In this paper, we address the identification of an unknown inclusion in an elliptic equation, which arises in the context of electrical impedance tomography and multiphase problems. We present a novel approach by formulating the overdetermined system as an optimal design problem based on the unknown inclusion and two state solutions, which is derived from by introducing a least square functional. We establish the existence results and first optimality conditions, and employ the finite element method to discretize the involved variational equations. Furthermore, we update the unknown inclusion using the level-set method. To demonstrate the effectiveness and validity of our proposed algorithm, we investigate simple and complex geometries numerically. Our results showcase the efficiency and accuracy of the method, making it a promising tool for solving similar inverse problems in various engineering applications.

在本文中，我们解决了椭圆方程中未知夹杂物的识别问题，该夹杂物是在电阻抗断层扫描和多相问题的背景下出现的。我们提出了一种新颖的方法，将超定系统表示为基于未知包含和两种状态解的最优设计问题，这是通过引入最小二乘函数得出的。建立了存在性结果和第一最优性条件，并采用有限元方法对涉及的变分方程进行离散化。此外，我们使用水平集方法更新未知包含物。为了证明我们提出的算法的有效性和有效性，我们对简单和复杂的几何形状进行了数值研究。我们的结果展示了该方法的效率和准确性，使其成为解决各种工程应用中类似反问题的有前途的工具。