This thesis is devoted to solving the inverse source problem arising in image reconstruction problems. In general, the solution is non-unique and the problem is severely ill-posed. Therefore, small perturbations, such as the noise in the data, and the modeling error in the forward problem, will cause huge errors in the computations. In practice, the most widely used method to tackle the problem is based on Tikhonov-type regularizations, which minimizes a cost function combining a regularization term and a data fitting term. However, because the two tasks, namely regularization and data fitting, are coupled together in Tikhonov regularization, they are difficult to solve. It happens even if each task can be efficiently solved when they are separate. We propose a method to overcome the major difficulties, namely the non-uniqueness of the solution and noisy data fitting, separately. First we find a particular solution called the orthogonal solution that satisfies the data fitting term. Then we add to it a correction function in the kernel space so that the final solution fulfills the regularization and other physical requirements. The key idea is that the correction function in the kernel has no impact to the data fitting, and the regularization is imposed in a smaller space. Moreover, there is no parameter needed to balance the data fitting and regularization terms. As a case study, we apply the proposed method to Fluorescence Tomography (FT), an emerging imaging technique well known for its ill-posedness and low image resolution in existing reconstruction techniques. We demonstrate by theory and examples that the proposed algorithm can drastically improve the computation speed and the image resolution over existing methods.

Lau Kin Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 80-83). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.3 / Chapter 1.1 --- Heat conduction problem --- p.3 / Chapter 1.2 --- Direct problem --- p.4 / Chapter 1.3 --- Inverse problem --- p.4 / Chapter 1.4 --- Difficulty of the inverse problems --- p.5 / Chapter 1.5 --- A simple but important example for instability --- p.5 / Chapter 1.6 --- The purpose of this thesis --- p.7 / Chapter 2 --- Stability of the inverse problem --- p.9 / Chapter 2.1 --- Conditional stability results --- p.9 / Chapter 2.2 --- Stability of the inverse problems --- p.11 / Chapter 3 --- The continuous formulation --- p.30 / Chapter 3.1 --- Constrained minimization problem --- p.30 / Chapter 3.2 --- Existence of minimizers to the minimization problem --- p.31 / Chapter 4 --- Discretization and its convergence --- p.36 / Chapter 4.1 --- Finite element space --- p.36 / Chapter 4.2 --- Two important discrete projection operators --- p.37 / Chapter 4.3 --- Finite element problem --- p.39 / Chapter 4.4 --- Existence of minimizers to the finite element problem --- p.39 / Chapter 4.5 --- Discrete minimizers and global minimizers --- p.42 / Chapter 5 --- Numerical algorithms --- p.51 / Chapter 5.1 --- Gateaux derivative --- p.51 / Chapter 5.2 --- Nonlinear single-grid gradient method --- p.53 / Chapter 5.3 --- Nonlinear multigrid gradient method --- p.55 / Chapter 6 --- Numerical experiments --- p.60 / Chapter 6.1 --- One dimensional examples --- p.60 / Chapter 6.2 --- Two dimensional examples --- p.66

