基于递归线性化的未知阻抗边界条件下多频逆障碍物散射
Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization
摘要：
我们考虑重建的形状和阻抗函数的障碍物从测量的散射场在接收器外的对象。数据被假定为由平面波从多个方向和以多个频率撞击在障碍物上产生。这个逆问题被重新表述为寻找带限形状和阻抗函数的最优化问题，其最小化接收器处的散射场的计算值与数据之间的$L^2$距离。该最优化问题是非线性、非凸和不适定的。此外，评估目标函数在计算上是昂贵的。递归线性化方法（RLA）提出的陈已经成功地解决了这些问题的背景下，恢复域的声速或形状的声音软障碍。我们提出了一个扩展的RLA的形状和阻抗功能的恢复。RLA是一种频率上的延拓方法，其中求解一系列单频逆问题。在每一个更高的频率，尝试使用假设为足够小的步长来恢复递增的更高分辨率的特征，以确保在前一频率处获得的初始猜测位于新频率处的牛顿法吸引区域中。我们证明了几个数值例子的方法的有效性。令人惊讶的是，我们发现，即使测量是从声音硬或声音软的对象，也可以恢复形状的高精度。虽然该方法是有效的，在获得高质量的重建复杂的几何形状和阻抗函数，一些有趣的悬而未决的问题。我们提出的数值实验，建议潜在的机制的成功和失败，显示领域的改进，可以帮助导致强大的和自动化的工具。
英文摘要：
We consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at receivers outside the object. The data is assumed to be generated by plane waves impinging on the obstacle from multiple directions and at multiple frequencies. This inverse problem is reformulated as the optimization problem of finding band-limited shape and impedance functions which minimize the $L^2$ distance between the computed value of the scattered field at the receivers and the data. The optimization problem is non-linear, non-convex, and ill-posed. Moreover, the objective function is computationally expensive to evaluate. The recursive linearization approach (RLA) proposed by Chen has been successful in addressing these issues in the context of recovering the sound speed of a domain or the shape of a sound-soft obstacle. We present an extension of the RLA for the recovery of both the shape and impedance functions. The RLA is a continuation method in frequency where a sequence of single frequency inverse problems is solved. At each higher frequency, one attempts to recover incrementally higher resolution features using a step assumed to be small enough to ensure that the initial guess obtained at the preceding frequency lies in the basin of attraction for Newton's method at the new frequency. We demonstrate the effectiveness of the method with several numerical examples. Surprisingly, we find that one can recover the shape with high accuracy even when the measurements are from sound-hard or sound-soft objects. While the method is effective in obtaining high quality reconstructions for complicated geometries and impedance functions, a number of interesting open questions remain. We present numerical experiments that suggest underlying mechanisms of success and failure, showing areas where improvements could help lead to robust and automatic tools.
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