This thesis reports on the research carried out in the area of Microwave Tomography (MWT) where the study aims to develop inversion algorithms to obtain cheap and stable solutions of MWT inverse scattering problems which are mathematically formulated as nonlinear ill posed problems. The study develops two algorithms namely Inexact Newton Backtracking Method (INBM) and Newton Iterative-Conjugate Gradient on Normal Equation (NI-CGNE) which are based on Newton method. These algorithms apply implicit solutions of the Newton equations with unspecific manner functioning as the regularized step size of the Newton iterative. The two developed methods were tested by the use of numerical examples and experimental data gained by the MWT system of the University of Manchester. The numerical experiments were done on samples with dielectric contrast objects containing different kinds of materials and lossy materials. Meanwhile, the quality of the methods is evaluated by comparingthem with the Levenberg Marquardt method (LM). Under the natural assumption that the INBM is a regularized method and the CGNE is a semi regularized method, the results of experiments show that INBM and NI-CGNE improve the speed, the spatial resolutions and the quality of direct regularization method by means of the LM method. The experiments also show that the developed algorithms are more flexible to theeffect of noise and lossy materials compared with the LM algorithm.

In this thesis we develop inverse scattering algorithms towards the ultimate goal of online diagnostic methods. The aim is to detect structural changes inside power transformers and other major power grid components, like generators, shunt reactors etc. Power grid components, such as large power transformers, are not readily available from the manufacturers as standard designs. They are generally optimized for specific functions at a specific position in the power grid. Their replacement is very costly and takes a long time. Online methods for the diagnostics of adverse changes of the mechanical structure and the integrity of the dielectric insulation in power transformers and other power grid components, are therefore essential for the continuous operation of a power grid. Efficient online diagnostic methods can provide a real-time monitoring of mechanical structures and dielectric insulation in the active parts of power grid components. Microwave scattering is a candidate that may detect these early adverse changes of the mechanical structure or the dielectric insulation. Upon early detection, proper actions to avoid failure or, if necessary, to prepare for the timely replacement of the damaged component can be taken. The existing diagnostic methods lack the ability to provide online reliable information about adverse changes inside the active parts. More details about the existing diagnostic methods, both online and offline, and their limitations can be found in the licentiate thesis preceding the present PhD thesis. We use microwave scattering together with the inverse scattering algorithms, developed in the present work, to reconstruct the shapes of adverse mechanical structure changes. We model the propagation environment as a waveguide, in which measurement data can be obtained only at two ends (ports). Since we want to detect the onset of some deformation, that only slightly alters the scattering situation (weak scattering), we have linearized the inverse problem with good results. We have calculated the scattering parameters of the waveguide in the first-order perturbation, where they have linear dependencies on the continuous deformation function. A linearized inverse problem with a weak scattering assumption typically results in an ill-conditioned linear equation system. This is handled using Tikhonov regularization, with the L-curve method for tuning regularization parameters. We show that localized one-dimensional and two-dimensional shape deformations, for rectangular and coaxial waveguide models, are efficiently reconstructed using the inverse scattering algorithms developed from the first principles, i.e. Maxwell’s theory of electromagnetism. An excellent agreement is obtained between the reconstructed and actual deformation shapes for a number of studied cases. These results show that it is possible to use the inverse algorithms, developed in the present thesis, as a theoretical basis for the design of a future diagnostic device. As a part of the future work, it remains to experimentally verify the results obtained so far, as well as to further study the theoretical limitations posed by linearization (first-order perturbation theory) and by the assumption of the continuity of the metallic waveguide boundaries and their deformations. / <p>QC 20160119</p>

