Dissertations / Theses on the topic 'Inverse Scattering Transform'

Inverse problems arise in numerous fields of science and engineering where one tries to find out the desired information of an unknown object or the cause of an observed effect. They are of fundamental importance in many areas including radar and sonar applications, nondestructive testing, image processing, medical imaging, remote sensing, geophysics and astronomy among others. This study is concerned with three issues in scattering theory, fractional operators, as well as some of their inverse problems. The first topic is scattering problems for electromagnetic waves governed by Maxwell equations. It will be proved in the current study that an inhomogeneous EM medium with a corner on its support always scatters by assuming certain regularity and admissible conditions. This result implies that one cannot achieve invisibility for such materials. In order to verify the result, an integral of solutions to certain interior transmission problem is to be analyzed, and complex geometry optics solutions to corresponding Maxwell equations with higher order estimate for the residual will be constructed. The second problem involves the linearized elastic or seismic wave scattering described by the Lamei system. We will consider the elastic or seismic body wave which is composed of two different type of sub-waves, that is, the compressional or primary (P-) and the shear or secondary (S-) waves. We shall prove that the P- and the S-components of the total wave can be completely decoupled under certain geometric and boundary conditions. This is a surprising finding since it is known that the P- and the S-components of the elastic or seismic body wave are coupled in general. Results for decoupling around local boundary pieces, for boundary value problems, and for scattering problems are to be established. This decoupling property will be further applied to derive uniqueness and stability for the associated inverse problem of identifying polyhedral elastic obstacles by an optimal number of scattering measurements. Lastly, we consider a type of fractional (and nonlocal) elliptic operators and the associated Calderoin problem. The well-posedness for a kind of forward problems concerning the fractional operator will be established. As a consequence, the corresponding Dirichlet to Neumann map with certain mapping property is to be defined. As for the inverse problem, it will be shown that a potential can be uniquely identified by local Cauchy data of the associated nonlocal operator, in dimensions larger than or equal to two.

Les tâches qui consistent à comprendre automatiquement le contenu d’un signal naturel, comme une image ou un son, sont en général difficiles. En effet, dans leur représentation naïve, les signaux sont des objets compliqués, appartenant à des espaces de grande dimension. Représentés différemment, ils peuvent en revanche être plus faciles à interpréter. Cette thèse s’intéresse à une représentation fréquemment utilisée dans ce genre de situations, notamment pour analyser des signaux audio : le module de la transformée en ondelettes. Pour mieux comprendre son comportement, nous considérons, d’un point de vue théorique et algorithmique, le problème inverse correspondant : la reconstruction d’un signal à partir du module de sa transformée en ondelettes. Ce problème appartient à une classe plus générale de problèmes inverses : les problèmes de reconstruction de phase. Dans un premier chapitre, nous décrivons un nouvel algorithme, PhaseCut, qui résout numériquement un problème de reconstruction de phase générique. Comme l’algorithme similaire PhaseLift, PhaseCut utilise une relaxation convexe, qui se trouve en l’occurence être de la même forme que les relaxations du problème abondamment étudié MaxCut. Nous comparons les performances de PhaseCut et PhaseLift, en termes de précision et de rapidité. Dans les deux chapitres suivants, nous étudions le cas particulier de la reconstruction de phase pour la transformée en ondelettes. Nous montrons que toute fonction sans fréquence négative est uniquement déterminée (à une phase globale près) par le module de sa transformée en ondelettes, mais que la reconstruction à partir du module n’est pas stable au bruit, pour une définition forte de la stabilité. On démontre en revanche une propriété de stabilité locale. Nous présentons également un nouvel algorithme de reconstruction de phase, non-convexe, qui est spécifique à la transformée en ondelettes, et étudions numériquement ses performances. Enfin, dans les deux derniers chapitres, nous étudions une représentation plus sophistiquée, construite à partir du module de transformée en ondelettes : la transformée de scattering. Notre but est de comprendre quelles propriétés d’un signal sont caractérisées par sa transformée de scattering. On commence par démontrer un théorème majorant l’énergie des coefficients de scattering d’un signal, à un ordre donné, en fonction de l’énergie du signal initial, convolé par un filtre passe-haut qui dépend de l’ordre. On étudie ensuite une généralisation de la transformée de scattering, qui s’applique à des processus stationnaires. On montre qu’en dimension finie, cette transformée généralisée préserve la norme. En dimension un, on montre également que les coefficients de scattering généralisés d’un processus caractérisent la queue de distribution du processus
Automatically understanding the content of a natural signal, like a sound or an image, is in general a difficult task. In their naive representation, signals are indeed complicated objects, belonging to high-dimensional spaces. With a different representation, they can however be easier to interpret. This thesis considers a representation commonly used in these cases, in particular for theanalysis of audio signals: the modulus of the wavelet transform. To better understand the behaviour of this operator, we study, from a theoretical as well as algorithmic point of view, the corresponding inverse problem: the reconstruction of a signal from the modulus of its wavelet transform. This problem belongs to a wider class of inverse problems: phase retrieval problems. In a first chapter, we describe a new algorithm, PhaseCut, which numerically solves a generic phase retrieval problem. Like the similar algorithm PhaseLift, PhaseCut relies on a convex relaxation of the phase retrieval problem, which happens to be of the same form as relaxations of the widely studied problem MaxCut. We compare the performances of PhaseCut and PhaseLift, in terms of precision and complexity. In the next two chapters, we study the specific case of phase retrieval for the wavelet transform. We show that any function with no negative frequencies is uniquely determined (up to a global phase) by the modulus of its wavelet transform, but that the reconstruction from the modulus is not stable to noise, for a strong notion of stability. However, we prove a local stability property. We also present a new non-convex phase retrieval algorithm, which is specific to the case of the wavelet transform, and we numerically study its performances. Finally, in the last two chapters, we study a more sophisticated representation, built from the modulus of the wavelet transform: the scattering transform. Our goal is to understand which properties of a signal are characterized by its scattering transform. We first prove that the energy of scattering coefficients of a signal, at a given order, is upper bounded by the energy of the signal itself, convolved with a high-pass filter that depends on the order. We then study a generalization of the scattering transform, for stationary processes. We show that, in finite dimension, this generalized transform preserves the norm. In dimension one, we also show that the generalized scattering coefficients of a process characterize the tail of its distribution

ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.

Thesis (Ph. D.)--University of Sydney, 1999.
Title from title screen (viewed Apr. 21, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.

Since the advent of the first ionizing radiation imaging devices initiated by Godfrey Newbold Hounsfield and Allan MacLeod Cormack, Nobel Prizes in 1979, the requirement for new non-invasive imaging techniques has grown. These techniques rely upon the properties of penetration in the matter of X and gamma radiation for detecting a hidden structure without destroying the illuminated environment. They are used in many fields ranging from medical imaging to non-destructive testing through. However, the techniques used so far suffer severe degradation in the quality of measurement and reconstructed images. Usually approximated by a noise, these degradations require to be compensated or corrected by collimating devices and often expensive filtering. These degradation is mainly due to scattering phenomena which may constitute up to 80% of the emitted radiation in biological tissue. In the 80's a new concept has emerged to circumvent this difficulty : the Compton scattering tomography (CST).This new approach proposes to measure the scattered radiation considering energy ranges ( 140-511 keV) where the Compton effect is the phenomenon of leading broadcast. The use of such imaging devices requires a deep understanding of the interactions between radiation and matter to propose a modeling, consistent with the measured data, which is essential to image reconstruction. In conventional imaging systems (which measure the primary radiation) the Radon transformdefined on the straight lines emerged as the natural modeling. But in Compton scattering tomography, the measured information is related to the scattering energy and thus the scattering angle. Thus the circular geometry induced by scattering phenomenon makes the classical Radon transform inadequate.In this context, it becomes necessary to provide such Radon transforms on broader geometric manifolds.The study of the Radon transform on new manifolds of curves becomes necessary to provide theoretical needs for new imaging techniques. Cormack, himself, was the first to extend the properties of the conventional Radon transform of a family of curves of the plane. Thereafter several studies have been done in order to study the Radon transform defined on different varieties of circles, spheres, broken lines ... . In 1994 S.J. Norton proposed the first modality in Compton scattering tomography modeled by a Radon transform on circular arcs, the CART1 here. In 2010, Nguyen and Truong established the inversion formula of a Radon transform on circular arcs, CART2, to model the image formation in a new modality in Compton scattering tomography. The geometry involved in the integration support of new modalities in Compton scattering tomography lead them to demonstrate the invertibility of the Radon transform defined on a family of Cormack-type curves, called C_alpha. They illustrated the inversion procedure in the case of a new transform, the CART3, modeling a new modeling of Compton scattering tomography. Based on the work of Cormack and Truong and Nguyen, we propose to establish several properties of the Radon transform on the family C_alpha especially on C1. We have thus demonstrated two inversion formulae that reconstruct the original image via its circular harmonic decomposition and itscorresponding transform. These formulae are similar to those established by Truong and Nguyen. We finally established the well-known filtered back projection and singular value decomposition in the case alpha = 1. All results established in this study provide practical problems of image reconstruction associated with these new transforms. In particular we were able to establish new inversion methods for transforms CART1,2,3 as well as numerical approaches necessary for the implementation of these transforms. All these results enable to solve problems of image formation and reconstruction related to three Compton scattering tomography modalities.In addition we propose to improve models and algorithms es

