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：我们提出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被视为对物理场景最精确的建模：在导电体表面放置电极，通过电极注入电流并测量相同电极上的电压响应。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。这使得我们能够将解重构为单层势能的形式，从而将问题转化为有限条光滑曲线上的积分方程组。我们讨论了一种自适应方法用于求解该弱奇异积分方程组，该求解器兼具快速性与高精度。随后我们提出以正问题求解器为核心的电阻抗层析成像（EIT）数值反演算法。为验证方法的适用性，我们使用芬兰反问题学会提供的公开电阻抗层析成像数据集对数值方法进行了测试。