The main topics of this dissertation are the hardware and the software developments in magnetic induction tomography imaging techniques. In the hardware sections, all the tomography systems developed by the author will be presented and discussed in detail. The developed systems can be divided into two categories, according to the property of the target imaging materials: high conductivity materials and low conductivity materials. Each system has its own suitable application, and each will thus be tested under different circumstances. In terms of the software development, the forward and inverse problems have been studied, including the eddy current problem modeling, sensitivity map formulae derivation and iterative/non-iterative inverse solvers equations. The Biot-Savart Theory was implemented in the ‘two-potential’ method that was used in the eddy current model in order to improve the system’s flexibility. Many different magnetic induction tomography schemes are proposed for the first time in this field of research, their aim being to improve the spatial and temporal resolution of the final reconstructed images. These novel schemes usually involve some modifications of the system hardware and forward/inverse calculations. For example, the rotational scheme can improve the ill-posedness and edge detectability of the system; the volumetric scheme can provide extra spatial resolution in the axial direction; and the temporal scheme can improve the temporal resolution by using the correlation between the consecutive datasets. Volumetric imaging requires an intensive amount of extra computational resources. To overcome the issue of memory constraints when solving large-scale inverse problems, a matrix-free method was proposed, also for the first time in magnetic induction tomography. All the proposed algorithms are verified by the experimental data obtained from suitable tomography systems developed by the author. Although magnetic induction tomography is a new imaging technique, it is believed that the technique is well developed for real-life applications. Several potential applications for magnetic induction tomography are suggested. The initial proof-of-concept study for a challenging low conductivity two-phase flow imaging process is provided. In this thesis, a range of contributions have been made in the field of magnetic induction tomography, which will help the magnetic induction tomography research to be carried on further.

Keung Yee Lo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 109-111). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Inverse problems and Parameter Identification --- p.1 / Chapter 1.2 --- Examples in inverse problems --- p.2 / Chapter 1.3 --- Applications in parameter identifications --- p.5 / Chapter 1.4 --- Difficulties arising in inverse problems --- p.7 / Chapter 2 --- Identifying Parameters in Parabolic Systems --- p.9 / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.2 --- An averaging-terminal status formulation and existence of its solutions --- p.12 / Chapter 2.3 --- Optimization approach and its convergence --- p.17 / Chapter 2.4 --- Unconstrained minimization problems --- p.26 / Chapter 2.5 --- Armijo algorithm --- p.28 / Chapter 2.6 --- Numerical experiments --- p.32 / Chapter 2.6.1 --- Convergence of the minimization problem --- p.40 / Chapter 2.7 --- Noisy data --- p.59 / Chapter 3 --- Identifying Parameters in Elliptic Systems --- p.68 / Chapter 3.1 --- Augmented Lagrangian Method --- p.68 / Chapter 3.2 --- The discrete saddle-point problem --- p.70 / Chapter 3.3 --- An Uzawa algorithm --- p.71 / Chapter 3.4 --- Formulation of the algorithm --- p.73 / Chapter 3.5 --- Numerical experiments --- p.76 / Chapter 3.6 --- Alternative formulation of the cost functional --- p.90 / Chapter 3.7 --- Iterative GMRES method --- p.102 / Bibliography --- p.109

