Regularized recursive Newton-type methods for inverse scattering problems using multifrequency measurements

Mourad Sini, Nguyen Trung Thành

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of scattered waves associated with incident plane waves sent from one incident direction but at multiple frequencies. We define, at each frequency, observable shapes as the ones which are described by finitely many modes and produce far field patterns close to the measured one. Our analysis consists of two steps. In the first step, we propose a regularized recursive Newton method for the reconstruction of an observable shape at the highest frequency knowing an estimate of an observable shape at the lowest frequency. We formulate conditions under which an error estimate in terms of the frequency step, the number of Newton iterations, and noise level can be proved. In the second step, we design a multilevel Newton method which has the same accuracy as the one described in the first step but with weaker assumptions on the quality of the estimate of the observable shape at the lowest frequency and a small frequency step (or a large number of Newton iterations). The performances of the proposed algorithms are illustrated with numerical results using simulated data.

Original languageEnglish (US)
Pages (from-to)459-480
Number of pages22
JournalESAIM: Mathematical Modelling and Numerical Analysis
Volume49
Issue number2
DOIs
StatePublished - Jan 1 2015

Fingerprint

Newton-type Methods
Recursive Method
Inverse Scattering Problem
Newton-Raphson method
Scattering
Newton Iteration
Newton Methods
Low Frequency
Acoustic waves
Far-field Pattern
Multilevel Methods
Far Field
Plane Wave
Estimate
Error Estimates
Numerical Results

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Modeling and Simulation
  • Computational Mathematics
  • Applied Mathematics

Cite this

@article{053d8c1041e64cbeabb1a2e079fcfdc9,
title = "Regularized recursive Newton-type methods for inverse scattering problems using multifrequency measurements",
abstract = "We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of scattered waves associated with incident plane waves sent from one incident direction but at multiple frequencies. We define, at each frequency, observable shapes as the ones which are described by finitely many modes and produce far field patterns close to the measured one. Our analysis consists of two steps. In the first step, we propose a regularized recursive Newton method for the reconstruction of an observable shape at the highest frequency knowing an estimate of an observable shape at the lowest frequency. We formulate conditions under which an error estimate in terms of the frequency step, the number of Newton iterations, and noise level can be proved. In the second step, we design a multilevel Newton method which has the same accuracy as the one described in the first step but with weaker assumptions on the quality of the estimate of the observable shape at the lowest frequency and a small frequency step (or a large number of Newton iterations). The performances of the proposed algorithms are illustrated with numerical results using simulated data.",
author = "Mourad Sini and Th{\`a}nh, {Nguyen Trung}",
year = "2015",
month = "1",
day = "1",
doi = "10.1051/m2an/2014040",
language = "English (US)",
volume = "49",
pages = "459--480",
journal = "ESAIM: Mathematical Modelling and Numerical Analysis",
issn = "0764-583X",
publisher = "EDP Sciences",
number = "2",

}

TY - JOUR

T1 - Regularized recursive Newton-type methods for inverse scattering problems using multifrequency measurements

AU - Sini, Mourad

AU - Thành, Nguyen Trung

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of scattered waves associated with incident plane waves sent from one incident direction but at multiple frequencies. We define, at each frequency, observable shapes as the ones which are described by finitely many modes and produce far field patterns close to the measured one. Our analysis consists of two steps. In the first step, we propose a regularized recursive Newton method for the reconstruction of an observable shape at the highest frequency knowing an estimate of an observable shape at the lowest frequency. We formulate conditions under which an error estimate in terms of the frequency step, the number of Newton iterations, and noise level can be proved. In the second step, we design a multilevel Newton method which has the same accuracy as the one described in the first step but with weaker assumptions on the quality of the estimate of the observable shape at the lowest frequency and a small frequency step (or a large number of Newton iterations). The performances of the proposed algorithms are illustrated with numerical results using simulated data.

AB - We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of scattered waves associated with incident plane waves sent from one incident direction but at multiple frequencies. We define, at each frequency, observable shapes as the ones which are described by finitely many modes and produce far field patterns close to the measured one. Our analysis consists of two steps. In the first step, we propose a regularized recursive Newton method for the reconstruction of an observable shape at the highest frequency knowing an estimate of an observable shape at the lowest frequency. We formulate conditions under which an error estimate in terms of the frequency step, the number of Newton iterations, and noise level can be proved. In the second step, we design a multilevel Newton method which has the same accuracy as the one described in the first step but with weaker assumptions on the quality of the estimate of the observable shape at the lowest frequency and a small frequency step (or a large number of Newton iterations). The performances of the proposed algorithms are illustrated with numerical results using simulated data.

UR - http://www.scopus.com/inward/record.url?scp=84924353422&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924353422&partnerID=8YFLogxK

U2 - 10.1051/m2an/2014040

DO - 10.1051/m2an/2014040

M3 - Article

AN - SCOPUS:84924353422

VL - 49

SP - 459

EP - 480

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 2

ER -