A conductivity inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from multistatic measurements. As a modification of GPTs, the Faber polynomial polarization tensors (FPTs) were recently introduced in two dimensions. In this study, we design two novel analytical non-iterative methods for recovering the shape of a simply connected inclusion from GPTs by employing the concept of FPTs. First, we derive an explicit expression for the coefficients of the exterior conformal mapping associated with an inclusion in a simple form in terms of GPTs, which allows us to accurately reconstruct the shape of an inclusion with extreme or near-extreme conductivity. Secondly, we provide an explicit asymptotic formula in terms of GPTs for the shape of an inclusion with arbitrary conductivity by considering the inclusion as a perturbation of its equivalent ellipse. With this formula, one can non-iteratively approximate an inclusion of general shape with arbitrary conductivity, including a straight or asymmetric shape. Numerical experiments demonstrate the validity of the proposed analytical approaches.

中文翻译：

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基于 Faber 多项式的电导夹杂物形状恢复解析

插入均匀背景中的电导率内含物会引起背景电位的扰动。这种扰动允许多极展开，其系数是所谓的广义极化张量 (GPT)。GPT 可以从多基地测量中获得。作为 GPT 的一种修改，最近在二维中引入了 Faber 多项式极化张量 (FPT)。在这项研究中，我们设计了两种新颖的分析非迭代方法，用于通过采用 FPT 的概念从 GPT 恢复简单连接的包含物的形状。首先，我们以 GPT 的形式以简单形式推导出与包裹体相关的外部共形映射系数的显式表达式，这使我们能够准确地重建具有极端或接近极端电导率的包裹体的形状。其次，我们通过将夹杂物视为其等效椭圆的扰动，为具有任意电导率的夹杂物的形状提供了关于 GPT 的明确渐近公式。使用该公式，可以非迭代地近似包含具有任意导电性的一般形状的包含物，包括直的或不对称的形状。数值实验证明了所提出的分析方法的有效性。