由線性彈性模型引起的多維系數反問題在很多範疇都有其應用，如斷層探測、油田檢測、鹽石檢測、礦石檢測及醫療成像等。瞬時成像技術是其中最有用的應用。它提供了一個快速及安全的醫療成像技術，可以用來檢測在身體內快速移動的器官的一些異常組織，如肝腫瘤。在這篇論文中，我們會重點討論兩個解決瞬時弹性成像反問題的數值方法，即水平集反演方法和近似全局收斂方法。我們會研究這兩種方法的推導和數值結果。 / 特別地，近似全局收斂方法是一種由Klibanov 新提出用來解決由雙曲偏微分引起的多維系數反問題的方法。因為這佪方法沒有使用求泛函極小值的步驟，因此能避免了一些眾所周知的問題，所以它特別穩定。數值結果顯示近似全局收斂方法對噪聲有很高的穩定性。這表明近似全局收斂方法是一個解決由線性彈性模型引起的多維系數反問題的其中一個有效方法。 / Multi-dimensional coefficient inverse problem (MCIP) in linear elasticity has found many applications, such as crack detection, oil/salt/ore detection, medical imaging. Transient elastography is among one of the most useful applications, providing a fast and safe medical imaging technique which can be used to detect tumors or abnormal tissue in “fast-moving“ organs such as the liver. In this thesis focus is casted on two of the numerical algorithms to solve inverse problems related to transient elastography, namely the level-set inversion method and the approximate globally convergent method. The derivations of both methods and numerical results are presented. In particular, the approximate globally convergent method is a newly developed stable method to solve coefficient determination inverse problem for hyperbolic partial differential equation proposed by Beilina and Klibanov in [6]. It achieves pproximately a global convergence by avoiding construction of a least squares functional, thus averting some of the well-known problems of trapping in the neighborhoods of local minima when one minimizes such a nonlinear functional. The results of the approximate globally convergent method have shown its strong stability and robustness. This suggests a good way for the reconstruction of the distribution of the shear modulus in the coefficient inverse problem of linear elasticity. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chow, Yat Tin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 98-102). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Linear Elasticity Model --- p.9 / Chapter 2.1 --- Introduction to Linear Elasticity Model --- p.9 / Chapter 2.2 --- Physical Meanings of Elasticity Equation --- p.12 / Chapter 2.3 --- Derivations of Linear Elasticity Equation --- p.14 / Chapter 2.4 --- Discussion of Christoffel’s Equation --- p.17 / Chapter 3 --- Formulations of the Forward and Inverse Problem --- p.27 / Chapter 3.1 --- The Forward Problem --- p.29 / Chapter 3.2 --- The Inverse Problem --- p.29 / Chapter 3.3 --- A Uniqueness Result --- p.30 / Chapter 4 --- Algorithms for Inverse Problems in Elasticity --- p.33 / Chapter 5 --- Level Set Inversion Method --- p.37 / Chapter 5.1 --- Arrival Time Acquisition: Cross-Correlation --- p.37 / Chapter 5.2 --- The Distance Inversion Method --- p.41 / Chapter 5.3 --- Solving the Forward Eikonal Equation --- p.43 / Chapter 5.3.1 --- Discretizing the eikonal equation --- p.43 / Chapter 5.3.2 --- A forward eikonal solver: fast marching algorithm --- p.47 / Chapter 5.3.3 --- A forward eikonal solver: fast sweeping algorithm --- p.50 / Chapter 5.4 --- Level Set Inversion Scheme --- p.54 / Chapter 5.5 --- Numerical Implementation --- p.57 / Chapter 5.6 --- Results of Reconstructions --- p.58 / Chapter 6 --- Approximate Globally Convergent Method --- p.63 / Chapter 6.1 --- The Forward Problem --- p.65 / Chapter 6.1.1 --- Forward problem in time domain --- p.65 / Chapter 6.1.2 --- Forward problem in Laplace domain --- p.67 / Chapter 6.2 --- The Inverse Problem --- p.67 / Chapter 6.2.1 --- Inverse problem in time domain --- p.67 / Chapter 6.2.2 --- Inverse problem in Laplace domain --- p.68 / Chapter 6.3 --- A Nonlinear Integral Differential Equation --- p.69 / Chapter 6.4 --- Approximation of the First Tail --- p.71 / Chapter 6.5 --- The Algorithm --- p.72 / Chapter 6.6 --- Notes About the Convergence Analysis --- p.76 / Chapter 6.6.1 --- Approximate global convergence --- p.76 / Chapter 6.6.2 --- Basic formulation of Theorem 2.9.4 of [6] --- p.78 / Chapter 6.6.3 --- Some ideas of the convergence analysis for the algorithm in section 6.5 --- p.80 / Chapter 6.7 --- Numerical Implementation --- p.81 / Chapter 6.8 --- Results of Reconstructions --- p.88 / Chapter 7 --- Conclusions --- p.96 / Bibliography --- p.98
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328613 |
| Date | January 2012 |
| Contributors | Chow, Yat Tin., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (102 leaves) : ill. (some col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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