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研究生：	高昱璋 Kao, Yu-Chang
論文名稱：	從橢圓區域的邊界上的測量去偵測區域內部的橢圓狀未知物之數值實驗—二維情況 Some Numerical Experiments on Detecting an Unknown Elliptical Obstacle from the Boundary Measurements in an Elliptical Region—2D Case
指導教授：	關汝琳 Kuan, Ru-Lin
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2026
畢業學年度：	114
語文別：	英文
論文頁數：	133
中文關鍵詞：	包圍法、反問題、邊界測量、未知物位置、電流密度、電壓、橢圓、狄利克雷邊界條件、紐曼邊界條件
外文關鍵詞：	enclosure method, inverse problem, boundary measurement, the position of unknown obstacle, current density, voltage, ellipse, Dirichlet boundary condition, Neumann boundary condition
相關次數：	點閱：15 下載：0
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鑒於 Masaru Ikehata 教授的論文 [5] 中所講述的包圍法（enclosure method）和蔡佩旻的論文 [12] 的啟發，本研究旨在探討如何利用在區域邊界測量的資訊來判斷鑲嵌於背景區域中的未知阻礙物之位置。具體來說，我們將背景區域設定為一個橢圓狀區域，並在其中嵌入另一個橢圓形阻礙物。對於該橢圓狀阻礙物，其位置、大小以及旋轉角度皆可自由調整。此外，我們亦考慮不含阻礙物之背景區域作為對照情況。針對上述兩種情況，我們分別在對應的區域上定義拉普拉斯方程（Laplace equation）。在兩個問題中，皆於背景區域的外邊界施加狄利克雷邊界條件（Dirichlet boundary condition）；而對於含有阻礙物的情況，則於此阻礙物的邊界設定紐曼邊界條件（Neumann boundary condition）。值得注意的是，所施加的狄利克雷邊界資訊（在物理學上可視為電壓，參考 [8] 第 12 章）會在背景區域邊界產生相對應的法向導數資訊（在物理學上可視為電流，參考 [8] 第 12 章）。透過結合上述兩大偏微分方程問題，我們進行大量數值實驗，以探討出邊界量測資訊與內部橢圓形阻礙物的位置之間的關聯性。藉由分析這些關聯，我們期望在未來能夠根據新給定的邊界量測資料來推測未知阻礙物在區域中的位置，是為反問題（inverse problem）。

In the light of [12] as well as Masaru Ikehata’s paper [5], the goal of my research is to find out the position of some unknown obstacle embedded in a background region. Particularly, we set up the background region to be an elliptical region containing an elliptical region (obstacle). For the elliptical obstacle, we can adjust its position, size and the angle of rotation. Besides, we also consider the counterpart that has no obstacle in the background region. For the two situations, we define Laplace equation in both domains. For the two problem, we set Dirichlet boundary conditions on the boundary of background region; for the problem associated with the domain containing an obstacle, we specify a Neumann boundary condition on the boundary of the obstacle. It is important that the input Dirichlet data (in physics, voltage, see the chapter 12 of [8]) will generate normal derivative data on the boundary of background (in physics, current density, see the chapter 12 of [8]). Combining the two, we carry out a great amount of numerical experiments to find out some relationship between boundary measurement information and the position of the elliptical obstacle. With the relationship, in the future when we’re given a number of boundary measurement data, we can accordingly deduce the position of the obstacle. This is a kind of inverse problem.

