论文标题
局部恢复EIT的分段常数各向异性电导率在裸露的角落的域上
Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners
论文作者
论文摘要
我们研究了某些有界的Lipschitz域上的EIT（电阻抗）中未知分段常数各向异性电导率的局部回收率，$ \ Mathbb {r}^2 $带有角落。该测量是在包含角落的$ω$的边界$ \ part的连接子集上进行的，并作为局部neumann to-dirichlet地图给出。以上未知的电导率是通过$ω$分解为多边形细胞来定义的。具体而言，我们考虑基于平行四边形的分解和基于梯形的分解。我们假设分解是已知的，但是每个细胞上的电导率尚不清楚。我们证明，在已知的分段常数各向异性电导率$γ_0$附近，局部恢复几乎肯定是正确的。我们这样做是通过证明frender $ f'（γ_0）$ forward Map $ f $（例如$γ_0$）的fréchet导数$ f'（γ_0）$的注射率几乎肯定是正确的。在这里提供的证明涉及以适当的方式定义不同类别的分解，以及以$ \ \ mbox {Supp} H $的单元或双重角度的扰动或对比度$ h $，以便在单元内找到$γ_0$的单个或双角，用于以前的分解和前一个decomptosition和strate decomptosition和strate decomptosition和strate ancompositionally上。然后，通过调整此类角附近的常规证明，我们建立了上述注射率。
We study the local recovery of an unknown piecewise constant anisotropic conductivity in EIT (electric impedance tomography) on certain bounded Lipschitz domains $Ω$ in $\mathbb{R}^2$ with corners. The measurement is conducted on a connected open subset of the boundary $\partialΩ$ of $Ω$ containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of $Ω$ into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity $γ_0$. We do so by proving that the injectivity of the Fréchet derivative $F'(γ_0)$ of the forward map $F$, say, at $γ_0$ is almost surely true. The proof presented, here, involves defining different classes of decompositions for $γ_0$ and a perturbation or contrast $H$ in a proper way so that we can find in the interior of a cell for $γ_0$ exposed single or double corners of a cell of $\mbox{supp}H$ for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.
