlike te integreer. In die tweede gedeelte bestudeer ons ’n Hessenberg ontbindingsmetode om geskuifde lineêre stelsels met ’n vol matriks op te los, en ons rapporteer numeriese vergelykings daarvan met die ingeboude direkte oplosser vir lineêre stelsels in SciPy. Aangesien beide metodes direk is lewer hulle dieselfde resultate in die afwesigheid van afrondingsfoute. Ons het egter bevind dat die Hessenberg ontbindingsmetode meer effektief is in terme van rekenaartyd in vergelyking met die direkte oplosser. As toepassing los ons ’n een-dimensionele weergawe van die hittevergelyking op. In die derde gedeelte beskou ons effektiewe tegnieke om geskuifde stelsels met ’n yl matriks op te los, met Krylov subruimte-metodes. As gevolg van hul skuifinvariansie eienskap, laat die Krylov metodes mens toe om benaderde oplossings te verkry vir alle waardes van die parameter, deur slegs een benaderingsruimte voort te bring. Krylov metodes toegepas op lineêre stelsels is in die algemeen stadig konvergerend, en gevolglik is prekondisionering nodig om die konvergensie te verbeter. Die gebruik van prekondisionering gebasseer op skuif-en-omkeer word bespreek en numeriese vergelykings met direkte oplossers word aangebied. As toepassing los ons ’n twee-dimensionele weergawe van die hittevergelyking op, met ’n konveksie term en daarsonder. Ons numeriese eksperimente dui aan dat die Krylov metodes met prekondisionering effektief is, beide in terme van berekeningstyd en rekenaargeheue, in vergelyking met die direkte metodes.
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Fels, Mark Eric. "Some applications of Cartan's method of equivalence to the geometric study of ordinary and partial differential equations." Thesis, McGill University, 1993. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=41274.
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Cartan's method of equivalence is used to prove that there exists two fundamental tensorial invariants which determine the geometry of systems of n ($ ge$ 2) second order ordinary differential equations. These invariants allow us prove that there exist a unique equivalence class of second order equations which admit a Lie point symmetry group of maximal dimension, the dimension being $n sp2 + 4n + 3$. For third order systems of ordinary differential equations, we prove that the possible dimension of the point symmetry group is bounded by $n sp2 + 3n + 3$. As well we find that there is a unique third order system whose symmetry group has dimension n$ sp2$ + 3n + 3.We also characterize invariantly under point transformations some equivalence classes of parabolic quasi-linear second order partial differential equations, and examine their point symmetry groups. We are able to make our characterizations by proving a reduction theorem for principal fibre bundles.
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Røvik, Camilla. "Fast Tensor-Product Solvers for the Numerical Solution of Partial Differential Equations : Application to Deformed Geometries and to Space-Time Domains." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2010. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-10814.
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Spectral discretization in space and time of the weak formulation of a partial differential equations (PDE) is studied. The exact solution to the PDE, with either Dirichlet or Neumann boundary conditions imposed, is approximated using high order polynomials. This is known as a spectral Galerkin method. The main focus of this work is the solution algorithm for the arising algebraic system of equations. A direct fast tensor-product solver is presented for the Poisson problem in a rectangular domain. We also explore the possibility of using a similar method in deformed domains, where the geometry of the domain is approximated using high order polynomials. Furthermore, time-dependent PDE's are studied. For the linear convection-diffusion equation in $mathbb{R}$ we present a tensor-product solver allowing for parallel implementation, solving $mathcal{O}(N)$ independent systems of equations. Lastly, an iterative tensor-product solver is considered for a nonlinear time-dependent PDE. For most algorithms implemented, the computational cost is $mathcal O (N^{p+1})$ floating point operations and a memory required of $mathcal O (N^{p})$ floating point numbers for $mathcal O (N^{p})$ unknowns. In this work we only consider $p=2$, but the theory is easily extended to apply in higher dimensions. Numerical results verify the expected convergence for both the iterative method and the spectral discretization. Exponential convergence is obtained when the solution and domain geometry are infinitely smooth.
