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

Yu 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

Fingerprint

Helmholtz equation
Helmholtz equations
Inverse Scattering Problem
inverse scattering
Helmholtz Equation
One Dimension
Scattering
stripping
scattering
Derivatives
Derivative
Trace Formula
Riccati equations
Riccati Equation
Riccati equation
Impedance
Frequency Domain
Directly proportional
Perturbation
Costs

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

On the inverse scattering problem for the Helmholtz equation in one dimension. / Chen, Yu; Rokhlin, V.

In: Inverse Problems, Vol. 8, No. 3, 002, 1992, p. 365-391.

Research output: Contribution to journalArticle

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