Yip Chi Ming = 守恆及開放系統的能譜反問題 / 葉志明. / Thesis submitted in 2000. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves [244]-247). / Text in English; abstracts in English and Chinese. / Yip Chi Ming = Shou heng ji kai fang xi tong de neng pu fan wen ti / Ye Zhiming. / Abstract --- p.i / Acknowledgements --- p.ii / Contents --- p.iii / List of Figures --- p.viii / List of Tables --- p.xxi / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- The Sturm-Liouville Problem --- p.3 / Chapter 1.2 --- Historical review of inverse problems --- p.7 / Chapter 1.3 --- Conservative systems --- p.10 / Chapter 1.4 --- Open systems --- p.10 / Chapter 1.5 --- Organization of the following chapters --- p.11 / Chapter Chapter 2. --- Conservative Spectral Problem --- p.12 / Chapter 2.1 --- The system --- p.12 / Chapter 2.2 --- Properties of conservative systems --- p.13 / Chapter 2.2.1 --- Asymptotic expansion of eigenvalues --- p.14 / Chapter 2.3 --- Forward spectral problem --- p.16 / Chapter 2.3.1 --- FDM and FEM --- p.17 / Chapter 2.3.2 --- Solving transcendental equation --- p.20 / Chapter 2.4 --- Phase shift problem --- p.20 / Chapter 2.4.1 --- Square well potential --- p.22 / Chapter Chapter 3. --- Forward Spectral Problem for Open Systems --- p.25 / Chapter 3.1 --- The system --- p.26 / Chapter 3.2 --- Properties of open systems --- p.28 / Chapter 3.2.1 --- Asymptotic behaviour of QNM eigenvalues --- p.28 / Chapter 3.2.2 --- Doubling of modes --- p.33 / Chapter 3.2.3 --- Generalized norm of QNMs --- p.34 / Chapter 3.2.4 --- Completeness --- p.37 / Chapter 3.2.5 --- Eigenfunction expansion for QNMs - two component formalism --- p.39 / Chapter 3.3 --- Forward spectral problem --- p.45 / Chapter Chapter 4. --- Conservative Inverse Problem --- p.50 / Chapter 4.1 --- Sun-Young-Zou (SYZ) method --- p.51 / Chapter 4.1.1 --- Perturbative inversion --- p.53 / Chapter 4.1.2 --- The regulators (δn) --- p.54 / Chapter 4.1.3 --- Total inversion (TI) --- p.59 / Chapter 4.1.4 --- Numerical results --- p.60 / Chapter 4.2 --- Rundell and Sacks method (RS method) --- p.74 / Chapter 4.2.1 --- Completeness --- p.75 / Chapter 4.2.2 --- The integral equation --- p.78 / Chapter 4.2.3 --- Uniqueness --- p.82 / Chapter 4.2.4 --- RS formalism --- p.84 / Chapter 4.2.5 --- Numerical results and difficulties --- p.89 / Chapter 4.2.6 --- Summary --- p.110 / Chapter 4.3 --- Phase shift problem --- p.112 / Chapter 4.3.1 --- Reduction to spectral problem --- p.113 / Chapter 4.3.2 --- Modified RS algorithm for finite-range phase shift problem --- p.116 / Chapter 4.3.3 --- Discussion --- p.130 / Chapter 4.4 --- Bound states --- p.131 / Chapter Chapter 5. --- Open Inverse Problem --- p.136 / Chapter 5.1 --- SYZ method --- p.136 / Chapter 5.1.1 --- Perturbative Inversion (PI) and Total Inversion (TI) --- p.137 / Chapter 5.1.2 --- Numerical results --- p.138 / Chapter 5.1.3 --- Other choices of (δn) --- p.156 / Chapter 5.2 --- RS method --- p.158 / Chapter 5.2.1 --- The integral equation --- p.159 / Chapter 5.2.2 --- Cauchy data --- p.160 / Chapter 5.2.3 --- Completeness conjecture --- p.162 / Chapter 5.2.4 --- Numerical verification of completeness condition --- p.163 / Chapter 5.2.5 --- Inversion for Cauchy data --- p.166 / Chapter 5.2.6 --- Cauchy data on 0 < x≤ α --- p.167 / Chapter 5.2.7 --- Comparison system --- p.169 / Chapter Chapter 6. --- Conclusions and Further Studies --- p.188 / Chapter 6.1 --- Conclusions of this thesis --- p.188 / Chapter 6.2 --- Further studies --- p.189 / Chapter Appendix A. --- Singular Value Decomposition --- p.199 / Chapter Appendix B. --- Asymptotic Behaviour of Phase Shifts --- p.203 / Chapter B.1 --- Asymptotic behaviour of phase shift data --- p.203 / Chapter B.2 --- Levinson's theorem --- p.204 / Chapter Appendix C. --- Forward Problem for Conservative Systems --- p.207 / Chapter C.1 --- Finite difference method --- p.207 / Chapter C.2 --- Finite element method --- p.209 / Chapter C.2.1 --- Solving transcendental equation --- p.215 / Chapter Appendix D. --- FDM and FEM for Open Systems --- p.220 / Chapter D.1 --- Finite difference method --- p.220 / Chapter D.2 --- Finite element method --- p.222 / Chapter Appendix E. --- Asymptotic Behaviour of NM Eigenvalues --- p.226 / Chapter Appendix F. --- Asymptotic Behaviour of QNM Eigenvalues --- p.232 / Chapter Appendix G. --- QNM Forward Problem 一 Transcendental Equation --- p.239 / Chapter Appendix H. --- Forward Problem - Calculation of Phase Shifts --- p.243 / Bibliography --- p.245

by Hau-Leung Chung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 41-42). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Paper I --- p.5 / Chapter 1.2 --- Paper II --- p.6 / Chapter 2 --- The Inexact Newton-Like Method for Inverse Eigen- value Problem --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- The Newton-Like Method --- p.9 / Chapter 2.3 --- The Inexact Newton-Like Method --- p.11 / Chapter 2.4 --- Convergence Analysis --- p.14 / Chapter 2.5 --- Numerical Experiments --- p.22 / Chapter 3 --- A DCT Based Watermarking Scheme for Copyright Protection of Images --- p.26 / Chapter 3.1 --- Introduction --- p.26 / Chapter 3.2 --- Preliminary --- p.28 / Chapter 3.2.1 --- Gray-level image --- p.28 / Chapter 3.2.2 --- Color image --- p.29 / Chapter 3.2.3 --- The Discrete Cosine transform --- p.30 / Chapter 3.3 --- Watermarking Approaches --- p.31 / Chapter 3.3.1 --- Insertion procedures --- p.31 / Chapter 3.3.2 --- Retrieval procedures --- p.33 / Chapter 3.4 --- Experimental results --- p.34 / Chapter 3.5 --- Other Applications --- p.38 / Chapter 3.5.1 --- Data Hiding --- p.38 / Chapter 3.5.2 --- Authentication --- p.39 / Chapter 3.5.3 --- Fingerprinting --- p.39 / Chapter 3.5.4 --- Copy Control --- p.39 / Chapter 3.6 --- Conclusion --- p.40 / Bibliography --- p.41

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

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