Chan Yuet Tai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 107-111). / Chapter 1 --- Introduction to Sturm-Liouville Problem --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Introductory background --- p.2 / Chapter 1.3 --- The Liouville transformation --- p.3 / Chapter 1.4 --- The Sturm-Liouville problem 一 A historical look --- p.4 / Chapter 1.5 --- Where Sturm-Liouville problems come from? --- p.6 / Chapter 1.6 --- Inverse problems of interest --- p.8 / Chapter 2 --- Reconstruction Method I --- p.10 / Chapter 2.1 --- Perturbative inversion --- p.10 / Chapter 2.1.1 --- Inversion problem via Fredholm integral equation --- p.10 / Chapter 2.1.2 --- Output least squares method for ill-posed integral equations --- p.15 / Chapter 2.1.3 --- Numerical experiments --- p.17 / Chapter 2.2 --- Total inversion --- p.38 / Chapter 2.3 --- Summary --- p.45 / Chapter 3 --- Reconstruction Method II --- p.46 / Chapter 3.1 --- Computation of q --- p.47 / Chapter 3.2 --- Computation of the Cauchy data --- p.48 / Chapter 3.2.1 --- Recovery of Cauchy data for K --- p.51 / Chapter 3.2.2 --- Numerical implementation for computation of the Cauchy data . --- p.51 / Chapter 3.3 --- Recovery of q from Cauchy data --- p.52 / Chapter 3.4 --- Iterative procedure --- p.53 / Chapter 3.5 --- Numerical experiments --- p.60 / Chapter 3.5.1 --- Eigenvalues without noised data --- p.64 / Chapter 3.5.2 --- Eigenvalues with noised data --- p.69 / Chapter 4 --- Appendices --- p.79 / Chapter A --- Tikhonov regularization --- p.79 / Chapter B --- Basic properties of the Sturm-Liouville operator --- p.80 / Chapter C --- Asymptotic formulas for the eigenvalues --- p.86 / Chapter C.1 --- Case 1: h ≠ ∞ and H ≠ ∞ --- p.87 / Chapter C.2 --- Case 2: h= ∞ and H ≠∞ --- p.90 / Chapter C.3 --- Case 3: h = ∞ and H = ∞ --- p.91 / Chapter D --- Completeness of the eigenvalues --- p.92 / Chapter E --- d'Alembert solution formula for the wave equation --- p.97 / Chapter E.1 --- "The homogeneous solution uH(x,t)" --- p.98 / Chapter E.2 --- "The particular solution up(x, t)" --- p.99 / Chapter E.3 --- "The standard d'Alembert solution u(x,t)" --- p.101 / Chapter E.4 --- Applications to our problem --- p.101 / Chapter F --- Runge-Kutta method for solving eigenvalue problems --- p.104 / Bibliography --- p.107

