published_or_final_version / Physics / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Hong Kong, 1988.

This thesis is in the area of ultrasound tomography, which is a non-destructive imaging method that attempts to create quantitative images of the acoustical properties of an object of interest (OI). Specifically, three quantitative images per OI are created in this thesis, two of which correspond to the complex compressibility profile of the OI, and the other corresponds to its density profile. The focus of this thesis is on the development of an appropriate two-dimensional inverse scattering algorithm to create these quantitative images. The core of this algorithm is the Born iterative method that is used in conjunction with a fast and efficient method of moments forward solver, a Krylov subspace regularization technique, and a balancing method. This inversion algorithm is capable of simultaneous inversion of multiple-frequency data, and can handle a large imaging domain. This algorithm is finally tested against synthetic and measured data.

This thesis reports on research undertaken in the area of microwave tomography (MWT) where the goal is to find the dielectric profile of an object of interest using microwave measurements collected outside the object. The main focus of this research is on the development of inversion algorithms which solve the electromagnetic inverse scattering problem associated with MWT. Various regularization techniques for the Gauss-Newton inversion algorithm are studied and classified. It is shown that these regularization techniques can be viewed from within a single consistent framework after applying some modifications. Within the framework of the two-dimensional MWT problem, the inversion of transverse magnetic and transverse electric data sets are considered and compared in terms of computational complexity, image quality and convergence rate. A new solution to the contrast source inversion formulation of the microwave tomography problem for the case where the MWT chamber consists of a circular conductive enclosure is introduced. This solution is based on expressing the unknowns of the problem as truncated eigenfunction expansions corresponding to the Helmholtz operator for a homogeneous background medium with appropriate boundary conditions imposed at the chamber walls. The MWT problem is also formulated for MWT chambers made of conducting cylinders of arbitrary shapes. It is shown that collecting microwave scattered-field data inside MWT setups with different boundary conditions can provide a robust set of useful information for the reconstruction of the dielectric profile. This leads to a novel MWT setup wherein a rotatable conductive triangular enclosure is used to generate scattered-field data. Antenna arrays, with as few as only four elements, that are fixed with respect to the object of interest can provide sufficient data to give good reconstructions, if the triangular enclosure is rotated a sufficient number of times. Preliminary results of using the algorithms presented herein on data collected using two different MWT prototypes currently under development at the University of Manitoba are reported. Using the open-region MWT prototype, a resolution study using the Gauss-Newton inversion method was performed using various cylindrical targets.

This thesis reports on research undertaken in the area of microwave tomography (MWT) where the goal is to find the dielectric profile of an object of interest using microwave measurements collected outside the object. The main focus of this research is on the development of inversion algorithms which solve the electromagnetic inverse scattering problem associated with MWT. Various regularization techniques for the Gauss-Newton inversion algorithm are studied and classified. It is shown that these regularization techniques can be viewed from within a single consistent framework after applying some modifications. Within the framework of the two-dimensional MWT problem, the inversion of transverse magnetic and transverse electric data sets are considered and compared in terms of computational complexity, image quality and convergence rate. A new solution to the contrast source inversion formulation of the microwave tomography problem for the case where the MWT chamber consists of a circular conductive enclosure is introduced. This solution is based on expressing the unknowns of the problem as truncated eigenfunction expansions corresponding to the Helmholtz operator for a homogeneous background medium with appropriate boundary conditions imposed at the chamber walls. The MWT problem is also formulated for MWT chambers made of conducting cylinders of arbitrary shapes. It is shown that collecting microwave scattered-field data inside MWT setups with different boundary conditions can provide a robust set of useful information for the reconstruction of the dielectric profile. This leads to a novel MWT setup wherein a rotatable conductive triangular enclosure is used to generate scattered-field data. Antenna arrays, with as few as only four elements, that are fixed with respect to the object of interest can provide sufficient data to give good reconstructions, if the triangular enclosure is rotated a sufficient number of times. Preliminary results of using the algorithms presented herein on data collected using two different MWT prototypes currently under development at the University of Manitoba are reported. Using the open-region MWT prototype, a resolution study using the Gauss-Newton inversion method was performed using various cylindrical targets.

Neste trabalho são apresentados alguns resultados matemáticos relevantes para a aplicação do método de espalhamento inverso à resolução de uma classe de equações de evolução não-lineares. É demonstrada a propriedade isoespectral para certas famílias de operadores lineares não auto-adjuntos. Esta propriedade tem um papel central na aplicação do método acima a equações de evolução não-lineares de interesse físico, tais como a equação de sine-Gordon e a equação de Schrödinger não-linear. É feito também, uma teoria de espalhamento inverso rigorosa para sistemas do tipo Zakharov-Shabat, o que inclui uma análise qualitativa do espectro de operadores deste tipo. / This Thesis presents some mathematical results relevant in applications of the inverse scattering transform to the solution of a class of non-linear evolution equations. First, it is shown that certain families of non-selfadjoint linear operators have the isospectral property, which is fundamental for the above applications. These families include various operators related to no-linear equations of great physical interest, such as the sine-Gordon and the non-linear Schrödinger equations. In the sequel, a rigorous inverse scattering theory, including a qualitative spectral analysis, is developed for systems of Zakharov-Shabat type.

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

Neste trabalho são apresentados alguns resultados matemáticos relevantes para a aplicação do método de espalhamento inverso à resolução de uma classe de equações de evolução não-lineares. É demonstrada a propriedade isoespectral para certas famílias de operadores lineares não auto-adjuntos. Esta propriedade tem um papel central na aplicação do método acima a equações de evolução não-lineares de interesse físico, tais como a equação de sine-Gordon e a equação de Schrödinger não-linear. É feito também, uma teoria de espalhamento inverso rigorosa para sistemas do tipo Zakharov-Shabat, o que inclui uma análise qualitativa do espectro de operadores deste tipo. / This Thesis presents some mathematical results relevant in applications of the inverse scattering transform to the solution of a class of non-linear evolution equations. First, it is shown that certain families of non-selfadjoint linear operators have the isospectral property, which is fundamental for the above applications. These families include various operators related to no-linear equations of great physical interest, such as the sine-Gordon and the non-linear Schrödinger equations. In the sequel, a rigorous inverse scattering theory, including a qualitative spectral analysis, is developed for systems of Zakharov-Shabat type.

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 (Sc. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1987. / "January 1987." Bibliography: p. 193-200.