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.

本論文主要研究污染源追蹤和重構的反問題以及倒向隨機微分方程的數值求解。 / 論文的第一部份考慮污染源追蹤及重構的反問題。它的目的是重構反應對流擴散系統中的未知污染源的位置以及強度。污染源的追蹤和重構在工程、化學、生物以及環境等領域有廣泛的應用。我們將同時重構靜態單點污染源的位置以及強度。在本論文中，我們提出了一個基於對偶概率的算法，它將污染源追蹤重構的反問題轉化為Volterra積分反問題。對於污染源的位置和污染物釋放強度的可重構性，文中也進行了理論上的分析和討論。數值結果表明此方法是高效穩定的。隨後，我們將對偶概率方法推廣應用與追蹤和重構動態單點污染源隨時間的軌跡以及強度。數值結果顯示，我們的方法要比多數現有的方法為有效，計算成本也大大降低。 / 論文的第二部份討論倒向隨機微分方程的數值求解。倒向隨機微分方程在隨機控制、生物、化學反應，尤其是數理金融上有重要的應用。論文中所提出的數值方法，主要是基於倒向隨機微分方程的置換解的概念。置換解的適定性分析不涉及鞅表示論，從而更靈活，更容易推廣。利用置換解的理論，文中所涉及的誤差分析都不需要用到鞅表示論。對於一般的倒向隨機微分方程，我們提出了一種簡單的倒向算法，并證明了它是半階收斂的。但是，在算法的實際應用中只可能選取有限個基函數，從而帶入了截斷誤差。截斷誤差在簡單倒向算法中會隨時間累加，導致誤差是半階增長的。為了克服這個缺點，我們提出了一種新的算法。這種算法無需進行皮卡迭代，並且在理論上我們證明了，使用這個新的算法，截斷誤差是可控的，它不會隨時間增加。隨後，我們對馬爾科夫情況的倒向隨機微分方程提出了幾個高階的數值算法，並且給出了嚴格的誤差分析。我們的數值實驗結果表明，文中所提出的方法精度高，穩定性強，且計算成本小。 / In this thesis, we shall propose some numerical methods for solving two important classes of application problems, namely the inverse heat source problems and the backward stochastic differential equations. / The inverse heat source problems are to recover the source terms in a convection-diffusion- reaction system. These inverse problems have wide applications in many areas, such as engineering, chemistry, biology, pollutant tracking, and so on. We shall first investigate the simultaneous reconstruction of the location and strength of a static singular source. An adjoint probabilistic algorithm is proposed, which turns the inverse heat source problem into an inverse Volterra integral problem. The identifiability of the location and strength of a singular source is also discussed, and numerical results are presented to show the robustness and effectiveness of the method. Then we extend the adjoint probabilistic method to reconstruct the source trace and release history of a singular moving point source. Numerical examples show that the adjoint probabilistic method is more efficient and less expensive than most existing efficient numerical methods. / The second part of the thesis is devoted to numerical solutions of some nonlinear backward stochastic differential equations (BSDEs). BSDEs are widely used in various fields like stochastic control, biology, chemistry reaction, especially mathematical finance. Our numerical methods are based on a new framework about the transposition solution to BSDEs. The proof of the well-posedness of the transposition solution does not involve Martingale representation, neither does our error analysis for the numerical schemes proposed in this thesis. For general BSDEs, we first propose a simple backward scheme, which is proved to have an accuracy of half order. However, in the real application of the scheme, it is only possible to choose a finite subset of basis functions, which will generate truncation error. The truncation error accumulates backward in time, leading to the increment of the numerical error up to a half order. To overcome this drawback, we propose a new numerical scheme without Picard iterations and prove that the truncation error is bounded independent of time partitions. Afterwards, we propose some higher order schemes for Markovian BSDEs with rigorous error analysis. Finally, numerical simulations are presented to demonstrate that the proposed methods are accurate, stable and less expensive than most existing ones. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Shiping. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 126-133). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.v / Chapter 1 --- Introduction to inverse heat source problems and BSDEs --- p.1 / Chapter 1.1 --- Inverse heat source problems --- p.2 / Chapter 1.2 --- Backward stochastic differential equations --- p.7 / Chapter 1.3 --- Outline of the thesis --- p.11 / Chapter Part I: --- Numerical Method for Inverse Heat Source Problem --- p.13 / Chapter 2 --- Inverse heat source: static point source --- p.14 / Chapter 2.1 --- Reformulation of the forward problem --- p.15 / Chapter 2.2 --- Inverse source problem and its identifiability --- p.21 / Chapter 2.2.1 --- Identifiability of partial time in one dimensional cases --- p.21 / Chapter 2.2.2 --- Identifiability of two dimensional cases --- p.25 / Chapter 2.3 --- Algorithm to solve the inverse problem --- p.26 / Chapter 2.4 --- Numerical experiments --- p.29 / Chapter 3 --- Inverse heat source: moving point source --- p.41 / Chapter 3.1 --- Reformulation of the problem --- p.42 / Chapter 3.2 --- Algorithm and numerical examples --- p.43 / Chapter 3.2.1 --- Algorithm to recover source trace and strength --- p.44 / Chapter 3.2.2 --- Numerical examples --- p.45 / Chapter Part II: --- Numerical Methods to Backward Stochastic Differential Equations --- p.55 / Chapter 4 --- Preliminaries --- p.56 / Chapter 4.1 --- Notations and definitions --- p.56 / Chapter 4.2 --- Useful lemmas and theorems --- p.60 / Chapter 4.3 --- Existing schemes for forward SDEs --- p.66 / Chapter 5 --- Numerical algorithms to BSDEs and error estimates --- p.68 / Chapter 5.1 --- A simple backward algorithm for BSDEs and its error estimate --- p.69 / Chapter 5.1.1 --- A simple backward algorithm --- p.69 / Chapter 5.1.2 --- Error estimate for simple backward scheme --- p.71 / Chapter 5.2 --- A new explicit backward algorithm for BSDEs and its error estimates --- p.85 / Chapter 5.2.1 --- A new explicit backward algorithm --- p.85 / Chapter 5.2.2 --- Error estimate for explicit backward scheme --- p.86 / Chapter 6 --- Higher order schemes of Markovian cases and error estimates --- p.91 / Chapter 6.1 --- Error estimate of 1-order scheme for Markovian BSDEs --- p.92 / Chapter 6.2 --- 2-order scheme for Markovian BSDEs and its error estimate --- p.100 / Chapter 6.2.1 --- 2-order scheme for Markovian BSDEs --- p.100 / Chapter 6.2.2 --- Error estimate of 2-order scheme --- p.102 / Chapter 7 --- Simulation results for BSDEs --- p.106 / Chapter 7.1 --- Basis functions --- p.107 / Chapter 7.2 --- Numerical simulations --- p.108 / Chapter 7.2.1 --- Application on option pricing --- p.108 / Chapter 7.2.2 --- Numerical examples on Markovian BSDEs --- p.114 / Chapter 8 --- Conclusions and future work --- p.123 / Chapter 8.1 --- Conclusions --- p.123 / Chapter 8.2 --- Future work --- p.124 / Bibliography --- p.126

