在這篇文章中我們討論一個反問題的數值實驗，我們知道生活上有許多反問題的應用，例如使用地震波測量來了解地表下是否有石油；在無損害人體的情形下，使用 X 射線進行醫學診斷等等。而我們的問題是如何通過邊界測量找到該地區的未知物或孔洞。在連續的反問題中，若想藉由邊界值測量得到內部未知物或孔洞的資訊。這裡有一個方法，我們可以從 Masaru Ikehata 教授發表的論文”The Enclosure Method and its applications” 中得知。在這個方法中，我們需要制定一組特殊的邊界值 (CGO solutions)，並在每個方向做測量，進而找出未知物或孔洞的位置。
理論上，Enclosure Method 看似完美，但在現實生活中，我們要制定出這組特殊邊界卻很困難。所以我們在想是否有比較簡單一點的方法，可以找到內部的未知孔洞? 因此我們想先把問題放在離散的反問題中，也就是在含有未知孔洞的矩形電路上，能否從任一邊界點加壓即得知未知孔洞的資訊或確切位置。此外，矩形電阻網路中可能存在許多種類的未知孔洞。但在本文中，我們只討論單一點破洞的例子，即在矩形電阻網路中挖一個內部點。
在這篇論文中，我們簡介 Masaru Ikehata 教授的論文”The Enclosure Method and its applications” 及針對我們的問題做猜測，並對我們的猜測進行數值實驗。我們想知道能否透過觀察 Λ0 ´ Λ1 這個矩陣即得知未知孔洞的位置?(Λ 表示將邊界電壓轉換為邊界電流的矩陣。其中，Λ0 表示電路尚未含破洞的矩陣；Λ1 表示電路含有破洞的矩陣，詳細定義會在 1.2 小節呈現)。然而實驗結果發現，在 Λ0 ´ Λ1 矩陣中，裡面的最大差值，其所對應的邊界點，雖然不是最靠近未知孔洞，但也沒有偏離太遠。我們會做所有100 ˆ 100 以下的矩形電路實驗，列出一些代表性的結果，並且在文章的最後給出結論。

In this article, we discuss a numerical experiment with an inverse problem. There are many inverse problem applications in life, such as using seismic wave measurements to know if there is oil on the surface, and X-ray scanning of the human body for medical diagnosis without damage, and so on. And our problem is how to find the unknown cavities in the area through boundary measurement. In a continuous inverse problem, if we want to get information about internal unknowns or cavities by boundary measurement. Here's an approach that we can learn from professor Masaru Ikehata’s paper ”The Enclosure Method and its applications”. In this method, we need to formulate a set of special boundary values(CGO solutions) and measure in each direction to find out the position of the unknown or cavities.
Theoretically, the Enclosure Method looks perfect, but in real life, it seems very
difficult for us to work out this special set of boundaries. So we are wondering if there is a simpler way to find an unknown cavity? Therefore, we want to put the problem in the discrete inverse problem first, that is, on a rectangular resistor network with unknown cavities, we ask whether we can know the information or exact position of the unknown cavity by applying a voltage at any boundary point. In addition, there might be many kinds of unknown cavities in a rectangular resistor network. But in this paper, we will only discuss the cavity of the unknown single internal point, which is to dig out an internal point in a rectangular resistor network.
In this paper, we briefly introduce Professor Masaru Ikehata’s paper ”The Enclosure Method and its applications” and make guesses for our problems, and conduct numerical experiments on our guesses. We wonder if we can know the location of the unknown cavity by looking at the matrix Λ0-Λ1(Λ represents the matrix that takes the boundary voltage to the boundary current. Among them, Λ0 represents the matrix without cavities in the circuit; Λ1 represents the matrix that contains cavities in the circuit. The detailed definition will be presented in section 1.2). However, the experimental results show that the boundary node corresponding to the maximum difference in the matrix Λ0-Λ1, although not the closest to the unknown cavity, did not deviate too far. We will do all the rectangular circuit experiments below 100 X 100, list some representative results, and give conclusions at the end of the article.

[1] E.B. Curtis and J.A. Morrow. Inverse Problems for Electrical Networks. Series on applied mathematics. World Scientific, 2000.
[2] Masaru Ikehata. The Enclosure Method and its Applications, pages 87–103. Springer US, Boston, MA, 2001.