La caractérisation tridimensionnelle de matériaux anciens plats est restée une activité non évidente à accomplir par des méthodes classiques de tomographie à rayons X en raison de leur morphologie anisotrope et de leur géométrie aplatie.Pour surmonter les limites de ces méthodologies, une modalité d'imagerie basée sur le rayonnement diffusé Compton est étudiée dans ce travail. La tomographie classique aux rayons X traite les données de diffusion Compton comme du bruit ajouté au processus de formation d'image, tandis que dans la tomographie du rayonnement diffusé, les conditions sont définies de sorte que la diffusion inélastique devienne le phénomène dominant dans la formation d'image. Dans ces conditions, les rotations relatives entre l'échantillon et la configuration d'imagerie ne sont plus nécessaires. Mathématiquement, ce problème est résolu par la transformée de Radon conique. Le problème direct où la sortie du système est l'image spectrale obtenue à partir d'un objet d'entrée est modélisé. Dans le problème inverse une estimation de la distribution tridimensionnelle de la densité électronique de l'objet d'entrée à partir de l'image spectrale est proposée. La faisabilité de cette méthodologie est supportée par des simulations numériques
Three-dimensional characterization of flat ancient material objects has remained a challenging activity to accomplish by conventional X-ray tomography methods due to their anisotropic morphology and flattened geometry.To overcome the limitations of such methodologies, an imaging modality based on Compton scattering is studied in this work. Classical X-ray tomography treats Compton scattering data as noise in the image formation process, while in Compton scattering tomography the conditions are set such that Compton data become the principal image contrasting agent. Under these conditions, we are able to avoid relative rotations between the sample and the imaging setup. Mathematically this problem is addressed by means of the conical Radon transform. A model of the direct problem is presented where the output of the system is the spectral image obtained from an input object. The inverse problem is addressed to estimate the 3D distribution of the electronic density of the input object from the spectral image. The feasibility of this methodology is supported by numerical simulations

The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schr\"odinger and the Gross-Pitaevskii equations.Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Transform (IST) can be exploited to obtain asymptotic information about solutions.This thesis will describe the IST of the Manakov system, and its asymptotic behavior at short times. I will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, I will show that the continuous spectrum gives the dominant contribution at short-times.

We clear up two issues regarding the eigenvalue problem for the Manakov system; these problems relate directly to the existence of the soliton [\emph{sic}] effect in fiber optic cables. The first issue is a bound on the eigenvalues of the Manakov system: \emph{if} the parameter $\xi$ is an eigenvalue, \emph{then} it must lie in a certain region in the complex plane. The second issue has to do with a chirped Manakov system. We show that if a system is chirped too much, the soliton effect disappears. While this has been known for some time experimentally, there has not yet been a theoretical result along these lines for the Manakov system.
Ph. D.

Doctor of Philosophy
Department of Mathematics
Alexander G. Ramm
In this dissertation, the author presents two independent researches on inverse problems: (1) creating materials in which heat propagates a long a line and (2) 3D inverse scattering problem with non-over-determined data. The theories of these methods were developed by Professor Alexander Ramm and are presented in Chapters 1 and 3. The algorithms and numerical results are taken from the papers of Professor Alexander Ramm and the author and are presented in Chapters 2 and 4.

