Huygens' principle and iterative methods in inverse obstacle scattering
type: Item
instance of: scholarly article
title: Huygens' principle and iterative methods in inverse obstacle scattering
zbMATH Open document ID: 1205.65304
author: Rainer Kress
author: Pedro Serranho
author: Olha Ivanyshyn Yaman
published in: Advances in Computational Mathematics
publication date: 2010-12-01T00:00:00Z
review text: The inverse problem the authors consider is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, Huygens' principle will be interpreted as a system of two integral equations. The scattering of a time harmonic incident wave \(u^{i}\) by a sound-soft obstacle which is a bounded domain \(D\subset \mathbb{R}^{m}, m=2,3\) with a connected complement is treated. This is modeled by an exterior boundary value problem for the Helmholtz equation \[ \triangle u^{s}+k^{2}u^{s}=0 \quad\text{in} \quad \mathbb{R}^{m}\setminus \overline{D} \qquad k>0, \] with the Dirichlet condition \[ u^{i}+u^{s}=0 \quad \text{on}\quad \partial D \] and the Sommerfeld radiation condition, uniformly in all directions. \[ \frac{\partial u^{s}}{\partial r}-iku^{s}=o\biggr(\frac{1}{r^{\frac{m-1}{2}}}\biggl), \quad r=|x|\rightarrow\infty. \] It is supposed that \(u^{i}=e^{ik x}\cdot d\) where the unit vector \(d\) is the direction of propagation. So the Huygens' principle can be viewed as a system of two integral equations for the unknown boundary \(\partial D\) and the induced surface flux: \[ -\frac{\partial u}{\partial \nu} \quad \text{on} \quad \partial D, \] where \(\nu\) is the interior normal vector. Next sections deal with the iteration scheme, which is of Newton type.
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G15
zbMATH DE Number: 5822004
zbMATH Keywords: inverse scattering
zbMATH Keywords: system of nonlinear integral equations
zbMATH Keywords: sound-soft obstacle
zbMATH Keywords: Newton iteration
zbMATH Keywords: inverse problem
zbMATH Keywords: Huygens' principle
zbMATH Keywords: exterior boundary value problem
zbMATH Keywords: Helmholtz equation
MaRDI profile type: Publication
full work available at URL: s10444-009-9135-6
OpenAlex ID: W2002400157
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DOI: 10.1007/S10444-009-9135-6
description: scientific article; zbMATH DE number 5822004