Uniqueness and numerical methods in inverse obstacle scattering
Uniqueness and numerical methods in inverse obstacle scattering
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DOI：
10.1088/1742-6596/73/1/012003
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发表时间：
2007-06
期刊：
影响因子：
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通讯作者：
R. Kress
中科院分区：
文献类型：
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作者：
R. Kress
在本教程中，我们考虑的逆问题是从时谐平面波散射的远场图案的知识来确定障碍物的形状。在第一部分中，我们将集中讨论独特性问题，即，我们将研究在什么条件下可以从入射平面波的远场图案的知识识别障碍物及其边界条件。我们将回顾一些经典的和最近的结果，并提请注意开放的问题。在第二部分中，我们将综述求解障碍物散射反问题的数值方法。粗略地说，这些方法可以分为三类。迭代方法将逆障碍物散射问题解释为非线性不适定算子方程，并应用诸如正则化牛顿法、Landweber迭代或共轭梯度法的迭代方案来求解。分解方法原则上将逆散射问题分离成不适定的线性问题，以从其远场重建散射波，并随后从边界条件确定散射体的边界。最后，第三类包括最近开发的抽样方法。这些都是基于数值评估的标准的指标功能，决定一个点是否位于内部或外部的散射。本教程将通过描述每个组的一个或两个代表来进行调查，包括对各种优点和缺点的讨论。
The inverse problem we consider in this tutorial is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the first part we will concentrate on the issue of uniqueness, i.e., we will investigate under what conditions an obstacle and its boundary condition can be identified from a knowledge of its far field pattern for incident plane waves. We will review some classical and some recent results and draw attention to open problems. In the second part we will survey on numerical methods for solving inverse obstacle scattering problems. Roughly speaking, these methods can be classified into three groups. Iterative methods interpret the inverse obstacle scattering problem as a nonlinear ill-posed operator equation and apply iterative schemes such as regularized Newton methods, Landweber iterations or conjugate gradient methods for its solution. Decomposition methods, in principle, separate the inverse scattering problem into an ill-posed linear problem to reconstruct the scattered wave from its far field and the subsequent determination of the boundary of the scatterer from the boundary condition. Finally, the third group consists of the more recently developed sampling methods. These are based on the numerical evaluation of criteria in terms of indicator functions that decide whether a point lies inside or outside the scatterer. The tutorial will give a survey by describing one or two representatives of each group including a discussion on the various advantages and disadvantages.