In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is recovered from the surface measurements of voltage and current flux. We first propose a new optimization problem based on the recovery of the conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between the predicted current and the measured current on the boundary. And we further prove the existence of the minimizer. Numerically the optimization problem is solved by a third order DG method with quadratic polynomials. Numerical results for several two-dimensional problems with both single and multiple inclusions are demonstrated to show the high {accuracy and efficiency} of the proposed high order DG method. Analysis and computation for discontinuous conductivities are also studied in this work.

翻译：本文提出了一种高效的高阶不连续伽辽金方法，用于求解电阻抗成像问题。电阻抗成像是一个高度非线性的病态逆问题，其目标是通过物体表面的电压与电流通量测量值恢复内部电导率分布。我们首先基于狄利克雷-诺伊曼映射的电导率恢复，建立了一个新的优化问题，以最小化边界上预测电流与实测电流之间的误差，并进一步证明了该优化问题极小值的存在性。数值计算中，采用具有二次多项式的三阶不连续伽辽金方法求解该优化问题。通过对多个含单个及多个夹杂物的二维问题的数值实验，验证了所提出的高阶不连续伽辽金方法的高精度与高效率。此外，本文还针对不连续电导率情形进行了理论分析与数值计算。