Title: Random walks in frequency and the reconstruction of obstacles with cavities from multi-frequency data Show Chinese title

Title: 频率中的随机游走以及从多频数据中重建带有空腔的障碍物

Abstract: Inverse obstacle scattering is the recovery of an obstacle boundary from the scattering data produced by incident waves. This shape recovery can be done by iteratively solving a PDE-constrained optimization problem for the obstacle boundary. While it is well known that this problem is typically non-convex and ill-posed, previous investigations have shown that in many settings these issues can be alleviated by using a continuation-in-frequency method and introducing a regularization that limits the frequency content of the obstacle boundary. It has been recently observed that these techniques can fail for obstacles with pronounced cavities, even in the case of penetrable obstacles where similar optimization and regularization methods work for the equivalent problem of recovering a piecewise constant wave speed. The present work investigates the recovery of obstacle boundaries for impenetrable, sound-soft media with pronounced cavities, given multi-frequency scattering data. Numerical examples demonstrate that the problem is sensitive to the choice of iterative solver used at each frequency and the initial guess at the lowest frequency. We propose a modified continuation-in-frequency method which follows a random walk in frequency, as opposed to the standard monotonically increasing path. This method shows some increased robustness in recovering cavities, but can also fail for more extreme examples. An interesting phenomenon is observed that while the obstacle reconstructions obtained over several random trials can vary significantly near the cavity, the results are consistent for non-cavity parts of the boundary. Show Chinese abstract

Abstract: 逆障碍散射是从入射波产生的散射数据中恢复障碍边界。 这种形状恢复可以通过对障碍边界进行迭代求解PDE约束优化问题来完成。 虽然众所周知，这个问题通常是非凸且不适定的，但以前的研究表明，在许多情况下，通过使用频率延续方法并引入限制障碍边界频率内容的正则化，可以缓解这些问题。 最近观察到，对于具有明显空腔的障碍物，这些技术可能会失效，即使在可穿透障碍物的情况下，类似的优化和正则化方法对于恢复分段常数波速的等效问题是有效的。 本研究调查了在多频散射数据下，不可穿透的、声学软介质中具有明显空腔的障碍物边界的恢复。 数值例子表明，该问题对每个频率使用的迭代求解器的选择以及最低频率的初始猜测非常敏感。 我们提出了一种修改后的频率延续方法，该方法在频率上进行随机游走，而不是标准的单调递增路径。 这种方法在恢复空腔方面表现出一些增强的鲁棒性，但对于更极端的例子也可能失败。 观察到一个有趣的现象，即在多次随机试验中获得的障碍物重建在空腔附近可能有显著差异，但对于非空腔部分的边界结果是一致的。