High resolution inverse scattering in two dimensions using recursive linearization

Carlos Borges, Adrianna Gillman, Leslie Greengard

Research output: Contribution to journalArticle

Abstract

We describe a fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions. Given full aperture far field measurements of the scattered field for multiple angles of incidence, we use Chen’s method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime. Despite the fact that the underlying optimization problem is formally ill-posed and nonconvex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed profile, we ensure that each least squares calculation is well-conditioned so that an iterative solver can be effectively applied. Each matrix-vector product involves the solution of a large number of forward scattering problems, for which we have created a new, spectrally accurate, fast direct solver. For the largest problems considered, involving 19,600 unknowns, approximately 1 million partial differential equations were solved, requiring approximately 2 days to compute using a parallel MATLAB implementation on a multicore workstation.

Original languageEnglish (US)
Pages (from-to)641-664
Number of pages24
JournalSIAM Journal on Imaging Sciences
Volume10
Issue number2
DOIs
StatePublished - 2017

Fingerprint

Inverse Scattering
Scattering Problems
Linearization
Two Dimensions
High Resolution
Acoustic waves
Scattering
Forward scattering
Iterative Solver
Unknown
Forward Problem
Acoustic Scattering
Linear Least Squares
Cross product
Matrix Product
Fully Nonlinear
Least Squares Problem
Far Field
MATLAB
Partial differential equations

Keywords

  • Acoustics
  • Electromagnetics
  • Fast direct solvers
  • Inverse scattering
  • Recursive linearization

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

High resolution inverse scattering in two dimensions using recursive linearization. / Borges, Carlos; Gillman, Adrianna; Greengard, Leslie.

In: SIAM Journal on Imaging Sciences, Vol. 10, No. 2, 2017, p. 641-664.

Research output: Contribution to journalArticle

@article{44790552e1ef44c090229320662b4986,
title = "High resolution inverse scattering in two dimensions using recursive linearization",
abstract = "We describe a fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions. Given full aperture far field measurements of the scattered field for multiple angles of incidence, we use Chen’s method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime. Despite the fact that the underlying optimization problem is formally ill-posed and nonconvex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed profile, we ensure that each least squares calculation is well-conditioned so that an iterative solver can be effectively applied. Each matrix-vector product involves the solution of a large number of forward scattering problems, for which we have created a new, spectrally accurate, fast direct solver. For the largest problems considered, involving 19,600 unknowns, approximately 1 million partial differential equations were solved, requiring approximately 2 days to compute using a parallel MATLAB implementation on a multicore workstation.",
keywords = "Acoustics, Electromagnetics, Fast direct solvers, Inverse scattering, Recursive linearization",
author = "Carlos Borges and Adrianna Gillman and Leslie Greengard",
year = "2017",
doi = "10.1137/16M1093562",
language = "English (US)",
volume = "10",
pages = "641--664",
journal = "SIAM Journal on Imaging Sciences",
issn = "1936-4954",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - High resolution inverse scattering in two dimensions using recursive linearization

AU - Borges, Carlos

AU - Gillman, Adrianna

AU - Greengard, Leslie

PY - 2017

Y1 - 2017

N2 - We describe a fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions. Given full aperture far field measurements of the scattered field for multiple angles of incidence, we use Chen’s method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime. Despite the fact that the underlying optimization problem is formally ill-posed and nonconvex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed profile, we ensure that each least squares calculation is well-conditioned so that an iterative solver can be effectively applied. Each matrix-vector product involves the solution of a large number of forward scattering problems, for which we have created a new, spectrally accurate, fast direct solver. For the largest problems considered, involving 19,600 unknowns, approximately 1 million partial differential equations were solved, requiring approximately 2 days to compute using a parallel MATLAB implementation on a multicore workstation.

AB - We describe a fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions. Given full aperture far field measurements of the scattered field for multiple angles of incidence, we use Chen’s method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime. Despite the fact that the underlying optimization problem is formally ill-posed and nonconvex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed profile, we ensure that each least squares calculation is well-conditioned so that an iterative solver can be effectively applied. Each matrix-vector product involves the solution of a large number of forward scattering problems, for which we have created a new, spectrally accurate, fast direct solver. For the largest problems considered, involving 19,600 unknowns, approximately 1 million partial differential equations were solved, requiring approximately 2 days to compute using a parallel MATLAB implementation on a multicore workstation.

KW - Acoustics

KW - Electromagnetics

KW - Fast direct solvers

KW - Inverse scattering

KW - Recursive linearization

UR - http://www.scopus.com/inward/record.url?scp=85021674780&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021674780&partnerID=8YFLogxK

U2 - 10.1137/16M1093562

DO - 10.1137/16M1093562

M3 - Article

AN - SCOPUS:85021674780

VL - 10

SP - 641

EP - 664

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

SN - 1936-4954

IS - 2

ER -