Lee Wai Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 129-130). / Text in English; abstract also in Chinese. / Lee Wai Shing. / Contents --- p.i / List of Figures --- p.v / Abstract --- p.vii / Acknowledgement --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Background of this research project --- p.2 / Chapter 1.3 --- Conservative systems and open systems -normal modes (NM's) vs quasi-normal modes (QNM's) --- p.3 / Chapter 1.4 --- Appetizer ´ؤ What our problems are like --- p.6 / Chapter 1.5 --- A brief overview of the following chapters --- p.7 / Chapter Chapter 2. --- Inversion of conservative systems- perturbative inversion --- p.9 / Chapter 2.1 --- Overview --- p.9 / Chapter 2.2 --- Way to introduce the additional information --- p.9 / Chapter 2.3 --- General Formalism --- p.11 / Chapter 2.4 --- Example --- p.15 / Chapter 2.5 --- Further examples --- p.19 / Chapter 2.6 --- Effects of noise --- p.23 / Chapter 2.7 --- Conclusion --- p.25 / Chapter Chapter 3. --- Inversion of conservative systems - total inversion --- p.26 / Chapter 3.1 --- Overview --- p.26 / Chapter 3.2 --- Asymptotic behaviour of the eigenfrequencies --- p.26 / Chapter 3.3 --- General formalism --- p.28 / Chapter 3.3.1 --- Evaluation of V(0) --- p.28 / Chapter 3.3.2 --- Squeezing the interval - evaluation of the potential at other positions --- p.32 / Chapter 3.4 --- Remarks --- p.36 / Chapter 3.5 --- Conclusion --- p.37 / Chapter Chapter 4. --- Theory of Quasi-normal Modes (QNM's) --- p.38 / Chapter 4.1 --- Overview --- p.38 / Chapter 4.2 --- What is a Quasi-normal Mode (QNM) system? --- p.38 / Chapter 4.3 --- Properties of QNM's in expectation --- p.40 / Chapter 4.4 --- General Formalism --- p.41 / Chapter 4.4.1 --- Construction of Green's function and the spectral represen- tation of the delta function --- p.42 / Chapter 4.4.2 --- The generalized norm --- p.45 / Chapter 4.4.3 --- Completeness of QNM's and its justification --- p.46 / Chapter 4.4.4 --- Different senses of completeness --- p.48 / Chapter 4.4.5 --- Eigenfunction expansions with QNM's 一 the two-component formalism --- p.49 / Chapter 4.4.6 --- Properties of the linear space Γ --- p.51 / Chapter 4.4.7 --- Klein-Gordon equation - The delta-potential system --- p.54 / Chapter 4.5 --- Studies of other QNM systems --- p.54 / Chapter 4.5.1 --- Wave equation - dielectric rod --- p.55 / Chapter 4.5.2 --- Wave equation ´ؤ string-mass system --- p.57 / Chapter 4.6 --- Summary --- p.58 / Chapter Chapter 5. --- Inversion of open systems- perturbative inversion --- p.59 / Chapter 5.1 --- Overview --- p.59 / Chapter 5.2 --- General Formalism --- p.59 / Chapter 5.3 --- Example 1. Klein-Gordon equation ´ؤ delta-potential system --- p.66 / Chapter 5.3.1 --- Model perturbations --- p.66 / Chapter 5.4 --- Example 2. Wave equation ´ؤ dielectric rod --- p.72 / Chapter 5.5 --- Example 3. Wave equation ´ؤ string-mass system --- p.76 / Chapter 5.5.1 --- Instability of the matrix [d] = [c]-1 upon truncation --- p.79 / Chapter 5.6 --- Large leakage regime and effects of noise --- p.81 / Chapter 5.7 --- Conclusion . . . --- p.84 / Chapter Chapter 6. --- Transition from open systems to conservative counterparts --- p.85 / Chapter 6.1 --- Overview --- p.85 / Chapter 6.2 --- Anticipations of what is going to happen --- p.86 / Chapter 6.3 --- Some computational experiments --- p.86 / Chapter 6.4 --- Reason of breakdown - An intrinsic error of physical systems --- p.87 / Chapter 6.4.1 --- Mathematical derivation of the breakdown behaviour --- p.90 / Chapter 6.4.2 --- Two verifications --- p.93 / Chapter 6.5 --- Another source of errors - An intrinsic error of practical computations --- p.95 / Chapter 6.5.1 --- Vindications --- p.96 / Chapter 6.5.2 --- Mathematical derivation of the breakdown --- p.98 / Chapter 6.6 --- Further sources of errors --- p.99 / Chapter 6.7 --- Dielectric rod --- p.100 / Chapter 6.8 --- String-mass system --- p.103 / Chapter 6.9 --- Conclusion --- p.105 / Chapter Chapter 7. --- A first step to Total Inversion of QNM systems? --- p.106 / Chapter 7.1 --- Overview --- p.106 / Chapter 7.2 --- Derivation for F(0) --- p.106 / Chapter 7.3 --- Example 一 delta potential system --- p.108 / Chapter Chapter 8. --- Conclusion --- p.111 / Chapter 8.1 --- A summary on what have been achieved --- p.111 / Chapter 8.2 --- Further directions to go --- p.111 / Appendix A. A note on notation --- p.113 / Appendix B. Asymptotic series of NM eigenvalues --- p.114 / Appendix C. Evaluation of functions related to RHS(x) --- p.117 / Appendix D. Asymptotic behaviour of the Green's function --- p.119 / Appendix E. Expansion coefficient an --- p.121 / Appendix F. Asymptotic behaviour of QNM eigenvalues --- p.123 / Appendix G. Properties of the inverse matrix [d] = [c]-1 --- p.125 / Appendix H. Matrix inverse through the LU decomposition method --- p.127 / Bibliography --- p.129

