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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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A Translating Fluxmeter for Solenoid MeasurementsMattsson Kjellqvist, Ville January 2024 (has links)
At the European Institute for Nuclear Research, CERN, a new electron cooler is being commissioned for the Antiproton Decelerator experiment. In this experiment protons are shot into a block of metal, which creates anti protons. These anti protons will thereafter be focused into a particle beam, a process done in several steps. One of these steps is with an electron cooler. This cooler shoots electrons into the ion-beam path. These electrons then collide with the beam particles, and momentum is transferred from the beam particles to the electrons. The electrons are then steered away from the beam path, into an electron collector. In the beam path drift of the cooler, where the anti protons and electrons meet, a normal conducting solenoid magnet is used to orient the electron path. This magnet comes with strict requirements on field quality, such that the transversal magnetic field must be less than 10 ppm of the lateral field. In this thesis a metrological characterization of a prototype measurement system for solenoidal magnets is presented. Instead of winding measurement coils with wire, they are instead printed on a circuit board over ten layers. Of particular interest was the magnet alignment with respect to the beam aperture, so that the magnetic solenoid axis is in line with the aperture central axis. For this purpose, a mathematical model for solenoidal magnetic fields has been constructed. This model can be used to quantify the sensitivity of the measurement system for an unaligned magnet. Furthermore, some test measurements are presented, along with some simulation campaigns to further characterize the problem. A specific method where the magnetic field peaks are used to measure the alignment is evaluated. / På den Europeiska organisationen för Kärnforskning pågår just nu ett uppgraderingsarbete för AD-experimentet, (fullständigt namn på engelska: Antiproton Decelerator). I detta experiment skjuts protoner in i ett block med metall, vilket skapar antiprotoner. Dessa antiprotoner ska sedan fokuseras till en partikelstråle, vilket görs i en rad olika steg, däribland med vad som kallas för en elektronkylare. Elektronkylaren skjuter in elektroner i partikelstrålens väg, vilka kolliderar med antiprotonerna och på detta sätt reducerar temperaturen i partikelstrålen genom att överföra momentum till elektronerna. Elektronerna leds sedan bort ur strålens väg, in i en elektronsamlare.I strålaperturen, där elektronerna och antiprotonerna möts, används en normalledande solenoidmagnet för att styra elektronerna. Dennasolenoidmagnet kommer med strikta krav på den magnetiska fältprofilen,varför känslig mätutrustning krävs. Det magnetiska fältet måste vara av solenoid karaktär, så att det transversella fältet är mindre än 10 ppm av det longitudinella. I denna rapport presenteras en metrologisk karaktärisering av en ny prototyp på mätsystem för solenoidmagneter. Istället för att linda spolar som en mäter fältkvalitén med, så har dessa istället tryckts på ett kretskort över tio lager. Av speciellt intresse var att mäta magnetens justering, så att solenoidaxeln ligger i linje med strålaperturen. För detta ändamål så har en matematisk modell för solenoida magnetfält konstruerats. Denna modell kan användas för att kvantifiera känsligheten hos mätsystemet för en ojusterad solenoidmagnet. Vidare så presenteras testmätningar med systemet, samt en rad simulationer för att vidare karaktärisera problemet. En specifik mätmetodik där magnetfältstopparna används för att undersöka magnetens justering utvärderas.
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Generalized partial differential equations for interactive designUgail, Hassan January 2007 (has links)
Yes / This paper presents a method for interactive design by means of extending the PDE
based approach for surface generation. The governing partial differential equation is
generalized to arbitrary order allowing complex shapes to be designed as single patch
PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating
the geometry of shape that satisfying an arbitrary set of boundary conditions.
Both the boundary conditions which are defined as curves in 3-space and the spine of the
corresponding PDE are utilized as interactive design tools for creating and manipulating
geometry intuitively. In order to facilitate interactive design in real time, a compact
analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme
even in the case of general boundary conditions satisfies exactly the boundary conditions
where the resulting surface has an closed form representation allowing real time
shape manipulation. In order to enable users to appreciate the powerful shape design
and manipulation capability of the method, we present a set of practical examples.
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