Even though most of the properties of optical fields, such as wavelength, polarization, wavefront curvature or angular spectrum, have been commonly manipulated in a variety of remote sensing procedures, controlling the degree of coherence of light did not find wide applications until recently. Since the emergence of optical coherence tomography, a growing number of scattering techniques have relied on temporal coherence gating which provides efficient target selectivity in a way achieved only by bulky short pulse measurements. The spatial counterpart of temporal coherence, however, has barely been exploited in sensing applications. This dissertation examines, in different scattering regimes, a variety of inverse scattering problems based on variable spatial coherence gating. Within the framework of the radiative transfer theory, this dissertation demonstrates that the short range correlation properties of a medium under test can be recovered by varying the size of the coherence volume of an illuminating beam. Nonetheless, the radiative transfer formalism does not account for long range correlations and current methods for retrieving the correlation function of the complex susceptibility require cumbersome cross-spectral density measurements. Instead, a variable coherence tomographic procedure is proposed where spatial coherence gating is used to probe the structural properties of single scattering media over an extended volume and with a very simple detection system. Enhanced backscattering is a coherent phenomenon that survives strong multiple scattering. The variable coherence tomography approach is extended in this context to diffusive media and it is demonstrated that specific photon trajectories can be selected in order to achieve depth-resolved sensing. Probing the scattering properties of shallow and deeper layers is of considerable interest in biological applications such as diagnosis of skin related diseases. The spatial coherence properties of an illuminating field can be manipulated over dimensions much larger than the wavelength thus providing a large effective sensing area. This is a practical advantage over many near-field microscopic techniques, which offer a spatial resolution beyond the classical diffraction limit but, at the expense of scanning a probe over a large area of a sample which is time consuming, and, sometimes, practically impossible. Taking advantage of the large field of view accessible when using the spatial coherence gating, this dissertation introduces the principle of variable coherence scattering microscopy. In this approach, a subwavelength resolution is achieved from simple far-zone intensity measurements by shaping the degree of spatial coherence of an evanescent field. Furthermore, tomographic techniques based on spatial coherence gating are especially attractive because they rely on simple detection schemes which, in principle, do not require any optical elements such as lenses. To demonstrate this capability, a correlated lensless imaging method is proposed and implemented, where both amplitude and phase information of an object are obtained by varying the degree of spatial coherence of the incident beam. Finally, it should be noted that the idea of using the spatial coherence properties of fields in a tomographic procedure is applicable to any type of electromagnetic radiation. Operating on principles of statistical optics, these sensing procedures can become alternatives for various target detection schemes, cutting-edge microscopies or x-ray imaging methods.
Ph.D.
Other
Optics and Photonics
Optics

In this thesis we present new ideas and mathematical insights in the field of Compton Scattering Tomography (CST), an X-ray and gamma ray imaging technique which uses Compton scattered data to reconstruct an electron density of the target. This is an area not considered extensively in the literature, with only two dimensional gamma ray (monochromatic source) CST problems being analysed thus far. The analytic treatment of the polychromatic source case is left untouched and while there are three dimensional acquisition geometries in CST which consider the reconstruction of gamma ray source intensities, an explicit three dimensional electron density reconstruction from Compton scatter data is yet to be obtained. Noting this gap in the literature, we aim to make new and significant advancements in CST, in particular in answering the questions of the three dimensional density reconstruction and polychromatic source problem. Specifically we provide novel and conclusive results on the stability and uniqueness properties of two and three dimensional inverse problems in CST through an analysis of a disc transform and a generalized spindle torus transform. In the final chapter of the thesis we give a novel analysis of the stability of a spindle torus transform from a microlocal perspective. The practical application of our inversion methods to fields in X-ray and gamma ray imaging are also assessed through simulation work.

En utilisant l'electron hydrate comme sonde intramicellaire, etude du comportement de l'eau au sein des microphases aqueuses, et du transfert d'electron de la chlorophylle vers des viologenes dans la microemulsion

We are concerned with the quantum inverse scattering problem. The corresponding Marchenko integral equation is solved by using the collocation method together with piece-wise polynomials, namely, Hermite splines. The scarcity of experimental data and the lack of phase information necessitate the generation of the input reflection coefficient by choosing a specific profile and then applying our method to reconstruct it. Various aspects of the single and coupled channels inverse problem and details about the numerical techniques employed are discussed. We proceed to apply our approach to synthetic seismic reflection data. The transformation of the classical one-dimensional wave equation for elastic displacement into a Schr¨odinger-like equation is presented. As an application of our method, we consider the synthetic reflection travel-time data for a layered substrate from which we recover the seismic impedance of the medium. We also apply our approach to experimental seismic reflection data collected from a deep water location in the North sea. The reflectivity sequence and the relevant seismic wavelet are extracted from the seismic reflection data by applying the statistical estimation procedure known as Markov Chain Monte Carlo method to the problem of blind deconvolution. In order to implement the Marchenko inversion method, the pure spike trains have been replaced by amplitudes having a narrow bell-shaped form to facilitate the numerical solution of the Marchenko integral equation from which the underlying seismic impedance profile of the medium is obtained.
Physics
D.Phil.(Physics)