Hung Kin Ting. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 126-129). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Inverse Problems --- p.1 / Chapter 1.2 --- Applications of Inverse Problems --- p.2 / Chapter 1.3 --- Least-squares Solutions --- p.4 / Chapter 1.4 --- Discrete Systems --- p.4 / Chapter 1.5 --- "Discretization, Regularization and Regularization Pa- rameters" --- p.5 / Chapter 1.6 --- Outline of the Thesis --- p.6 / Chapter 2 --- Some Basic Concepts and Mathematical Tools --- p.8 / Chapter 2.1 --- Singular Value Decomposition (SVD) --- p.8 / Chapter 2.2 --- Generalized Singular Value Decomposition (GSVD) --- p.13 / Chapter 2.3 --- White Noises --- p.16 / Chapter 3 --- Regularized Solutions --- p.18 / Chapter 3.1 --- Derivation of Regularized Solutions --- p.18 / Chapter 3.2 --- Discrete Picard Condition --- p.20 / Chapter 3.3 --- Relationship between Discrete Picard Condition and Regularized Solution --- p.21 / Chapter 3.4 --- Checking for the Discrete Picard Condition --- p.22 / Chapter 4 --- Different Discretization Approaches --- p.23 / Chapter 4.1 --- Problem 1 - Volterra Integral Equation of the First Kind --- p.25 / Chapter 4.2 --- Examples of Problem 1 --- p.30 / Chapter 4.3 --- Problem 2 - Fredholm Integral Equation of the First Kind --- p.49 / Chapter 4.4 --- Examples of Problem 2 --- p.53 / Chapter 4.5 --- Conclusion --- p.57 / Chapter 5 --- Effect of Different Kinds of Observation Data and Differential Operators on Accuracy --- p.59 / Chapter 5.1 --- Pointwise Observation Data --- p.60 / Chapter 5.2 --- Pointwise Observation Data of Heat Fluxes at the Boundary --- p.69 / Chapter 5.3 --- Observation Data with Heat Fluxes --- p.80 / Chapter 5.4 --- Conclusion --- p.89 / Chapter 6 --- L-curve --- p.90 / Chapter 6.1 --- Properties of L-curve --- p.93 / Chapter 6.2 --- L-curve in Log-Log Scale --- p.100 / Chapter 6.3 --- Disadvantages of the L-curve Method --- p.100 / Chapter 7 --- Algorithms of Finding the Corner of L-curve --- p.105 / Chapter 7.1 --- Cubic Spline Curve Fitting --- p.105 / Chapter 7.2 --- Conic Section Fitting --- p.106 / Chapter 7.3 --- Triangle Method --- p.109 / Chapter 8 --- Implementation of the L-curve Method --- p.111 / Chapter 8.1 --- Our Algorithm --- p.111 / Chapter 8.2 --- Numerical Experiments --- p.112 / Chapter 8.3 --- Conclusion --- p.124 / Bibliography --- p.126

