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 accurate and efficient of the proposed high order DG method. A problem with discontinuous background is also presented to show the advantage of DG method over other conventional methods.
翻译:在这项工作中,我们开发了一种高效的高顺序不连续的Galerkin(DG)方法,以解决电气阻力成形学(EIT)问题。经济转型期是一个高度非线性不良的反向问题,一个物体的内部导电率是从对电压和当前通量的表面测量中恢复过来的。我们首先提出一个新的优化问题,其依据是从Drichlet-to-Neumann地图中恢复导电率,以尽量减少在边界上预测的电流和测量的电流之间的不匹配。我们进一步证明了最小化器的存在。从数字上看,优化问题通过第三顺序的DG方法用四面形多面体解算法解决。一些单面和多重包容的二维问题,其数值结果显示拟议的高顺序DG方法的高度准确和高效。还提出了不连续背景问题,以显示DG方法比其他常规方法的优势。