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Bedford, Stephen James. "Calculus of variations and its application to liquid crystals." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a2004679-5644-485c-bd35-544448f53f6a.
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The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function n ε W1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state n to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV2 (Ω,S2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.
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Cao, Xinlin. "Geometric structures of eigenfunctions with applications to inverse scattering theory, and nonlocal inverse problems." HKBU Institutional Repository, 2020. https://repository.hkbu.edu.hk/etd_oa/754.
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Inverse problems are problems where causes for desired or an observed effect are to be determined. They lie at the heart of scientific inquiry and technological development, including radar/sonar, medical imaging, geophysical exploration, invisibility cloaking and remote sensing, to name just a few. In this thesis, we focus on the theoretical study and applications of some intriguing inverse problems. Precisely speaking, we are concerned with two typical kinds of problems in the field of wave scattering and nonlocal inverse problem, respectively. The first topic is on the geometric structures of eigenfunctions and their applications in wave scattering theory, in which the conductive transmission eigenfunctions and Laplacian eigenfunctions are considered. For the study on the intrinsic geometric structures of the conductive transmission eigenfunctions, we first present the vanishing properties of the eigenfunctions at corners both in R2 and R3, based on microlocal analysis with the help of a particular type of planar complex geometrical optics (CGO) solution. This significantly extends the previous study on the interior transmission eigenfunctions. Then, as a practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter. For the study on the geometric structures of Laplacian eigenfunctions, we separately discuss the two-dimensional case and the three-dimensional case. In R2, we introduce a new notion of generalized singular lines of Laplacian eigenfunctions, and carefully investigate these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We provide an accurate and comprehensive quantitative characterization of the relationship. In R3, we study the analytic behaviors of Laplacian eigenfunctions at places where nodal or generalized singular planes intersect, which is much more complicated. These theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal (polyhedral) setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Our second topic is concerning the fractional partial differential operators and some related nonlocal inverse problems. We present some prelimilary knowledge on fractional Sobolev Spaces and fractional partial differential operators first. Then we focus on the simultaneous recovery results of two interesting nonlocal inverse problems. One is simultaneously recovering potentials and the embedded obstacles for anisotropic fractional Schrödinger operators based on the strong uniqueness property and Runge approximation property. The other one is the nonlocal inverse problem associated with a fractional Helmholtz equation that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. The main method utilized here is the low-frequency asymptotics combining with the variational argument. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.
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Evans, James Alexander. "Some analytical techniques for partial differential equations on periodic structures and their applications to the study of metamaterials." Thesis, Cardiff University, 2016. http://orca.cf.ac.uk/94875/.
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The work presented in this thesis is a study of homogenisation problems in electromagnetics and elasticity with potential applications to the development of metamaterials. In Chapter 1, I study the leading order frequency approximations of the quasi-static Maxwell equations on the torus. A higher-order asymptotic regime is used to derive a higher-order homogenised equation for the solution of an elliptic second-order partial differential equation. The equivalent variational approach to this problem is studied which leads to an equivalent higher-order homogenised equation. Finally, the derivation of higher-order constitutive laws relating the fields to their inductions is presented. In Chapter 2, I study the governing equations of linearised elasticity where the periodic composite material of interest is made up of a "critically" scaled "stiff" rod framework with the voids in between filled in with a "soft" material which is in high-contrast with the stiff material. Using results from two-scale convergence theory, a well posed homogenised model is presented with features reminiscent of both high-contrast and thin structure homogenised models with the additional feature of a linking relation of Wentzell type. The spectrum of the limiting operator is investigated and the establishment of the convergence of spectra from the initial problem is derived. In the final chapter, I investigate brie y three additional homogenisation problems. In the first problem, I study a periodic dielectric composite and show that there exists a critical scaling between the material parameter of the soft inclusion and the period of the composite. In the second problem, I use of two-scale convergence theory to derive a homogenised model for Maxwell's equations on thin rod structures and in the final problem I study Maxwell's equations in R^3 under a chiral transformation of the coordinates and derive a homogenised model in this special geometry.
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Maroofi, Hamed. "Applications of the Monge - Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29197.