by Bai Zheng Jian. / "May 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references. / 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.

Xie Jian Li. / "August 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 106-109). / 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.

Lo Ting Shek = 從開放系統頻譜或散射相移到逆有限支合集勢函數的研究 / 盧庭碩. / "April 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 144-146). / 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. / Lo Ting Shek = Cong kai fang xi tong pin pu huo san she xiang yi dao ni you xian zhi he ji shi han shu de yan jiu / Lu Tingshuo.

by Hao-min Zhou. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references. / Abstract --- p.1 / Introduction --- p.3 / Paper I --- p.8 / Paper II --- p.28

A identificação de esforços dinâmicos vem despertando um contínuo interesse da comunidade modal e de teste de vibrações principalmente nas duas últimas décadas. De forma semelhante como na maioria das técnicas de identificação modal, a identificação de forças, representa um problema inverso. Problemas inversos são conhecidos pelo fato de serem naturalmente mau condicionados numericamente, uma vez que os mesmos representam uma operação de deconvolução no tempo. Dentre os métodos utilizados na identificação de forças a partir de medidas de movimento, pode-se destacar a técnica da pseudo inversa no domínio da frequência. A técnica da pseudo inversa requer o conhecimento das medidas das respostas da estrutura usualmente acelerações assim como das funções repostas em frequência da estrutura (FRF). Este método requer a inversão das matrizes de FRF dos sistemas para todas as linhas espectrais na faixa de frequência de interesse. Este procedimento de inversão usualmente apresenta dificuldades numéricas em algumas frequências. Será levantado suas principais vantagens e desvantagens quando comparado com um método no domínio do tempo no caso da técnica da SWAT, para alguns tipos de formas de excitação comumente empregada sem testes modais de vibração. São apresentados resultados de simulações numéricas e resultados experimentais para uma estrutura simples. / The main goal of this dissertation is to takle a difficult problem in experimental modal analysis that is the indirect identification of input forces based on the knowledge of the structure\'s output response and dynamic characteristics, such as impulse responses and/or FRF. This represents an inverse problem in mechanics and usually offers great numerical difficulties in the process of identification of the input forces. There are methods for input force identification in the lime and in the frequency domains. In lhe time domain the SWAT method (Sum of lhe Weighet Accelerations Technique) is a method that is based on the principle of motion of the mass center. This technique alows the estimation of the resulting force acting on the structure and is primarily used with impact force signals. In the frequency domain the pseudo-inverse technique is known to give estimates of the resulting forces acting on the structure under investigation. In this work numerically simulated as well as experimental results are presented for both methods and their major advantages and disadvantages are discussed. The pseudo-inverse method is further employed in the identification of experimental transient and random multiple inputs.

