Title: On the simultaneous recovery of boundary impedance and internal conductivity Show Chinese title

Title: 关于同时恢复边界阻抗和内部电导率

Abstract: Consider an inverse problem of the simultaneous recovery of boundary impedance and internal conductivity in the electrical impedance tomography (EIT) model using local internal measurement data, which is governed by a boundary value problem for an elliptic equation in divergence form with Robin boundary condition. We firstly express the solution to the forward problem by volume and surface potentials in terms of the Levi function. Then, for the inverse problem, we prove the uniqueness of the solution in an admissible set by unique extension of the solution under some {a-prior} assumption. Finally we establish the regularizing reconstruction schemes for boundary impedance and internal conductivity using noisy measurement data with rigorous error estimates. The mollification method is proposed to recover the boundary impedance from the boundary condition, and the internal conductivity with known boundary value is recovered from an integral system, where the Tikhonov regularization is applied to seek the stable solution, considering that the error involved in the boundary impedance coefficient reconstruction will propagate to the recovering process for internal conductivity. Numerical implementations are presented to illustrate the validity of the proposed method. Show Chinese abstract

Abstract: 考虑使用局部内部测量数据，在电导率断层扫描(EIT)模型中同时恢复边界阻抗和内部电导率的反问题，该模型由具有罗宾边界条件的散度形式椭圆方程的边值问题所控制。我们首先通过体积和表面势来表示正问题的解，这些势是用Levi函数表示的。然后，对于反问题，在某种{先验}假设下，通过解的唯一延拓证明了可允许集中的解的唯一性。最后，我们利用带有严格误差估计的噪声测量数据建立了边界阻抗和内部电导率的正则化重建方案。提出了一种模糊方法来从边界条件中恢复边界阻抗，而已知边界值的内部电导率则通过一个积分系统来恢复，在此过程中应用了Tikhonov正则化以寻找稳定解，考虑到边界阻抗系数重构中的误差会传播到内部电导率的恢复过程中。数值实现被提出以说明所提出方法的有效性。