心臟表面電位能夠真實反映心肌的活動，因此以重建心臟表面電位為目標的心臟電生理反問題被廣泛研究。心臟電生理反問題是一個不適定問題，因此輸入數據中一個小的噪聲也有可能導致一個高度不穩定的解。因此，通常基於2 範數的規則化方法被用於解決這個病態問題。但是2 範數的懲罰函數會導致一定程度的模糊，使得分辨和定位心臟表面一些不正常或者病變部位不準確。而直接使用1 範數的懲罰函數，會由於其不可微分而增加計算復雜度。 / 我們首先提出一種基於 1 範數的方法來減少計算復雜度和能夠快速收斂。在這個方法中，使用變量分離技術使得1 範數的懲罰函數可微分。然後這個反問題被構造成一個有界約束二次優化問題，從而可以很容易地利用梯度映射法叠代求解。在試驗中，使用合成數據和真實數據來評估提出的方法。實驗表明，提出的方法可以很好地處理測量噪聲和幾何噪聲，而且能夠獲得比以前的1、2 範數方法更準確的實驗結果。 / 盡管提出的 1 範數方法能夠有效克服2 範數存在的問題，但是1 範數方法仍然只是0 範數的近似。因此我們采用了一種平滑0 範數的方法來求解心臟電生理反問題。平滑0 範數使用平滑函數，使得0 範數連續，從而能夠直接求解0 範數的反問題。實驗結果表明，使用平滑0範數方法可以獲得比1、2 範數更好、更準確的心臟表面電位。 / 在以往的心臟反問題研究中，使用的心臟幾何模型都是靜態的，與實際跳動的心臟不符，從而使得反問題方法難以進入臨床。因此我們提出了從動態心臟模型中重建心臟表面電位。動態心臟模型是從一系列核磁共振圖像中重建得到的。體表電位也同步獲得。仿真實驗獲得了很好的心臟表面電位結果。 / 在論文最後，我們提出一個基於心臟電生理反問題的系統，來輔助束支傳導阻滯的治療。在這個系統中，心臟模型和體表模型都從病人的數據中重建獲得，體表電位也得到收集。通過電生理反問題方法，在心臟表面重建電位及其分布。醫生通過觀察重建結果來輔助束支傳導阻滯的診斷和治療。 / The epicardial potentials (EPs) targeted inverse problem of electrocardiography (ECG) has been widely investigated as it is demonstrated that EPs reflect underlying myocardial activity. It is a wellknown ill-posed problem as small noises in input data may yield a highly unstable solution. Traditionally, L2-norm regularization methods have been proposed to solve this ill-posed problem. But L2-norm penalty function inherently leads to considerable smoothing of the solution, which reduces the accuracy of distinguishing abnormalities and locating diseased regions. In this thesis, we propose three new techniques in order to achieve more accurate reconstruction results of EPs and applied these techniques to a clinical application. We first propose a L1-norm regularization method in order to reduce the computational complexity and make rapid convergence possible. Variable splitting is employed to make the L1- norm penalty function differentiable based on the observation that both positive and negative potentials exist on the epicardial surface. Then, the inverse problem of ECG is further formulated as a boundconstrained quadratic problem, which can be efficiently solved by gradient projection in an iterative manner. Extensive experiments conducted on both synthetic data and real data demonstrate that the proposed method can handle both measurement noise and geometry noise and obtain more accurate results than previous L2- and L1- norm regularization methods, especially when the noises are large. / Although L1 norm regularization achieves better reconstructed results compared with L2 norm regularization, L1 norm is still an approximation of L0 norm which is more accurate than L1 norm. We further presented a smoothed L0 norm technique in order to directly solve the L0 norm constrained problem. Our method employs a smoothing function to make the L0 norm continuous. Extensive experiments showed that the proposed method reconstructs more accurate epicardial potentials compared with L1 norm and L2 norm. / In current research of ECG inverse problem, epicardial potentials are reconstructed from a static heart model which blocks the techniques to clinic applications. A novel strategy is presented to recovii er epicardial potentials using a dynamic heart model built from MRI image sequences and ECG data. We used MRI images to estimate the current density and visualize it on the surface of the heart model. The ECG data also be used to achieve the time synchronization when the propagation of the current density. Experiments are conducted on a set of real time MRI images, also with the real ECG data, and we get favorable results. / Finally, a non-invasive system is presented for enhancing the diagnosis of Bundle Branch Block (BBB). In this system, epicardial potential is estimated and visualized in the 3D heart model to improve the diagnosis of BBB. Using patient CT and BSPM data, the system is able to reconstruct details of the complete electrical activity of BBB on the 3D heart model. Through the analysis of the epicardial potential mapping in this system, patients with BBB are easily and accurately distinguished instead of from empirically checking ECG. Therefore the diagnosis of BBB is improved using this system. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Liansheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 103-124). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Inverse Problem of ECG --- p.6 / Chapter 2.1 --- Background --- p.6 / Chapter 2.2 --- Problem Formulations --- p.8 / Chapter 2.2.1 --- Potential Reconstruction Problem --- p.8 / Chapter 2.2.2 --- Coefficient Reconstruction Problem --- p.11 / Chapter 2.3 --- Solving Methods --- p.11 / Chapter 2.3.1 --- Regularization Methods --- p.11 / Chapter 2.3.2 --- Non-quadratic Regularization --- p.12 / Chapter 2.3.3 --- Activation Wavefronts Solution --- p.14 / Chapter 3 --- L1-Norm to EPs Reconstruction --- p.16 / Chapter 3.1 --- Related Work --- p.16 / Chapter 3.2 --- Method --- p.21 / Chapter 3.3 --- Experimental Results and Validation --- p.24 / Chapter 3.3.1 --- Error Evaluation --- p.26 / Chapter 3.3.2 --- Synthetic Data Cases --- p.26 / Chapter 3.3.3 --- Real Data Cases --- p.32 / Chapter 3.4 --- Discussion --- p.44 / Chapter 3.5 --- Summary --- p.48 / Chapter 4 --- L0-Norm to EPs Reconstruction --- p.49 / Chapter 4.1 --- Related Work --- p.49 / Chapter 4.2 --- Smoothed L0-norm Method --- p.54 / Chapter 4.3 --- Experimental Results and Protocols --- p.57 / Chapter 4.3.1 --- Data --- p.57 / Chapter 4.3.2 --- Evaluation Protocol --- p.60 / Chapter 4.3.3 --- Experiments and Results --- p.60 / Chapter 4.4 --- Discussion --- p.68 / Chapter 4.5 --- Summary --- p.69 / Chapter 5 --- EPs Reconstruction in A Dynamic Model --- p.71 / Chapter 5.1 --- Related Work --- p.71 / Chapter 5.2 --- Forward Model --- p.73 / Chapter 5.3 --- Parameters Estimation for Inverse Problem of ECG --- p.75 / Chapter 5.4 --- Experiments and Results --- p.77 / Chapter 5.5 --- Summary --- p.80 / Chapter 6 --- Diagnosis of BBB: an Application --- p.82 / Chapter 6.1 --- Related Work --- p.82 / Chapter 6.2 --- Method --- p.84 / Chapter 6.2.1 --- Data --- p.85 / Chapter 6.2.2 --- Signal Preprocessing of BSPM --- p.87 / Chapter 6.2.3 --- Epicardial Potential Estimation and Imaging --- p.88 / Chapter 6.3 --- Experiments and Results --- p.89 / Chapter 6.3.1 --- Population Under Study --- p.89 / Chapter 6.3.2 --- Results --- p.89 / Chapter 6.4 --- Summary --- p.92 / Chapter 7 --- Conclusion --- p.94 / Chapter 7.1 --- Summary of Contributions --- p.94 / Chapter 7.2 --- Future Works --- p.96 / Chapter A --- Barzilai and Borwein Approach --- p.97 / Chapter B --- List of Publications --- p.99 / Bibliography --- p.103

The main goal of this thesis is to investigate the frequentist asymptotic properties of nonparametric Bayesian procedures in inverse problems and the Gaussian white noise model. In the first part, we study the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. This rate provides a quantitative measure of the quality of statistical estimation of the procedure. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting, minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform. In the second part of this thesis, we investigate Bernstein--von Mises type results for adaptive nonparametric Bayesian procedures in both the Gaussian white noise model and the mildly ill-posed inverse setting. The Bernstein--von Mises theorem details the asymptotic behaviour of the posterior distribution and provides a frequentist justification for the Bayesian approach to uncertainty quantification. We establish weak Bernstein--von Mises theorems in both a Hilbert space and multiscale setting, which have applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets using a Bayesian approach. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the different geometries involved.

