Working within the class of piecewise constant conductivities, the inverse problem of electrical impedance tomography can be recast as a shape optimization problem where the discontinuity interface is the unknown. Using Groger's $W^{1}_p$-estimates for mixed boundary value problems, the averaged adjoint method is extended to the case of Banach spaces, which allows to compute the derivative of shape functionals involving point evaluations. We compute the corresponding distributed expression of the shape derivative and show that it may contain Dirac measures in addition to the usual domain integrals. We use this distributed shape derivative to devise a numerical algorithm, show various numerical results supporting the method, and based on these results we discuss the influence of the point measurements patterns on the quality of the reconstructions.

中文翻译：

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具有点测量的电阻抗断层扫描的形状优化方法

在分段恒定电导率类别中工作，电阻抗断层扫描的逆问题可以重新定义为形状优化问题，其中不连续界面是未知的。使用 Groger 的 $W^{1}_p$ 估计混合边界值问题，平均伴随方法扩展到 Banach 空间的情况，它允许计算涉及点评估的形状泛函的导数。我们计算了形状导数的相应分布表达式，并表明除了通常的域积分之外，它可能还包含狄拉克测度。我们使用这种分布式形状导数来设计数值算法，显示支持该方法的各种数值结果，