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Birkhoff Normal Form with Application to Gross Pitaevskii EquationYan, Zhenbin 10 1900 (has links)
<p>L^p is supposed to be L with a superscript lower case 'p.'</p> / <p>This thesis investigates a 1-dimensional Gross-Pitaevskii (GP) equation from the viewpoint of a system of Hamiltonian partial differential equations (PDEs). A theorem on Birkhoff normal forms is a particularly important goal of this study. The resulting system is a perturbed system of a completely resonant system, which we analyze, using several forms of perturbation theory.</p> <p>In chapter two, we study estimates 011 integrals of products of four Hermite functions, which represent coefficients of mode coupling, and play an important role in the proof of the Birkhoff normal form theorem. This is a basic problem, which has a close relationship with a problem of Besicovitch, namely the behavior of the L^p norms of L² -normalized Hermite functions.</p> <p>In chapter three we carefully reconsider the linear Schrodinger equation with a harmonic potential, and we introduce a family of Hilbert spaces for studying the GP equation, which generalize the traditional energy spaces in which one works. One unexpected fact is that these function spaces have a close relationship with the former works for the tempered distributions, in particular the N-representation theory due to B. Simon, and V. Bargmann's theory, which uncovers relationship between the tempered distributions and his function spaces through the so-called Segal-Bargmann transformation. In addition, our function spaces have a nice relationship with the Sobolev spaces. In this chapter, a few other questions regarding these function spaces are discussed.</p> <p>In chapter four the proof of the Birkhoff normal form theorem on spaces we have introduced are provided. The analysis is divided into two cases according to the regularity of the related function space. After proving the Birkhoff normal form theorem, we made an analysis of the impact of the perturbation on the main part of the GP system, which we remark is completel:y resonant.</p> / Doctor of Philosophy (PhD)
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OpenLB-Open source lattice Boltzmann codeKrause, M.J., Kummerländer, A., Avis, S.J., Kusumaatmaja, H., Dapelo, Davide, Klemens, F., Gaedtke, M., Hafen, N., Mink, A., Marquardt, J.E., Maier, M.-L., Haussmann, M., Simonis, S. 25 November 2020 (has links)
Yes / We present the OpenLB package, a C++ library providing a flexible framework for lattice Boltzmann simulations. The code is publicly available and published under GNU GPLv2, which allows for adaption and implementation of additional models. The extensibility benefits from a modular code structure achieved e.g. by utilizing template meta-programming. The package covers various methodical approaches and is applicable to a wide range of transport problems (e.g. fluid, particulate and thermal flows). The built-in processing of the STL file format furthermore allows for the simple setup of simulations in complex geometries. The utilization of MPI as well as OpenMP parallelism enables the user to perform those simulations on large-scale computing clusters. It requires a minimal amount of dependencies and includes several benchmark cases and examples. The package presented here aims at providing an open access platform for both, applicants and developers, from academia as well as industry, which facilitates the extension of previous implementations and results to novel fields of application for lattice Boltzmann methods. OpenLB was tested and validated over several code reviews and publications. This paper summarizes the findings and gives a brief introduction to the underlying concepts as well as the design of the parallel data structure.
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Asymptotic and Factorization Analysis for Inverse Shape Problems in Tomography and Scattering TheoryGovanni Granados (18283216) 01 April 2024 (has links)
<p dir="ltr">Developing non-invasive and non-destructive testing in complex media continues to be a rich field of study (see e.g.[22, 28, 36, 76, 89] ). These types of tests have applications in medical imaging, geophysical exploration, and engineering where one would like to detect an interior region or estimate a model parameter. With the current rapid development of this enabling technology, there is a growing demand for new mathematical theory and computational algorithms for inverse problems in partial differential equations. Here the physical models are given by a boundary value problem stemming from Electrical Impedance Tomography (EIT), Diffuse Optical Tomography (DOT), as well as acoustic scattering problems. Important mathematical questions arise regarding existence, uniqueness, and continuity with respect to measured surface data. Rather than determining the solution of a given boundary value problem, we are concerned with using surface data in order to develop and implement numerical algorithms to recover unknown subregions within a known domain. A unifying theme of this thesis is to develop Qualitative Methods to solve inverse shape problems using measured surface data. These methods require very few a priori assumptions on the regions of interest, boundary conditions, and model parameter estimation. The counterpart to qualitative methods, iterative methods, typically require a priori information that may not be readily available and can be more computationally expensive. Qualitative Methods usually require more data.</p><p dir="ltr">This thesis expands the library of Qualitative Methods for elliptic problems coming from tomography and scattering theory. We consider inverse shape problems where our goal is to recover extended and small volume regions. For extended regions, we consider applying a modified version of the well-known Factorization Method [73]. Whereas for the small volume regions, we develop a Multiple Signal Classification (MUSIC)-type algorithm (see for e.g. [3, 5]). In all of our problems, we derive an imaging functional that will effectively recover the region of interest. The results of this thesis form part of the theoretical forefront of physical applications. Furthermore, it extends the mathematical theory at the intersection of mathematics, physics and engineering. Lastly, it also advances knowledge and understanding of imaging techniques for non-invasive and non-destructive testing.</p>
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Numerical Methods for Accurate Computation of Design SensitivitiesStewart, Dawn L. 23 July 1998 (has links)
This work is concerned with the development of computational methods for approximating sensitivities of solutions to boundary value problems. We focus on the continuous sensitivity equation method and investigate the application of adaptive meshing and smoothing projection techniques to enhance the basic scheme. The fundamental ideas are first developed for a one dimensional problem and then extended to 2-D flow problems governed by the incompressible Navier-Stokes equations. Numerical experiments are conducted to test the algorithms and to investigate the benefits of adaptivity and smoothing. / Ph. D.
