论文标题
内部数据的逆电导率问题
Inverse conductivity problem with internal data
论文作者
论文摘要
本文涉及从内部数据中构成的域上提出的二阶椭圆方程的标量系数的重建。该理论在多波成像，贪婪的方法中发现了应用于近似参数依赖性椭圆问题的应用，以及具有部分微分方程的图像处理。我们首先表明，平滑系数的逆问题可以作为线性传输方程重写。假设系数是在边界附近已知的，我们使用不连续的Galerkin方法研究了相关运输方程的适当性及其数值分辨率。我们提出了一个正规运输方程，使我们能够根据多项式近似和正则化参数的顺序得出数值方法的严格合并速率。我们最终为倒置提供了数值示例，假设系数的规律性较低并使用合成数据。
This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.