Zhang, Hai.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.
Includes bibliographical references (leaves 106-109).
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.5
Chapter 2 --- Preliminaries --- p.13
Chapter 2.1 --- Maxwell equations --- p.13
Chapter 2.2 --- Reflection principle --- p.15
Chapter 3 --- Scattering by General Polyhedral Obstacle --- p.19
Chapter 3.1 --- Direct problem --- p.19
Chapter 3.2 --- Inverse problem and statement of main results --- p.21
Chapter 3.3 --- Proof of the main results --- p.22
Chapter 3.3.1 --- Preliminaries --- p.23
Chapter 3.3.2 --- Properties of perfect planes --- p.24
Chapter 3.3.3 --- Proofs --- p.33
Chapter 4 --- Scattering by Bi-periodic Polyhedral Grating (I) --- p.35
Chapter 4.1 --- Direct problem --- p.36
Chapter 4.2 --- Inverse problem and statement of main results --- p.38
Chapter 4.3 --- Preliminaries --- p.39
Chapter 4.4 --- Classification of unidentifiable periodic structures --- p.41
Chapter 4.4.1 --- Observations and auxiliary tools --- p.41
Chapter 4.4.2 --- First class of unidentifiable gratings --- p.45
Chapter 4.4.3 --- Preparation for finding other classes of unidentifiable gratings --- p.47
Chapter 4.4.4 --- A simple transformation --- p.52
Chapter 4.4.5 --- Second class of unidentifiable gratings --- p.53
Chapter 4.4.6 --- Third class of unidentifiable gratings --- p.58
Chapter 4.4.7 --- Excluding the case with L --- p.61
Chapter 4.4.8 --- Summary on all unidentifiable gratings --- p.65
Chapter 4.5 --- Proof of Main results --- p.65
Chapter 5 --- Scattering by Bi-periodic Polyhedral Grating (II) --- p.69
Chapter 5.1 --- Preliminaries --- p.70
Chapter 5.2 --- Classification of unidentifiable periodic structures --- p.72
Chapter 5.2.1 --- First class of unidentifiable gratings --- p.72
Chapter 5.2.2 --- Preparation for finding other classes of unidentifiable gratings --- p.73
Chapter 5.2.3 --- Studying of the case L --- p.76
Chapter 5.2.4 --- Study of the case with L --- p.89
Chapter 5.2.5 --- Study of the case with L --- p.95
Chapter 5.2.6 --- Summary on all unidentifiable gratings --- p.104
Chapter 5.3 --- Unique determination of bi-periodic polyhedral grating --- p.104
Bibliography --- p.106

Deng, Xiaomao.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.
Includes bibliographical references (leaves 98-104).
Chapter 1 --- Introduction to Inverse Scattering Problems --- p.6
Chapter 1.1 --- Direct Problems --- p.6
Chapter 1.1.1 --- Far-field Patterns --- p.10
Chapter 1.2 --- Inverse Problems --- p.16
Chapter 1.2.1 --- Introduction --- p.16
Chapter 2 --- Numerical Methods in Inverse Obstacle Scattering --- p.19
Chapter 2.1 --- Linear Sampling Method --- p.19
Chapter 2.1.1 --- History Review --- p.19
Chapter 2.1.2 --- Numerical Scheme of LSM --- p.21
Chapter 2.1.3 --- Theoretic Justification --- p.25
Chapter 2.1.4 --- Summarize --- p.38
Chapter 2.2 --- Point Source Method --- p.38
Chapter 2.2.1 --- Historical Review --- p.38
Chapter 2.2.2 --- Superposition of Plane Waves --- p.40
Chapter 2.2.3 --- Approximation of Domains --- p.42
Chapter 2.2.4 --- Algorithm --- p.44
Chapter 2.2.5 --- Summarize --- p.49
Chapter 2.3 --- Singular Source Method --- p.49
Chapter 2.3.1 --- Historical Review --- p.49
Chapter 2.3.2 --- Algorithm --- p.51
Chapter 2.3.3 --- Far-field Data --- p.54
Chapter 2.3.4 --- Summarize --- p.55
Chapter 2.4 --- Probe Method --- p.57
Chapter 2.4.1 --- Historical Review --- p.57
Chapter 2.4.2 --- Needle --- p.58
Chapter 2.4.3 --- Algorithm --- p.59
Chapter 3 --- Numerical Experiments --- p.61
Chapter 3.1 --- Discussions on Linear Sampling Method --- p.61
Chapter 3.1.1 --- Regularization Strategy --- p.61
Chapter 3.1.2 --- Cut off Value --- p.70
Chapter 3.1.3 --- Far-field data --- p.76
Chapter 3.2 --- Numerical Verification of PSM and SSM --- p.80
Chapter 3.3 --- Inverse Medium Scattering --- p.83
Bibliography --- p.98

