We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, and currents are injected through the electrodes and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single layer potentials; the problem is thus re-cast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography (EIT) which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.

翻译：我们发展了一种基于边界积分方程的数值方法，用于求解二维空间中具有分段常数电导率介质内的静电位，其边界条件由完整电极模型（CEM）给定。CEM被视为对物理场景的最精确建模：电极置于导电体表面，电流通过电极注入，并在同一电极上测量产生的电压。该积分方程公式基于将静电位表示为有限个拉普拉斯方程的解，这些方程通过边界匹配条件耦合。由此，我们可将解重新表示为单层势函数形式；从而将问题转化为定义在若干光滑曲线上的积分方程组。我们提出了一种自适应方法求解该弱奇异积分方程组，该求解器兼具快速性与高精度。进而我们给出了电阻抗成像（EIT）的数值反演求解器，其核心采用所发展的正问题求解器。为验证结果适用性，我们在芬兰逆问题协会提供的公开电阻抗成像数据集上进行了数值测试。