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Karasev, Peter A. "Feedback augmentation of pde-based image segmentation algorithms using application-specific exogenous data." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/50257.
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This thesis is divided into five chapters. The scope of problems considered is defined in chapter I.
Next, chapter II provides background material on image processing with
partial differential equations
and a review of prior work in the
field. Chapter III covers the medical imaging portion of the research;
the key contribution is a control-based algorithm for interactive image
segmentation. Applications of the feedback-augmented level set method to fracture reconstruction and surgical planning are shown. Problems in vision-based control are considered in Chapters IV and V. A method of improving performance in closed-loop target tracking using level set
segmentation is developed, with unmanned aerial vehicle or
next-generation missile guidance being the primary applications of interest.
Throughout this thesis, the two application types are connected into a
unified viewpoint of open-loop systems that are augmented by exogenous data.
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Heryudono, Alfa R. H. "Adaptive radial basis function methods for the numerical solution of partial differential equations, with application to the simulation of the human tear film." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 178 p, 2008. http://proquest.umi.com/pqdweb?did=1601513551&sid=5&Fmt=2&clientId=8331&RQT=309&VName=PQD.
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Shek, Cheuk-man Edmond, and 石焯文. "The continuous and discrete extended Korteweg-de Vries equations and their applications in hydrodynamics and lattice dynamics." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2006. http://hub.hku.hk/bib/B36925585.
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Butt, Muhammad Akmal. "Continuous and discrete approaches to morphological image analysis with applications : PDEs, curve evolution, and distance transforms." Diss., Georgia Institute of Technology, 1998. http://hdl.handle.net/1853/15465.
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Li, Siqing. "Kernel-based least-squares approximations: theories and applications." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/539.
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Kernel-based meshless methods for approximating functions and solutions of partial differential equations have many applications in engineering fields. As only scattered data are used, meshless methods using radial basis functions can be extended to complicated geometry and high-dimensional problems. In this thesis, kernel-based least-squares methods will be used to solve several direct and inverse problems. In chapter 2, we consider discrete least-squares methods using radial basis functions. A general l^2-Tikhonov regularization with W_2^m-penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases in which the function being approximated is within and is outside of the native space of the kernel. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In chapter 3, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on three other factors: proportion of the measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved to be convergent at the H^2 (Ω) norm. Numerical experiments in two and three dimensions show that we can obtain desired convergent results under different boundary conditions and different domain shapes. In chapter 4, we use a kernel-based least-squares method to solve ill-posed Cauchy problems for elliptic partial differential equations. We construct stable methods for these inverse problems. Numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints. A convergence analysis with respect to noise levels and fill distances of data points is provided, from which a Tikhonov regularization strategy is obtained. A nonlinear algorithm is proposed to obtain stable solutions of the resulting nonlinear problems. Numerical experiments are provided to verify our convergence results. In the final chapter, we apply meshless methods to the Gierer-Meinhardt activator-inhibitor model. Pattern transitions in irregular domains of the Gierer-Meinhardt model are shown. We propose various parameter settings for different patterns appearing in nature and test these settings on some irregular domains. To further simulate patterns in reality, we construct different kinds of domains and apply proposed parameter settings on different patches of domains found in nature.
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Ježková, Jitka. "Modelování dopravního toku." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2015. http://www.nusl.cz/ntk/nusl-232180.
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Tato diplomová práce prezentuje problematiku dopravního toku a jeho modelování. Zabývá se především několika LWR modely, které následně rozebírá a hledá řešení pro počáteční úlohy. Ukazuje se, že ne pro všechny počáteční úlohy lze řešení definovat na celém prostoru, ale jen v určitém okolí počáteční křivky. Proto je dále odvozena metoda výpočtu velikosti tohoto okolí a to nejen zcela obecně, ale i pro dané modely. Teoretický rozbor LWR modelů a řešení počátečních úloh jsou demonstrovány několika příklady, které zřetelně ukazují, jak se dopravní tok simulovaný danými modely chová.
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Heuer, Christof. "High-order compact finite difference schemes for parabolic partial differential equations with mixed derivative terms and applications in computational finance." Thesis, University of Sussex, 2014. http://sro.sussex.ac.uk/id/eprint/49800/.
