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研究生： 徐嘉群
Hsu, Chia-Chun
論文名稱： 以費雪訊息量處理有雜訊的拉普拉斯逆變換問題
Inverse Laplace transformation of noisy data using Fisher information
指導教授： 楊緒濃
Nyeo, Su-Long
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2014
畢業學年度： 102
語文別： 英文
論文頁數： 41
中文關鍵詞： 逆問題 、不適定問題 、Tikhonov正規化 、費雪訊息量
外文關鍵詞： Inverse Problem, Ill-Posedness, Tikhonov Regularization, Fisher Information
相關次數： 點閱：93  下載：3 
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    • 逆問題的研究起源於科學和工程。其中具有第一類Fredholm積分方程中帶有指數遞減形式的積分變換核的問題是一類不適定問題。在這份論文中，我們藉由Tikhonov正規化方法來探討這類問題。
    本文將列出針對第一類Fredholm積分方程求出良適定義解的數值演算法，並且利用L-曲線的方式決定最佳解。
    最後，我們會把上述方法應用在對數常態分佈函數所模擬的單峰值與雙峰值函數的資料，並且得出本文所用的方法可以恰當的重建單峰值情形，並且相比於其他演算法下得到具有相當精確程度的雙峰值情形。

    Inverse problems arise in many areas of research and applications. The Fredholm integral equation of the first kind with an exponential decay kernel is an ill-posed inverse problem. In this thesis, the equation is studied using the Tikhonov regularization method with Fisher information as a regularization function.
    A numerical algorithm for solving the Fredholm integral equation is outlined to obtain well-defined solutions and an optimal, unique solution is determined by the L-curve criterion.
    In our study, several sets of simulated data are created from the log-normal distribution function to evaluate the performance of our algorithm. We show that the algorithm can efficiently recover broad single-peak distributions and double-peak distributions with higher accuracy than the well-known algorithms of the maximum-entropy method and CONTIN.

    1 Introduction 2 2 Inverse Problems 4 2.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Hadamard’s conditions for well-posedness . . . . . . . . . . . . . . . . . 5 2.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Analytic Solution of the Linear Inverse Problem without Noise . . . . . 7 2.5 Least Square Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Laplace Transform Inversion 9 3.1 The Inversion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Methods for Solving Least-Square Problem . . . . . . . . . . . . . . . . 10 3.2.1 Truncated Singular Value Decomposition . . . . . . . . . . . . . 10 3.2.2 Eigenfunctions of the Kernel . . . . . . . . . . . . . . . . . . . . 10 3.2.3 Non-Negative Singular Value Decomposition . . . . . . . . . . . 11 3.2.4 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . 12 4 Fisher Information Regularization 13 4.1 The Iteration Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Criterion for Choosing λ . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Evaluation of Algorithm 16 5.1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Comparing Auto-Correlation Functions for Different Cases . . . . . . . 18 5.3 Effect of Different Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.4 Fitting Single-Peak Distribution . . . . . . . . . . . . . . . . . . . . . . 22 5.5 Fitting Double-Peak Distribution . . . . . . . . . . . . . . . . . . . . . 25 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Conclusion 29 Appendix A Fisher Information 30 Appendix B Codes Used in This Thesis 33

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