### Abstract

We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill-posed and nonlinear. It is common practice to overcome the ill-posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton’s method. When multiple frequency data is available, we supplement Newton’s method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.

Original language | English (US) |
---|---|

Pages (from-to) | 280-298 |

Number of pages | 19 |

Journal | SIAM Journal on Imaging Sciences |

Volume | 8 |

Issue number | 1 |

DOIs | |

State | Published - Jan 27 2015 |

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### Keywords

- Acoustics
- Inverse scattering
- Newton’s method
- Recursive linearization
- Regularization

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)

### Cite this

**Inverse obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence.** / Borges, Carlos; Greengard, Leslie.

Research output: Contribution to journal › Article

*SIAM Journal on Imaging Sciences*, vol. 8, no. 1, pp. 280-298. https://doi.org/10.1137/140982787

}

TY - JOUR

T1 - Inverse obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence

AU - Borges, Carlos

AU - Greengard, Leslie

PY - 2015/1/27

Y1 - 2015/1/27

N2 - We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill-posed and nonlinear. It is common practice to overcome the ill-posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton’s method. When multiple frequency data is available, we supplement Newton’s method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.

AB - We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill-posed and nonlinear. It is common practice to overcome the ill-posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton’s method. When multiple frequency data is available, we supplement Newton’s method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.

KW - Acoustics

KW - Inverse scattering

KW - Newton’s method

KW - Recursive linearization

KW - Regularization

UR - http://www.scopus.com/inward/record.url?scp=84926374677&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84926374677&partnerID=8YFLogxK

U2 - 10.1137/140982787

DO - 10.1137/140982787

M3 - Article

AN - SCOPUS:84926374677

VL - 8

SP - 280

EP - 298

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

SN - 1936-4954

IS - 1

ER -