Random Walks in Frequency and the Reconstruction of Obstacles with Cavities from Multi-frequency Data

Travis Askham, Carlos Borges, Jeremy Hoskins, Manas Rachh

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number15
JournalJournal of Scientific Computing
Issue number1
StatePublished - Jan 2024

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


  • Continuation method
  • Inverse scattering
  • Trapping region


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