论文标题
来自当前密度的成像各向异性电导率
Imaging Anisotropic Conductivities from Current Densities
论文作者
论文摘要
在本文中，我们提出和分析了一种重建算法，用于在二阶椭圆PDE中成像各向异性电导率张量，并具有内部电流密度的非零dirichlet边界条件。它基于具有标准$ l^2（ω）^{d，d} $惩罚的正规输出最小二乘公式公式，然后由标准的Galerkin有限元方法离散。我们在$ l^p（ω）^{d，d} $ - 规范中建立了向前图的连续性和可不同性。此外，我们使用H-Convergence的离散对应物提供了离散问题的详细分析，尤其是离散近似相对于网格大小的融合。此外，我们开发了一种预测的牛顿算法，用于解决一阶最优系统。我们提出了广泛的二维数值示例，以显示该方法的效率。
In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(Ω)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(Ω)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.