碩士
國立成功大學
化學工程研究所
82
This thesis studies the analytic solutions to nonlinear wave equations by using the nonlinear Fourier transform, which is also called the inverse scattering transform. It also explores the applications of nonlinear Fourier transform to the analysis of nonlinear wave data. According to the characteristics of solutions on the domain boundaries, the method is applied to the cases of infinite interval and periodic boundary conditions. Besides, the application of nonlinear Fourier transform to the signal processing is studied. Moreover, the differences between the linear and nonlinear Fourier analysis are highlighted. In general, nonlinear Fourier transform is useful for nonlinear problems. It needs a broader mathematical knowledge. The required mathematical background involves the Riemann surface, Riemann .theta.-function, Cauchy's residue theorem, and even the Floquet theory. All of these mathematical pre-requsite are reviewed. Besides giving tedious mathematical derivations, the thesis also gives a numerical algorithm to obtain the nonlinear Fourier spectra. Furthermore, the reconstruction of potential functions from these spectra is achieved by using the inverse scattering transform. From the standpoint of signal processing, this problem is equivalent to signal decomposition and reconstruction. As a result, the nonlinear Fourier transform would serve as a supplement for the original Fourier transform.

Wang, Yiran.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.
Includes bibliographical references (leaves 70-73).
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.6
Chapter 1.1 --- Overview of the subject --- p.6
Chapter 1.2 --- Motivation --- p.8
Chapter 2 --- Inverse Medium Scattering Problem --- p.11
Chapter 2.1 --- Mathematical Formulation --- p.11
Chapter 2.1.1 --- Absorbing Boundary Conditions --- p.12
Chapter 2.1.2 --- Applications --- p.14
Chapter 2.2 --- Preliminary Results --- p.17
Chapter 2.2.1 --- Weak Formulation --- p.17
Chapter 2.2.2 --- About the Unique Determination --- p.21
Chapter 3 --- Unconstrained Optimization: Steepest Decent Method --- p.25
Chapter 3.1 --- Recursive Linearization Method Revisited --- p.25
Chapter 3.1.1 --- Frechet differentiability --- p.26
Chapter 3.1.2 --- Initial guess --- p.28
Chapter 3.1.3 --- Landweber iteration --- p.30
Chapter 3.1.4 --- Numerical Results --- p.32
Chapter 3.2 --- Steepest Decent Analysis --- p.35
Chapter 3.2.1 --- Single Wave Case --- p.36
Chapter 3.2.2 --- Multiple Wave Case --- p.39
Chapter 3.3 --- Numerical Experiments and Discussions --- p.43
Chapter 4 --- Constrained Optimization: Augmented Lagrangian Method --- p.51
Chapter 4.1 --- Method Review --- p.51
Chapter 4.2 --- Problem Formulation --- p.54
Chapter 4.3 --- First Order Optimality Condition --- p.56
Chapter 4.4 --- Second Order Optimality Condition --- p.60
Chapter 4.5 --- Modified Algorithm --- p.62
Chapter 5 --- Conclusions and Future Work --- p.68
Bibliography --- p.70

Thesis (Ph.D.)--University of Alberta, 2009.
Title from PDF file main screen (viewed on Apr. 1, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Electrical and Computer Engineering, University of Alberta. Includes bibliographical references.