论文标题
基于Faber多项式的电导率包容性的分析形状恢复
Analytical shape recovery of a conductivity inclusion based on Faber polynomials
论文作者
论文摘要
插入均匀背景的电导率包含在背景潜力中引起扰动。这种扰动允许其系数是所谓的广义极化量（GPTS）的多极扩展。可以从多阶段测量值获得GPT。作为GPT的修改，最近在二维中引入了Faber多项式极化张量（FPT）。在这项研究中，我们设计了两种新型的分析方法，用于通过采用FPT概念从GPT中恢复简单连接的包含的形状。首先，我们得出了一个明确的表达，以根据GPT的简单形式与包含相关的外部共形映射的系数，这使我们能够准确地重建具有极端或近距离电导率的包含的形状。其次，我们通过将包含作为其等效椭圆形的扰动，以GPT的形式提供了与任意电导率的包含形状的明确渐近公式。使用该公式，可以非静态地近似具有任意电导率的一般形状，包括直形状或不对称形状。数值实验证明了所提出的分析方法的有效性。
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.