by Yu Zhenli. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (p. 193-202). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

在本文中， 我們將主要討論一種在求解一類波動方程反問題中很有價值的數值方法：線性化方法。 / 在介紹上述的數值方法之前， 我們將首先討論波動方程的一些重要的特質，主要包括四類典型的波動方程模型，方程的基本解和一般解，以及波動方程解的性質。 / 接下來，在本文的第二部分中，我們會首先介紹所求解的模型以及其反問題。此反問題主要研究求解波動方程[附圖]中的系數c. 線性化方法的主要思想在於將速度c分解成兩部分：c₁ 和c₂ ，並且滿足關系式：[附圖]，其中c₁ 是一個小的擾動量。另一方面，上述波動方程的解u 可以被線性表示：u = u₀ + u₁ ，其中u₀ 和u₁ 分別是一維問題和二維問題的解。相應的，我們將運用有限差分方法和傅利葉變換方法求解上述一維問題和二維問題，從而分別求解c₁ 和c₂ ，最終求解得到係數c. 在本文的最後，我們將進行一些數值試驗，從而驗證此線性化方法的有效性和可靠性。 / In this thesis, we will discuss a numerical method of enormous value, a linearized method for solving a certain kind of inverse wave equations. / Before the introduction of the above-mentioned method, we shall discuss some important features of the wave equations in the first part of the thesis, consisting of four typical mathematical models of wave equations, there fundamental solutions, general solutions and the properties of those general solutions. / Next, we shall present the model and its inverse problem of recovering the coefficient c representing the propagation velocity of wave from the wave equation [with mathematic formula] The linearized method aims at dividing the velocity c into two parts, c₀ and c₁, which satisfying the relation [with mathematic formula], where c₁ is a tiny perturbation. On the other hand, the solution u can be represented in the linear form, u = u₀ + u₁, where u₀ and u₁ are the solutions to one-dimensional problem and two- dimensional problem respectively. Accordingly, we can use the numerical methods, finite difference method and Fourier transform method to solve the one-dimensional forward problem and two-dimensional inverse problem respectively, thus we can get c₀ and c₁, a step before we recover the velocity c. In the numerical experiments, we shall test the proposed linearized numerical method for some special examples and demonstrate the effectiveness and robustness of the method. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Xu, Xinyi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 64-65). / Abstracts also in Chinese. / Chapter 1 --- Fundamental aspects of wave equations --- p.6 / Chapter 1.1 --- Introduction --- p.6 / Chapter 1.1.1 --- Four important wave equations --- p.6 / Chapter 1.1.2 --- General form of wave equations --- p.10 / Chapter 1.2 --- Fundamental solutions --- p.11 / Chapter 1.2.1 --- Fourier transform --- p.11 / Chapter 1.2.2 --- Fundamental solution in three-dimensional space --- p.14 / Chapter 1.2.3 --- Fundamental solution in two-dimensional space --- p.16 / Chapter 1.3 --- General solution --- p.19 / Chapter 1.3.1 --- One-dimensional wave equations --- p.19 / Chapter 1.3.2 --- Two and three dimensional wave equations --- p.26 / Chapter 1.3.3 --- n dimensional case --- p.28 / Chapter 1.4 --- Properties of solutions to wave equation --- p.31 / Chapter 1.4.1 --- Properties of Kirchhoff’s solutions --- p.31 / Chapter 1.4.2 --- Properties of Poisson’s solutions --- p.33 / Chapter 1.4.3 --- Decay of the solutions to wave equation --- p.34 / Chapter 2 --- Linearized method for wave equations --- p.36 / Chapter 2.1 --- Introduction --- p.36 / Chapter 2.1.1 --- Background --- p.36 / Chapter 2.1.2 --- Forward and inverse problem --- p.38 / Chapter 2.2 --- Basic ideas of Linearized Method --- p.39 / Chapter 2.3 --- Theoretical analysis on linearized method --- p.41 / Chapter 2.3.1 --- One-dimensional forward problem --- p.42 / Chapter 2.3.2 --- Two-dimensional forward problem --- p.43 / Chapter 2.3.3 --- Existence and uniqueness of solutions to the inverse problem --- p.45 / Chapter 2.4 --- Numerical analysis on linearized method --- p.45 / Chapter 2.4.1 --- Discrete analog of the inverse problem --- p.46 / Chapter 2.4.2 --- Fourier transform --- p.48 / Chapter 2.4.3 --- Direct methods for inverse and forward problems --- p.52 / Chapter 2.5 --- Numerical Simulation --- p.54 / Chapter 2.5.1 --- Special Case --- p.54 / Chapter 2.5.2 --- General Case --- p.59 / Chapter 3 --- Conclusion --- p.63 / Bibliography --- p.64

