Zhang Kai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 99-102). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- The Generalized PML Theory --- p.6 / Chapter 1.1.1 --- Background --- p.6 / Chapter 1.1.2 --- Derivation --- p.8 / Chapter 1.1.3 --- Reflection Properties --- p.11 / Chapter 1.2 --- Unified Formulation --- p.12 / Chapter 1.2.1 --- "Face-, Edge- and Corner-PMLs" --- p.12 / Chapter 1.2.2 --- Unified PML Equations in 3D --- p.15 / Chapter 1.2.3 --- Unified PML Equations in 2D --- p.16 / Chapter 1.2.4 --- Examples of PML Formulations --- p.16 / Chapter 1.3 --- Inhomogeneous Initial Conditions --- p.23 / Chapter 2 --- Numerical Analysis of PMLs --- p.25 / Chapter 2.1 --- Continuous PMLs --- p.26 / Chapter 2.1.1 --- PMLs for Wave Equations --- p.27 / Chapter 2.1.2 --- Finite PMLs for Wave Equations --- p.31 / Chapter 2.1.3 --- Berenger's PMLs for Maxwell Equations --- p.33 / Chapter 2.1.4 --- Finite Berenger's PMLs for Maxwell Equations --- p.35 / Chapter 2.1.5 --- PMLs for Acoustic Equations --- p.38 / Chapter 2.1.6 --- Berenger's PMLs for Acoustic Equations --- p.39 / Chapter 2.1.7 --- PMLs for 1-D Hyperbolic Systems --- p.42 / Chapter 2.2 --- Discrete PMLs --- p.44 / Chapter 2.2.1 --- Discrete PMLs for Wave Equations --- p.44 / Chapter 2.2.2 --- Finite Discrete PMLs for Wave Equations --- p.51 / Chapter 2.2.3 --- Discrete Berenger's PMLs for Wave Equations --- p.53 / Chapter 2.2.4 --- Finite Discrete Berenger's PMLs for Wave Equations --- p.56 / Chapter 2.2.5 --- Discrete PMLs for 1-D Hyperbolic Systems --- p.58 / Chapter 2.3 --- Modified Yee schemes for PMLs --- p.59 / Chapter 2.3.1 --- Stability of the Yee Scheme for Wave Equation --- p.61 / Chapter 2.3.2 --- Decay of the Yee Scheme Solution to the Berenger's PMLs --- p.62 / Chapter 2.3.3 --- Stability and Convergence of the Yee Scheme for the Berenger's PMLs --- p.67 / Chapter 2.3.4 --- Decay of the Yee Scheme Solution to the Hagstrom's PMLs --- p.70 / Chapter 2.3.5 --- Stability and Convergence of the Yee Scheme for the Hagstrom's PMLs --- p.75 / Chapter 2.4 --- Modified Lax-Wendroff Scheme for PMLs --- p.80 / Chapter 2.4.1 --- Exponential Decays in Parabolic Equations --- p.80 / Chapter 2.4.2 --- Exponential Decays in Hyperbolic Equations --- p.82 / Chapter 2.4.3 --- Exponential Decays of Modified Lax-Wendroff Solutions --- p.86 / Chapter 3 --- Numerical Simulation --- p.93 / Bibliography --- p.99