Cheung, Man Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 89-91). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to Inverse Problems --- p.1 / Chapter 1.1 --- Major properties --- p.1 / Chapter 1.2 --- Typical examples --- p.3 / Chapter 1.3 --- Thesis outline --- p.5 / Chapter 2 --- Some Operator Theory --- p.6 / Chapter 2.1 --- Fredholm integral equation of the first kind --- p.6 / Chapter 2.2 --- Compact operator theory --- p.8 / Chapter 2.3 --- Singular system --- p.12 / Chapter 2.4 --- Moore-Penrose generalized inverse --- p.14 / Chapter 3 --- Regularization Theory for First Kind Equations --- p.19 / Chapter 3.1 --- General regularization theory --- p.19 / Chapter 3.2 --- Tikhonov regularization --- p.24 / Chapter 3.3 --- Landweber iteration --- p.26 / Chapter 3.4 --- TSVD --- p.28 / Chapter 4 --- Multilevel Algorithms for Ill-posed Problems --- p.30 / Chapter 4.1 --- Basic assumptions and definitions --- p.31 / Chapter 4.2 --- Multilevel analysis --- p.33 / Chapter 4.3 --- Applications --- p.37 / Chapter 4.3.1 --- Preconditioned iterative methods with nonzero regularization parameter --- p.38 / Chapter 4.3.2 --- Preconditioned iterative methods with zero regularization parameter --- p.38 / Chapter 4.3.3 --- Full multilevel algorithm --- p.40 / Chapter 5 --- Numerical Experiments --- p.41 / Chapter 5.1 --- Integral equations --- p.41 / Chapter 5.1.1 --- Discretization --- p.42 / Chapter 5.1.2 --- Test problems --- p.43 / Chapter 5.1.3 --- "Singular values, singular vectors and condition numbers" --- p.45 / Chapter 5.1.4 --- Effect of condition numbers on numerical accuracies --- p.49 / Chapter 5.2 --- Differential equations --- p.50 / Chapter 5.2.1 --- Discretization --- p.51 / Chapter 5.2.2 --- "Singular values, singular vectors and condition numbers" --- p.53 / Chapter 5.3 --- Numerical experiments by classical methods --- p.55 / Chapter 5.3.1 --- Tikhonov regularization --- p.55 / Chapter 5.3.2 --- TSVD --- p.56 / Chapter 5.3.3 --- Landweber iteration --- p.63 / Chapter 5.4 --- Numerical experiments by multilevel methods --- p.63 / Chapter 5.4.1 --- General convergence --- p.63 / Chapter 5.4.2 --- Numerical results --- p.65 / Chapter 5.4.3 --- Effect of multilevel parameters on convergence --- p.76 / Bibliography --- p.89

Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².

This dissertation describes research conducted on the development and improvement of microwave tomography algorithms for imaging the bulk-electrical parameters of unknown objects. The full derivation of a new inversion algorithm based on the state-of-the-art contrast source inversion (CSI) algorithm coupled to a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The algorithm is applied to two-dimensional (2D) scalar and vectorial configurations, as well as three-dimensional (3D) full-vectorial problems. The unknown electrical properties of the object are distributed on the elements of arbitrary meshes with varying densities. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The CSI algorithm is used in conjunction with multiplicative regularization (MR), as it is typical in most implementations of CSI. Due to the use of arbitrary meshes in the present implementation, new techniques are introduced to perform the necessary spatial gradient and divergence operators of MR. The approach is different from other MR-CSI implementations where the unknown variables are located on a uniform grid of rectangular cells and represented using pulse basis functions; with rectangular cells finite-difference operators can be used, but this becomes unwieldy in FEM-CSI. Furthermore, an improvement for MR is proposed that accounts for the imbalance between the real and imaginary parts of the electrical properties of the unknown objects. The proposed method is not restricted to any particular formulation of the contrast source inversion. The functionality of the new inversion algorithm with the different enhancements is tested using a wide range of synthetic datasets, as well as experimental data collected by the University of Manitoba electromagnetic imaging group and research centers in Spain and France.

This dissertation describes research conducted on the development and improvement of microwave tomography algorithms for imaging the bulk-electrical parameters of unknown objects. The full derivation of a new inversion algorithm based on the state-of-the-art contrast source inversion (CSI) algorithm coupled to a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The algorithm is applied to two-dimensional (2D) scalar and vectorial configurations, as well as three-dimensional (3D) full-vectorial problems. The unknown electrical properties of the object are distributed on the elements of arbitrary meshes with varying densities. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The CSI algorithm is used in conjunction with multiplicative regularization (MR), as it is typical in most implementations of CSI. Due to the use of arbitrary meshes in the present implementation, new techniques are introduced to perform the necessary spatial gradient and divergence operators of MR. The approach is different from other MR-CSI implementations where the unknown variables are located on a uniform grid of rectangular cells and represented using pulse basis functions; with rectangular cells finite-difference operators can be used, but this becomes unwieldy in FEM-CSI. Furthermore, an improvement for MR is proposed that accounts for the imbalance between the real and imaginary parts of the electrical properties of the unknown objects. The proposed method is not restricted to any particular formulation of the contrast source inversion. The functionality of the new inversion algorithm with the different enhancements is tested using a wide range of synthetic datasets, as well as experimental data collected by the University of Manitoba electromagnetic imaging group and research centers in Spain and France.

Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².