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

In mining engineering, blast-induced ground vibration has become one of the major concerns when production blasts are conducted, especially when the mining areas and the blast sites are near inhabited areas or infrastructure of interest. To comply with regulations, a vibration monitoring program should be developed for each mining operation. The vibration level, which is usually indicated by the peak particle velocity (PPV) of the vibration waveform, should fall below the maximum allowable values. Ideally, when blasting is near structures of interest (power towers, dams, houses, etc.), the vibration level (PPV) should be predicted prior to the actual production blasts. There are different techniques to predict the PPV, one in particular is the signature hole technique. This technique is based on signals and systems theory and uses a mathematical operation called convolution to assess the waveform of the production blast. This technique uses both the vibration waveform of an isolated hole and the timing function given by the timing used in the blast. The signature hole technique requires an isolated single-hole waveform to create a prediction. Sometimes this information is difficult to acquire, as it requires the synthesis of a single-hole vibration waveform from a production blast vibration signal. The topic of ground vibrations from mining blasts, and more specifically the synthesis of a single-hole vibration waveform, has been studied by researchers in past decades, but without any concrete success. This lack of success may be partially due to the complexity and difficulty of modelling and calculation. However, this inverse methodology can be very meaningful if successfully applied in blasting engineering. It provides a convenient and economical way to obtain the single-hole vibration waveform and make the prediction of a production blast waveform easier. This dissertation research involves the theories of deconvolution, linear superposition, and Fourier phases to recover single-hole vibration waveforms from a production waveform. Preliminary studies of deconvolution included spectral division deconvolution and Wiener filtering deconvolution. In addition to the adaptation of such methodologies to the blast vibrations problems, the effectiveness of the two deconvolution methods by the influence of delay interval and number of holes is also discussed. Additionally, a new statistical waveform synthesis method based on the theories of linear superposition, properties of Fourier phase, and group delays was developed. The validation of the proposed methodology was also conducted through several field blasting tests. Instead of synthesizing one normalized single-hole vibration waveform by deconvolution, the proposed statistical waveform synthesis methodology generates a different single-hole vibration waveform for each blast hole. This method is more effective and adaptable when synthesizing single-hole vibration waveforms. Recommendations for future work is also provided to improve the methodology and to study other inverse problems of blast vibrations.

Vibration-based structural health monitoring (SHM) seeks to detect, quantify, locate, and prognosticate damage by processing vibration signals measured while the structure is operational. The basic premise of vibration-based SHM is that damage will affect the stiffness, mass or energy dissipation properties of the structure and in turn alter its measured dynamic characteristics. In order to make SHM a practical technology it is necessary to perform damage assessment using only a minimum number of permanently installed sensors. Deducing damage at unmeasured regions of the structural domain requires solving an inverse problem that is underdetermined and(or) ill-conditioned. In addition, the effects of local damage on global vibration response may be overshadowed by the effects of modelling error, environmental changes, sensor noise, and unmeasured excitation. These theoretical and practical challenges render the damage identification inverse problem ill-posed, and in some cases unsolvable with conventional inverse methods. This dissertation proposes and tests a novel interpretation of the damage identification inverse problem. Since damage is inherently local and strictly reduces stiffness and(or) mass, the underdetermined inverse problem can be made uniquely solvable by either imposing sparsity or non-negativity on the solution space. The goal of this research is to leverage this concept in order to prove that damage identification can be performed in practical applications using significantly less measurements than conventional inverse methods require. This dissertation investigates two sparsity inducing methods, L1-norm optimization and the non-negative least squares, in their application to identifying damage from eigenvalues, a minimal sensor-based feature that results in an underdetermined inverse problem. This work presents necessary conditions for solution uniqueness and a method to quantify the bounds on the non-unique solution space. The proposed methods are investigated using a wide range of numerical simulations and validated using a four-story lab-scale frame and a full-scale 17 m long aluminum truss. The findings of this study suggest that leveraging the attributes of both L1-norm optimization and non-negative constrained least squares can provide significant improvement over their standalone applications and over other existing methods of damage detection.