Inverse problems are mathematically and numerically very challenging due to their inherent ill-posedness in the sense that a small perturbation of the data may cause an enormous deviation of the solution. Regularization methods have been established as the standard approach for their stable numerical solution thanks to the ground-breaking work of late Russian mathematician A.N. Tikhonov. However, existing studies mainly focus on general-purpose regularization procedures rather than exploiting mathematical structures of specific problems for designing efficient numerical procedures. Moreover, the stochastic nature of data noise and model uncertainties is largely ignored, and its effect on the inverse solution is not assessed. This thesis attempts to design some problem-specific efficient numerical methods for the Robin inverse problem and to quantify the associated uncertainties. It consists of two parts: Part I discusses deterministic methods for the Robin inverse problem, while Part II studies stochastic numerics for uncertainty quantification of inverse problems and its implication on the choice of the regularization parameter in Tikhonov regularization. / Key Words: Robin inverse problem, variational approach, preconditioning, Modica-Motorla functional, spectral stochastic approach, Bayesian inference approach, augmented Tikhonov regularization method, regularization parameter, uncertainty quantification, reduced-order modeling / Part I considers the variational approach for reconstructing smooth and nonsmooth coefficients by minimizing a certain functional and its discretization by the finite element method. We propose the L2-norm regularization and the Modica-Mortola functional from phase transition for smooth and nonsmooth coefficients, respectively. The mathematical properties of the formulations and their discrete analogues, e.g. existence of a minimizer, stability (compactness), convexity and differentiability, are studied in detail. The convergence of the finite element approximation is also established. The nonlinear conjugate gradient method and the concave-convex procedure are suggested for solving discrete optimization problems. An efficient preconditioner based on the Sobolev inner product is proposed for justifying the gradient descent and for accelerating its convergence. / Part II studies two promising methodologies, i.e. the spectral stochastic approach (SSA) and the Bayesian inference approach, for uncertainty quantification of inverse problems. The SSA extends the variational approach to the stochastic context by generalized polynomial chaos expansion, and addresses inverse problems under uncertainties, e.g. random data noise and stochastic material properties. The well-posedness of the stochastic variational formulation is studied, and the convergence of its stochastic finite element approximation is established. Bayesian inference provides a natural framework for uncertainty quantification of a specific solution by considering an ensemble of inverse solutions consistent with the given data. To reduce its computational cost for nonlinear inverse problems incurred by repeated evaluation of the forward model, we propose two accelerating techniques by constructing accurate and inexpensive surrogate models, i.e. the proper orthogonal decomposition from reduced-order modeling and the stochastic collocation method from uncertainty propagation. By observing its connection with Tikhonov regularization, we propose two functionals of Tikhonov type that could automatically determine the regularization parameter and accurately detect the noise level. We establish the existence of a minimizer, and the convergence of an alternating iterative algorithm. This opens an avenue for designing fully data-driven inverse techniques. / This thesis considers deterministic and stochastic numerics for inverse problems associated with elliptic partial differential equations. The specific inverse problem under consideration is the Robin inverse problem: estimating the Robin coefficient of a Robin boundary condition from boundary measurements. It arises in diverse industrial applications, e.g. thermal engineering and nondestructive evaluation, where the coefficient profiles material properties on the boundary. / Jin, Bangti. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3541. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 174-187). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In this thesis, we will address two most important topics in inverse acoustic and electromagnetic obstacle scattering problems: uniqueness and reconstruction algorithms. / The first part is devoted to the uniqueness issues. A detailed exposition of the background of this problem and a comprehensive discussion of the existing results are presented. The focus of this part is on our contribution to this field, especially on the unique determination of polygonal or polyhedral scatterers with a single or finitely many far-field measurements. In summary, we have shown the following results when the polyhedral type scatterers are concerned in inverse acoustic obstacle scattering: if the scatterer consists of finitely many solid polyhedral obstacles, which may be either sound-soft, sound-hard or two types mixed together, and it may also contain some crack-type obstacles but only sound-soft ones, then one can uniquely determine the scatterer by a single incident plane wave at some fixed k0 > 0 and d0 ∈ SN-1 . This statement is affirmatively verified in any dimensions whenever there is no any sound-hard obstacle present; when there is any sound-hard obstacle, the uniqueness is validated in the R2 case, but still incomplete in the RN case with N ≥ 3, which is proved to be true only by N different incident plane waves. Whenever the scatterer contains some sound-hard crack-type obstacles, we have constructed some examples to show that one cannot uniquely determine the scatterer by any less than N incident waves. So in the case with the additional presence of sound-hard crack-type obstacles, another result we have established that one can uniquely determine such a scatterer by N incident waves at any fixed wave number and arbitrary N linearly independent incident directions is optimal. We also consider more general polyhedral type scatterers with partially coated components, and some uniqueness results are established to determine the underlying physical properties. Besides, we have also collected a global uniqueness result for balls or discs. It is shown that in the resonance region, the shape of a sound-soft/sound-hard ball in R3 or a soundhard/sound-soft disc in R2 is uniquely determined by a single far-field datum measured at some fixed spot corresponding to a single incident plane wave. This seems to be an important result in the uniqueness study field as it is the first to establish the unique determination by a single far-field datum measured at one fixed spot. While all the other existing uniqueness results require far-field data observed at least in one open subset on the unit sphere with non-zero measure. To pave the way for the uniqueness study with such simple balls or discs, we also present a systematic and rather complete study of the interlacing character of the zeros for Bessel and spherical Bessel functions and their respective derivatives. Finally, all the uniqueness results for inverse acoustic obstacle scattering associated with general polyhedral scatterers have been extended to the inverse electromagnetic scattering. / The second part of this thesis is concerned with the reconstruction algorithms. We will present a novel multilevel linear sampling method (MLSM) which is developed in our recent work. The new method resembles the popular multi-level techniques in scientific computing and is shown to possess the asymptotically optimal computational complexity. For an n x n sampling mesh-grid in R2 or an n x n x n sampling mesh-grid in R3 , the proposed algorithm only requires to solve O (nN-1)( N = 2,3) far-field equations for a RN problem, and this is in sharp contrast to the original version of the linear sampling method which needs to solve n N far-field equations instead. Numerical experiments have illustrated the promising feature of the new algorithm in significantly reducing the computational costs. / Liu, Hongyu. / "June 2007." / Adviser: Jun Zou. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0354. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 161-168). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Tsang, Ka Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves [121]-123). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Inverse problems and formulations --- p.6 / Chapter 3 --- Review of some existing methods --- p.8 / Chapter 4 --- Trust Region Method --- p.16 / Chapter 4.1 --- Some Auxiliary Tools --- p.18 / Chapter 4.2 --- Trust Region Algorithm --- p.23 / Chapter 4.3 --- Convergence of trust region method --- p.28 / Chapter 4.3.1 --- Notations and Assumptions --- p.28 / Chapter 4.3.2 --- Convergence for exact data --- p.29 / Chapter 4.3.3 --- Regularity For Inexact Data --- p.36 / Chapter 4.4 --- Experiment On Trust Region Method --- p.39 / Chapter 4.4.1 --- Problem Setting --- p.39 / Chapter 4.4.2 --- Algorithm --- p.40 / Chapter 4.4.3 --- Experiment Results --- p.42 / Chapter 4.5 --- Trust Region Conjugate Gradient Method --- p.46 / Chapter 4.5.1 --- Notations and Assumptions --- p.49 / Chapter 4.5.2 --- Convergence Properties for Exact Data --- p.52 / Chapter 4.5.3 --- Regularity for Inexact Data --- p.57 / Chapter 5 --- Parameter Identification Problems --- p.60 / Chapter 5.1 --- Introduction --- p.60 / Chapter 5.1.1 --- Computation of VJ(x) --- p.67 / Chapter 5.2 --- Algorithm for Parameter Identification Problems --- p.72 / Chapter 5.2.1 --- "Finite Element Method in Two Dimensions:Ω =[0,1] x [0,1]" --- p.75 / Chapter 5.3 --- Experiments on Trust Region-CG Method for Parameter Identification Problems --- p.82 / Chapter 5.3.1 --- One Dimension Problem --- p.82 / Chapter 5.3.2 --- Two Dimensions Problem --- p.95 / Chapter 5.4 --- Conclusion --- p.119 / Bibliography --- p.121

Liu, Keji. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 97-98). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Background and models of inverse scattering --- p.1 / Chapter 1.1 --- Model I --- p.1 / Chapter 1.2 --- Model II --- p.3 / Chapter 1.3 --- Model III --- p.3 / Chapter 2 --- Direct and Inverse Problems of Three Models --- p.6 / Chapter 2.1 --- First Model: Direct Problem --- p.6 / Chapter 2.2 --- First Model: Inverse Problem --- p.10 / Chapter 2.2.1 --- Linear Sampling Method --- p.10 / Chapter 2.2.2 --- Strengthened Linear Sampling Method --- p.12 / Chapter 2.2.3 --- Multilevel Linear Sampling Method --- p.15 / Chapter 2.3 --- Second Model: Direct Problem --- p.19 / Chapter 2.4 --- Second Model: Inverse Problem --- p.21 / Chapter 2.4.1 --- Contrast Source Inversion Method --- p.21 / Chapter 2.4.2 --- Subspace-based Optimization Method --- p.25 / Chapter 2.4.3 --- Multiple Signal Classification Method --- p.31 / Chapter 2.5 --- Third Model: Direct Problem --- p.33 / Chapter 2.6 --- Third Model: Inverse Problem --- p.41 / Chapter 2.6.1 --- Generalized Dual Space Indicator Method --- p.41 / Chapter 3 --- Numerical Simulations --- p.44 / Chapter 3.1 --- Numerical Simulations of First Model --- p.44 / Chapter 3.1.1 --- Linear Sampling Method --- p.44 / Chapter 3.1.2 --- Strengthened Linear Sampling Method --- p.51 / Chapter 3.1.3 --- Multilevel Linear Sampling Method --- p.58 / Chapter 3.2 --- Numerical Simulations of Second Model --- p.68 / Chapter 3.2.1 --- Contrast Source Inversion Method --- p.68 / Chapter 3.2.2 --- Subspace-based Optimization Method --- p.74 / Chapter 3.2.3 --- Twofold Subspace-based Optimization Method --- p.79 / Chapter 3.2.4 --- Multiple Signal Classification Method --- p.85 / Chapter 3.3 --- Numerical Simulations of Third Model --- p.85 / Chapter 3.3.1 --- Boundary Integral Method --- p.86 / Chapter 3.3.2 --- Generalized Dual Space Indicator Method --- p.89 / Chapter 4 --- Conclusion