Introduction 1 
1 Our Background and Goal  1 
2 Motivations  4 
2.1 Enclosure Method from [5] and a viewpoint taken reference from chapter 12 of [8] and chapter 7 of [10]  4 
2.2 A difficulty of implementing enclosure method to practical problems  11 
2.3 Inspiration from [12]  12 
2.4 Our Experiments  12 
Our Goal and Plan 13 
Our Technique, Numerical Experiments and Observation 20 
1 Our Techniques — The Idea Behind Our MATLAB Codes and Strategy to Observe Numerical Results  20 
1.1 The settings and the procedures of our codes  21 
1.2 The Scheme of Observing Numerical Experiments  24 
2 Numerical Experiment I : a = 1.5, b = 0.675  46
2.1 First Path  46 
2.2 Second Path  49 
2.3 Third Path  51 
2.4 Fourth Path  52 
2.5 Fifth Path  54 
2.6 Sixth Path  56 
2.7 Seventh Path  57 
2.8 Eighth Path  59 
2.9 Graph  61 
3 Numerical Experiment II : a = 1, b = 0.45  61 
3.1 First Path  61 
3.2 Second Path  63 
3.3 Third Path  65 
3.4 Fourth Path  66 
3.5 Fifth Path  68 
3.6 Sixth Path  70 
3.7 Seventh Path  71 
3.8 Eighth Path  74 
3.9 Graph  76 
4 Numerical Experiment III : a = √1.2, b = √2.7  76 
4.1 First Path  76 
4.2 Second Path  78 
4.3 Third Path  80 
4.4 Fourth Path  81 
4.5 Fifth Path 83 
4.6 Sixth Path  85 
4.7 Seventh Path  86 
4.8 Eighth Path  88 
4.9 Graph  90 
5 Numerical Experiment IV : a = b = √1.8  91 
5.1 First Path  91 
5.2 Second Path  92 
5.3 Third Path  93 
5.4 Fourth Path  94 
5.5 Fifth Path  95 
5.6 Sixth Path  97 
5.7 Seventh Path  98 
5.8 Eighth Path  99 
5.9 Graph  100 
6 Numerical Experiment V : a = b = √1.0125  101 
6.1 First Path  101 
6.2 Second Path  102 
6.3 Third Path  103 
6.4 Fourth Path  104 
6.5 Fifth Path  105 
6.6 Sixth Path  107 
6.7 Seventh Path  108 
6.8 Eighth Path  109 
6.9 Graph  110 
Comparisons, Summary and Conjectures 111 
1 Comparison between Our Numerical Experiments—Similar Ellipses with Different Sizes  111 
2 Comparison between Our Numerical Experiments—Ellipses with SameArea  112 
3 Comparison Between our research with [12]  113 
3.1 Similarities and differences between our research and [12]  113 
3.2 View [12] as an analogue of our research  114 
4 Comparison between our research and partial contents in chapter 12 ∼ 13 of [8]  115 
5 Summaries and Conjectures  116 
5.1 Summaries  116 
5.2 Conjectures  117 
Future Research 119 
Reference 121
                                    

[1] Richard L. Burden, J. Douglas Faires, and Annette M. Burden. Numerical analysis. Cengage Learning Asia Pte Ltd, Singapore, 10th edition, asia edition edition. Section: xvi, 896 pages, 2016.
[2] John B. Garnett and Donald E. Marshall. Harmonic Measure. New Mathematical Monographs. Cambridge University Press, 2005.
[3] P. Grisvard. Elliptic problems in nonsmooth domains. Monographs and studies in mathematics 24. Pitman Advanced Publishing Program, Boston, 1985.
[4] David Halliday, Jearl Walker, and Robert Resnick. Principles of physics. Wiley, Singapore, tenth edition, international student version jearl walker, halliday & resnick edition, 2014 - 2014.
[5] Masaru Ikehata. Reconstruction of the support function for inclusion from boundary measurements. Journal of Inverse and Ill-Posed Problems, 8(4):367–378, 2000.
[6] Tomasz Klimsiak and Tomasz Komorowski. On the maximum principles and the quantitative version of the hopf lemma for uniformly elliptic integro-differential operators. Journal of Differential Equations, 298:346–386, 2021.
[7] William McLean. Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge, 2000.
[8] Jennifer L Mueller and Samuli Siltanen. Linear and nonlinear inverse problems with practical applications. SIAM, 2012.
[9] Kwang Suk Park and SpringerLink (Online service). Humans and electricity : understanding body electricity and applications. Springer International Publishing, Cham, 2023.
[10] Saburou Saitoh, Nakao. Hayashi, and Masahiro Yamamoto. Analytic extension formulas and their applications. International Society for Analysis, Applications, and Computation ; Volume 9. Kluwer Academic Publishers, Dordrecht, 2001.
[11] Walter A. Strauss. Partial Differential Equations: An Introduction. Wiley, Hoboken, New Jersey, 2008.
[12] 蔡佩旻. 從邊界測量去偵測矩形電阻網路中未知孔洞的位置的數值實驗 = numerical experiments on the detection of the unknown cavities in rectangular resistor network from boundary measurement. Master’s thesis, 國立成功大學, 2022.

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