基于单频数据重构散射障碍的新型线性采样方法

A Novel Linear Sampling Method for Reconstructing Scattering Obstacles from Data at One Frequency

散射及反散射的数学理论与计算一直是应用数学领域中的重要课题,其成果在地质勘探、无损探测、医学成像等领域都具有广泛的应用.线性采样方法(LSM)是近年来反散射理论中一类非常流行的非迭代型重建算法,但是这种方式很难推广到如半空间中障碍反散射等更为复杂的问题中.本文基于单频数据研究Dirichlet障碍反散射问题的数值重建算法.通过构造带有阻尼边界条件的辅助边值问题,提出了一类新型的线性采样方法,并在理论上严格证明了该方法在任意给定的波数下重构障碍形状及位置的有效性.

The theory and calculation of scattering and inverse scattering problems has always been an important subject in the field of applied mathematics,and its results have a wide range of applications in the fields of geological prospecting,non-destructive detection,medical imaging,etc.Linear sampling method(LSM)is a very popular non-iterative reconstruction algorithm in inverse scattering problems in recent years,but this method is difficult to generalize to more complex problems such as obstacle inverse scattering in half space.This paper is associated with the numerical reconstruction algorithm in the inverse Dirichlet obstacle scattering problem from single frequency data.By constructing an auxiliary boundary value problem with an impedance boundary condition,a novel linear sampling method is proposed.Then it is theoretically proved that the method is effective for any given wavenumber in reconstructing the shape and location of the obstacle.