This work extends the results of [Garde and Hyv\"onen, Math. Comp. 91:1925-1953] on series reversion for Calder\'on's problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the internal and contact admittivities to the linear electrode current-to-potential map, is first proven to be analytic. A reversion of the corresponding Taylor series yields a family of numerical methods of different orders for solving the inverse problem of electrical impedance tomography, with the possibility to employ different parametrizations for the unknown internal and boundary admittivities. The functionality and convergence of the methods is established only if the employed finite-dimensional parametrization of the unknowns allows the Fr\'echet derivative of the forward map to be injective, but we also heuristically extend the methods to more general settings by resorting to regularization motivated by Bayesian inversion. The performance of this regularized approach is tested via three-dimensional numerical examples based on simulated data. The effect of modeling errors related to electrode shapes and contact admittances is a focal point of the numerical studies.

翻译：本文将[Garde and Hyvönen, Math. Comp. 91:1925-1953]中关于Calderón问题级数反演的结果推广至实际电极测量情形，将待测物体内部导纳率与电极-物体界面接触导纳率均作为未知量处理。首先证明将内部导纳率与接触导纳率映射至线性电极电流-电位算子的正演算子具有解析性。通过相应泰勒级数的反演，可获得求解电阻抗层析成像反问题的一系列不同阶数值方法，且允许对未知内部导纳率与边界导纳率采用不同参数化形式。仅当所用未知量有限维参数化能保证正演映射的Fréchet导数具有单射性时，方能建立方法的可行性及收敛性证明。然而我们通过采用贝叶斯反演启发的正则化策略，启发式地将方法拓展至更一般的情形。基于模拟数据的三维数值算例测试了该正则化方法的性能，其中电极形状与接触导纳率相关的建模误差效应是数值研究的重点。