论文标题
范围和无响应测试之间的二元性及其在反问题上的应用
Duality between range and no-response tests and its application for inverse problems
论文作者
论文摘要
In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in $Ω\setminus\overline D$ with an obstacle $D\SubsetΩ$ whose boundary $\partial D$ is visible from the boundary $\partialΩ$ of $Ω$ and a measurement is given as a set of Cauchy data on $ \partialΩ$。在这里，cauchy数据由$ω\ setminus \ setminus \ edminus \叠加d $的唯一解决方案$ u $带有均质和不均匀的dirichlet边界条件，分别是$ \ partial d $和$ \ \partialΩ$。这些测试方法是使用测试域和相关指标函数估算障碍物位置的域采样方法。而且，这两种测试方法都可以测试$ u $的分析扩展到测试域的外部。由于这些方法是通过一些彼此双重的运算符定义的，因此我们可以期望两种方法之间存在双重性。我们将根据与指标函数相关的预调格函数的等效性来给出这种双重性。作为双重性的应用，使用RT的$ d $重建可以使用NRT重建$ D $，反之亦然。如果Cauchy数据的Dirichlet数据上的$ \partialΩ$的Dirichlet数据并不相同，并且对关联的前向问题的解决方案没有任何分析扩展，则我们还将在不使用双重性的情况下提供这些重建。此外，我们将证明，如果$ d $是凸多边形，这些方法仍然可以给出$ d $的重建，并且满足以下两个属性之一：其所有角度角度都是不合理的，并且其直径小于$ \ \partialΩ$。
In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in $Ω\setminus\overline D$ with an obstacle $D\SubsetΩ$ whose boundary $\partial D$ is visible from the boundary $\partialΩ$ of $Ω$ and a measurement is given as a set of Cauchy data on $\partialΩ$. Here the Cauchy data is given by a unique solution $u$ of the boundary value problem for the Laplace equation in $Ω\setminus\overline D$ with homogeneous and inhomogeneous Dirichlet boundary condition on $\partial D$ and $\partialΩ$, respectively. These testing methods are domain sampling method to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of $u$ to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of $D$ using the RT gives the reconstruction of $D$ using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on $\partialΩ$ is not identically zero and the solution to the associated forward problem does not have any analytic extension across $\partial D$. Moreover, we will show that these methods can still give the reconstruction of $D$ if $D$ is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to $\partialΩ$.
