### 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/a^{m}, 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(n^{2}), 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 language | English (US) |
---|---|

Article number | 002 |

Pages (from-to) | 365-391 |

Number of pages | 27 |

Journal | Inverse Problems |

Volume | 8 |

Issue number | 3 |

DOIs | |

State | Published - 1992 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Inverse Problems*,

*8*(3), 365-391. [002]. https://doi.org/10.1088/0266-5611/8/3/002

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

Research output: Contribution to journal › Article

*Inverse Problems*, vol. 8, no. 3, 002, pp. 365-391. https://doi.org/10.1088/0266-5611/8/3/002

}

TY - JOUR

T1 - On the inverse scattering problem for the Helmholtz equation in one dimension

AU - Chen, Yu

AU - Rokhlin, V.

PY - 1992

Y1 - 1992

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0000879452&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000879452&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/8/3/002

DO - 10.1088/0266-5611/8/3/002

M3 - Article

AN - SCOPUS:0000879452

VL - 8

SP - 365

EP - 391

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 3

M1 - 002

ER -