论文标题
在三维calderón问题II中的天生近似II：径向情况下的数值重建
The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case
论文作者
论文摘要
在这项工作中，我们通过数值实验说明了三维calderón反电导率问题中天生近似的许多特性。结果基于作者最近引入的天生近似值的显式表示公式。我们专注于球中的径向电导率的特定情况，$ b_r \ subset \ mathbb {r}^3 $ radius $ r $，其中calderón问题的线性化等同于Hausdorff Moment问题。我们给出数值证据，即出生的近似值是针对$ l^{\ infty} $电导率很好的，并且我们提出了一种新颖的数值算法，以在适当的小假设下从天生的近似值中重建径向电导率。我们还表明，天生的近似具有深度依赖性的唯一性和近似功能，具体取决于到边界$ \ partial b_r $的距离（深度）。然后，我们研究增加半径$ r $如何影响天生近似的质量，并且存在散射极限为$ r \ to \ infty $。 Schrödinger运算符$-δ+Q $的反边界问题中也说明了类似的属性，在这种情况下，观察到了奇异结果的强烈回收。
In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $B_R \subset \mathbb{R}^3 $ of radius $R$, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $L^{\infty}$ conductivities, and we present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $\partial B_R$. We then investigate how increasing the radius $R$ affects the quality of the Born approximation, and the existence of a scattering limit as $R\to \infty$. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $-Δ+q$, and strong recovery of singularity results are observed in this case.