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This thesis is concerned with the derivation, numerical analysis and implementation of high-order compact finite difference schemes for parabolic partial differential equations in multiple spatial dimensions. All those partial differential equations contain mixed derivative terms. The resulting schemes have been applied to equations appearing in computational finance. First, we develop and study essentially high-order compact finite difference schemes in a general setting with option pricing in stochastic volatility models on non-uniform grids as application. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In the numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiment a comparative standard second-order discretisation is significantly outperform. We conduct a numerical stability study which indicates unconditional stability of the scheme. Second, we derive and analyse high-order compact schemes with n-dimensional spatial domain in a general setting. We obtain fourth-order accuracy in space and second-order accuracy in time. A thorough von Newmann stability analysis is performed for spatial domain with dimensions two and three. We prove that a necessary stability condition holds unconditionally without additional restrictions on the choice of the discretisation parameters for vanishing mixed derivative terms. We also give partial results for non-vanishing mixed derivative terms. As first example Black-Scholes Basket options are considered. In all numerical experiments, where the initial conditions were smoothened using the smoothing operators developed by Kreiss, Thomée and Widlund, a comparative standard second-order discretisation is significantly outperformed. As second example the multi-dimentional Heston basket option is considered for n independent Heston processes, where for each Heston process there is a non-vanishing correlation between the stock and its volatility.
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Edquist, Anders. "Monotonicity formulas and applications in free boundary problems." Doctoral thesis, KTH, Matematik (Avd.), 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-12405.
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This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven.QC20100621
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Chen, Meng. "Intrinsic meshless methods for PDEs on manifolds and applications." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/528.
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Radial basis function (RBF) methods for partial differential equations (PDEs), either in bulk domains, on surfaces, or in a combination of the formers, arise in a wide range of practical applications. This thesis proposes numerical approaches of RBF-based meshless techniques to solve these three kinds of PDEs on stationary and nonstationary surfaces and domains. In Chapter 1, we introduce the background of RBF methods, some basic concepts, and error estimates for RBF interpolation. We then provide some preliminaries for manifolds, restricted RBFs on manifolds, and some convergence properties of RBF interpolation. Finally, implicit-explicit time stepping schemes are briefly presented. In Chapter 2, we propose methods to implement meshless collocation approaches intrinsically to solve elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, that is, Kansa methods. Our main theoretical contribution is a robust convergence analysis under some standard smoothness assumptions for high-order convergence. We simulate reaction-diffusion equations to generate Turing patterns and solve shallow water problems on manifolds. In Chapter 3, we consider convective-diffusion problems that model surfactants or heat transport along moving surfaces. We propose two time-space algorithms by combining the methods of lines and kernel-based meshless collocation techniques intrinsic to surfaces. We use a low-order time discretization for fair comparison, and higher-order schemes in time are possible. The proposed methods can achieve second-order convergence. They use either analytic or approximated spatial discretization of the surface operators, which do not require regeneration of point clouds at each temporal iteration. Thus, they are alternatively applied to handle models on two types of evolving surfaces, which are defined as prescribed motions and governed by geometric evolution laws, respectively. We present numerical examples on various evolving surfaces for the performance of our algorithms and apply the approximated one to merging surfaces. In Chapter 4, a kernel-based meshless method is developed to solve coupled second-order elliptic PDEs in bulk domains and on surfaces, subject to Robin boundary conditions. It combines a least-squares kernel-based collocation method with a surface-type intrinsic approach. We can thus use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, to search for least-squares solutions in bulks and on surfaces, respectively. We first analyze error estimates for a domain-type Robin-boundary problem. Based on this analysis and the existing results for surface PDEs, we discuss the theoretical requirements for the Sobolev kernels used. We then select the orders of smoothness for the kernels in bulks and on surfaces. Finally, several numerical experiments are demonstrated to test the robustness of the coupled method in terms of accuracy and convergence rates under different settings.
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Nabil, Aïssam. "Homogénéisation de l'équation de la chaleur et des ondes et application à la contrôlabilité approchée." Rouen, 1998. http://www.theses.fr/1998ROUES027.