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A Flexible Galerkin Finite Element Method with an A Posteriori Discontinuous Finite Element Error Estimation for Hyperbolic ProblemsMassey, Thomas Christopher 15 July 2002 (has links)
A Flexible Galerkin Finite Element Method (FGM) is a hybrid class of finite element methods that combine the usual continuous Galerkin method with the now popular discontinuous Galerkin method (DGM). A detailed description of the formulation of the FGM on a hyperbolic partial differential equation, as well as the data structures used in the FGM algorithm is presented. Some hp-convergence results and computational cost are included. Additionally, an a posteriori error estimate for the DGM applied to a two-dimensional hyperbolic partial differential equation is constructed. Several examples, both linear and nonlinear, indicating the effectiveness of the error estimate are included. / Ph. D.
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Shape morphing of complex geometries using partial differential equations.Gonzalez Castro, Gabriela, Ugail, Hassan January 2007 (has links)
An alternative technique for shape morphing
using a surface generating method using partial differential
equations is outlined throughout this work. The boundaryvalue
nature that is inherent to this surface generation
technique together with its mathematical properties are
hereby exploited for creating intermediate shapes between
an initial shape and a final one. Four alternative shape
morphing techniques are proposed here. The first one is
based on the use of a linear combination of the boundary
conditions associated with the initial and final surfaces,
the second one consists of varying the Fourier mode for
which the PDE is solved whilst the third results from a
combination of the first two. The fourth of these alternatives
is based on the manipulation of the spine of the surfaces,
which is computed as a by-product of the solution. Results
of morphing sequences between two topologically nonequivalent
surfaces are presented. Thus, it is shown that the
PDE based approach for morphing is capable of obtaining
smooth intermediate surfaces automatically in most of the
methodologies presented in this work and the spine has been
revealed as a powerful tool for morphing surfaces arising
from the method proposed here.
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Reconstruction of 3D human facial images using partial differential equations.Elyan, Eyad, Ugail, Hassan January 2007 (has links)
One of the challenging problems in geometric
modeling and computer graphics is the construction of
realistic human facial geometry. Such geometry are
essential for a wide range of applications, such as 3D face
recognition, virtual reality applications, facial expression
simulation and computer based plastic surgery application.
This paper addresses a method for the construction of 3D
geometry of human faces based on the use of Elliptic Partial
Differential Equations (PDE). Here the geometry
corresponding to a human face is treated as a set of surface
patches, whereby each surface patch is represented using
four boundary curves in the 3-space that formulate the
appropriate boundary conditions for the chosen PDE. These
boundary curves are extracted automatically using 3D data
of human faces obtained using a 3D scanner. The solution of
the PDE generates a continuous single surface patch
describing the geometry of the original scanned data. In this
study, through a number of experimental verifications we
have shown the efficiency of the PDE based method for 3D
facial surface reconstruction using scan data. In addition to
this, we also show that our approach provides an efficient
way of facial representation using a small set of parameters
that could be utilized for efficient facial data storage and
verification purposes.
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Method of numerical simulation of stable structures of fluid membranes and vesicles.Ugail, Hassan, Jamil, N., Satinoianu, R. January 2006 (has links)
In this paper we study a methodology for the numerical simulation of stable structures of fluid membranes and vesicles in biological organisms. In particular, we discuss the effects of spontaneous curvature on vesicle cell membranes under the bending energy for given volume and surface area. The geometric modeling of the vesicle shapes are undertaken by means of surfaces generated as Partial Differential Equations (PDEs). We combine PDE based geometric modeling with numerical optimization in order to study the stable shapes adopted by the vesicle membranes. Thus, through the PDE method we generate a generic template of a vesicle membrane which is then efficiently parameterized. The parameterization is taken as a basis to set up a numerical optimization procedure which enables us to predict a series of vesicle shapes subject to given surface area and volume.
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Efficient 3D data representation for biometric applicationsUgail, Hassan, Elyan, Eyad January 2007 (has links)
Yes / An 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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Time-dependent shape parameterisation of complex geometry using PDE surfacesUgail, Hassan January 2004 (has links)
Yes
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