在本文中，我們為了三維空間的馬克士威方程組(Maxwell’s equation)制定和分析了一套新種類的交錯間斷伽遼金(discontinuous Galerkin)方法，同時考慮了時間依賴性和時間諧波的馬克士威方程組。我們用了空間離散上交錯笛卡兒網格，這種方法具有許多良好的性質。首先，我們的方法所得出的數值解保留了電磁能量，並自動符合了高斯定律的離散版本。第二，質量矩陣是對角矩陣，從而時間推進是顯式和非常有效的。第三，我們的方法是高階準確，最佳收斂性在這裏會被嚴格地證明。第四，基於笛卡兒網格，它也很容易被執行，並可視為是典型的Yee’s Scheme的以及四邊形的邊有限元的推廣。最後，超收斂結果也會在這裏被證明。 / 在本文中，我們還提供了幾個數值結果驗證了理論的陳述。我們計算了時間依賴性和時間諧波的馬克士威方程組數值收斂結果。此外，我們計算時間諧波馬克士威方程組特徵值問題的數值特徵值，並與理論特徵值比較結果。最後，完美匹配層(Perfect Matching Layer)吸收邊界的問題也有實行其數值結果。 / We develop and analyze a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations in this paper. Both time-dependent and time-harmonic Maxwell’s equations are considered. The spatial discretization is based on staggered Cartesian grids which possess many good properties. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Second, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Third, our method is high order accurate and the optimal order of convergence is rigorously proved. Fourth, it is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Lastly, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. / In this paper, we also provide several numerical results to verify the theoretical statements. We compute the numerical convergence order using L2-norm and discrete-norm respectively for both the time-dependent and time-harmonic Maxwell’s equations. Also, we compute the numerical eigenvalues for the time-harmonic eigenvalue problem and compare the result with the theoretical eigenvalues. Lastly, applications to problems in unbounded domains with the use of PML are also presented. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yu, Tang Fei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 46-49). / Abstracts also in Chinese. / Chapter 1 --- Introduction and Model Problems --- p.1 / Chapter 2 --- Staggered DG Spaces --- p.4 / Chapter 2.1 --- Review on Gauss-Radau and Gaussisan points --- p.5 / Chapter 2.2 --- Basis functions --- p.6 / Chapter 2.3 --- Finite Elements space --- p.7 / Chapter 3 --- Method derivation --- p.14 / Chapter 3.1 --- Method --- p.14 / Chapter 3.2 --- Time discretization --- p.17 / Chapter 4 --- Energy conservation and Discrete Gauss law --- p.19 / Chapter 4.1 --- Energy conservation --- p.19 / Chapter 4.2 --- Discrete Gauss law --- p.22 / Chapter 5 --- Error analysis --- p.24 / Chapter 6 --- Numerical examples --- p.29 / Chapter 6.1 --- Convergence tests --- p.30 / Chapter 6.2 --- Diffraction by a perfectly conducting object --- p.30 / Chapter 6.3 --- Perfectly matched layers --- p.37 / Chapter 7 --- Time Harmonic Maxwell’s equations --- p.40 / Chapter 7.1 --- Model Problems --- p.40 / Chapter 7.2 --- Numerical examples --- p.40 / Chapter 7.2.1 --- Convergence tests --- p.41 / Chapter 7.2.2 --- Eigenvalues tests --- p.41 / Chapter 8 --- Conclusion --- p.45 / Bibliography --- p.46

Wong Chak-fu. / "September 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 451-475). / 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.

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

Dai, Lipeng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 73-75). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 1.0.1 --- Linear sampling method --- p.2 / Chapter 1.0.2 --- choice of cut-off values --- p.5 / Chapter 1.0.3 --- Underwater image problem --- p.7 / Chapter 2 --- Mathematical justification of LSM --- p.10 / Chapter 2.1 --- Some mathematical preparations --- p.11 / Chapter 2.2 --- Well-posedness of an interior transmission problem --- p.13 / Chapter 2.3 --- Linear sampling method: full aperture --- p.20 / Chapter 2.4 --- Linear sampling method: limited aperture --- p.23 / Chapter 3 --- Strengthened linear sampling method --- p.28 / Chapter 3.1 --- Proof of theorem 1.0.3 --- p.28 / Chapter 3.2 --- Several estimates in theory for strengthened LSM --- p.33 / Chapter 4 --- Underwater imaging problem --- p.38 / Chapter 4.1 --- Boundary integral method --- p.38 / Chapter 4.2 --- Approximation of the Integral Kernel in (4.12) --- p.40 / Chapter 4.3 --- Numerical solution of (4.12) --- p.44 / Chapter 4.4 --- Underwater image problem --- p.45 / Chapter 4.5 --- Imaging scheme without a reference object --- p.48 / Chapter 4.6 --- Numerical examples without a reference object --- p.49 / Chapter 4.7 --- Imaging scheme with a reference object --- p.59 / Chapter 4.8 --- Numerical examples with a reference object --- p.61

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University of Macau / Faculty of Science and Technology / Department of Electromechanical Engineering

A numerical method based on the the method of characteristics for hyperbolic systems of partial differential equations in four independent variables is developed and used for solving time domain Maxwell's equations. The method uses the characteristic hypersurfaces and the characteristic conditions to derive a set of independent equations relating the electric and magnetic field components on these hypersurfaces. A discretization scheme is developed to solve for the unknown field components at each time step. The method retains many of the good features of the original method of characteristics for hyperbolic systems in two independent variables, such as optimal time step, good behavior near data discontinuities and the ability to treat general boundary conditions. The method is exemplified by calculating the time domain response of a few typical planar interconnect structures to Gaussian and unit step excitations. Although the general emphasis is on interconnect problems, the method is applicable to a number of other transient electromagnetic field problems governed by Maxwell's equations. In addition to the method of characteristics a finite difference scheme, known in mathematic circles as the modified Richtmyer scheme, is applied to the time domain solution of Maxwell's equations. Both methods should be useful for efficient full wave analysis of three dimensional electromagnetic field problems. / Graduation date: 1994

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