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Cette thèse, constituée de trois parties, est consacrée à l'étude de l'homogénéisation de l'équation de la chaleur et des ondes, dans le cas de coefficients rapidement oscillants et d'un domaine périodiquement perforé. Dans la première partie, on étudie le comportement asymptotique de l'équation de la chaleur sous de faibles hypothèses de convergence des données initiales, quand la période tend vers zéro. On montre ensuite un résultat de correcteurs pour ce problème. La deuxième partie de la thèse, concerne l'étude du comportement asymptotique d'un problème de contrôlabilité approchée pour l'équation de la chaleur considérée dans la première partie. En utilisant les résultats obtenus précédemment et des techniques de -convergence, on décrit le comportement asymptotique de ce problème. Plus précisément, on montre que la contrôlabilité approchée du problème homogénéisé est pénalisée par une constante qui mesure la proportion de matériaux dans le domaine. La troisième partie est consacrée à l'étude des correcteurs de l'équation des ondes. On donne des conditions suffisantes sur les données initiales, qui assurent la convergence des énergies vers l'énergie du problème homogénéisé. On établit alors un résultat de correcteurs.
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Ugail, Hassan, and Eyad Elyan. "Efficient 3D data representation for biometric applications." IOS Press, 2007. http://hdl.handle.net/10454/2683.
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YesAn important issue in many of today's biometric applications is the development of efficient and accurate techniques for representing related 3D data. Such data is often available through the process of digitization of complex geometric objects which are of importance to biometric applications. For example, in the area of 3D face recognition a digital point cloud of data corresponding to a given face is usually provided by a 3D digital scanner. For efficient data storage and for identification/authentication in a timely fashion such data requires to be represented using a few parameters or variables which are meaningful. Here we show how mathematical techniques based on Partial Differential Equations (PDEs) can be utilized to represent complex 3D data where the data can be parameterized in an efficient way. For example, in the case of a 3D face we show how it can be represented using PDEs whereby a handful of key facial parameters can be identified for efficient storage and verification.
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Scotti, Simone. "Applications of the error theory using Dirichlet forms." Phd thesis, Université Paris-Est, 2008. http://tel.archives-ouvertes.fr/tel-00349241.
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This thesis is devoted to the study of the applications of the error theory using Dirichlet forms. Our work is split into three parts. The first one deals with the models described by stochastic differential equations. After a short technical chapter, an innovative model for order books is proposed. We assume that the bid-ask spread is not an imperfection, but an intrinsic property of exchange markets instead. The uncertainty is carried by the Brownian motion guiding the asset. We find that spread evolutions can be evaluated using closed formulae and we estimate the impact of the underlying uncertainty on the related contingent claims. Afterwards, we deal with the PBS model, a new model to price European options. The seminal idea is to distinguish the market volatility with respect to the parameter used by traders for hedging. We assume the former constant, while the latter volatility being an erroneous subjective estimation of the former. We prove that this model anticipates a bid-ask spread and a smiled implied volatility curve. Major properties of this model are the existence of closed formulae for prices, the impact of the underlying drift and an efficient calibration strategy. The second part deals with the models described by partial differential equations. Linear and non-linear PDEs are examined separately. In the first case, we show some interesting relations between the error and wavelets theories. When non-linear PDEs are concerned, we study the sensitivity of the solution using error theory. Except when exact solution exists, two possible approaches are detailed: first, we analyze the sensitivity obtained by taking "derivatives" of the discrete governing equations. Then, we study the PDEs solved by the sensitivity of the theoretical solutions. In both cases, we show that sharp and bias solve linear PDE depending on the solution of the former PDE itself and we suggest algorithms to evaluate numerically the sensitivities. Finally, the third part is devoted to stochastic partial differential equations. Our analysis is split into two chapters. First, we study the transmission of an uncertainty, present on starting conditions, on the solution of SPDE. Then, we analyze the impact of a perturbation of the functional terms of SPDE and the coefficient of the related Green function. In both cases, we show that the sharp and bias verify linear SPDE depending on the solution of the former SPDE itself