On the inverse scattering problem for the Helmholtz equation in one dimension

Y. Chen, V. Rokhlin

Research output: Contribution to journalArticle

Abstract

A scheme is presented for the solution of inverse scattering problems for the one-dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the potential, and can be viewed as a frequency-domain version of the layer-stripping approach. The principal advantage of the procedure is that if the scatterer is be reconstructed has m>or=1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/am, where a is the highest frequency for which scattering data are available. Thus a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost O(n2), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several examples.

Original languageEnglish (US)
Article number002
Pages (from-to)365-391
Number of pages27
JournalInverse Problems
Volume8
Issue number3
DOIs
StatePublished - 1992

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Mathematical Physics
  • Computer Science Applications
  • Applied Mathematics

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