scholarly journals Two-Step Extended Sampling Method for the Inverse Acoustic Source Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jiyu Sun ◽  
Yuhui Han
Keyword(s):  
Inverse Scattering ◽  
Sampling Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Indicator Function ◽  
Scattering Data ◽  
Acoustic Source ◽  
Scattering Problems ◽  
Linear Sampling Method ◽  
Ill Posed

Recently, a new method, called the extended sampling method (ESM), was proposed for the inverse scattering problems. Similar to the classical linear sampling method (LSM), the ESM is simple to implement and fast. Compared to the LSM which uses full-aperture scattering data, the ESM only uses the scattering data of one incident wave. In this paper, we generalize the ESM for the inverse acoustic source problems. We show that the indicator function of ESM, which is defined using the approximated solutions of some linear ill-posed integral equations, is small when the support of the source is contained in the sampling disc and is large when the source is outside. This behavior is similar to the ESM for the inverse scattering problem. Numerical examples are presented to show the effectiveness of the method.

scholarly journals Target signatures for thin surfaces

2021 ◽  
Author(s):  
Fioralba Cakoni ◽  
Peter B. Monk ◽  
Yangwen Zhang
Keyword(s):  
Sampling Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
Open Surface ◽  
Linear Sampling Method ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Theoretical Results ◽  
Linear Sampling

Abstract We investigate an inverse scattering problem for a thin inhomogeneous scatterer in ${\mathbb R}^m$, $m=2,3$, which we model as a $m-1$ dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results.

Features of solving the direct and inverse scattering problems for two sets of monopole scatterers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Konstantin V. Dmitriev ◽  
Olga D. Rumyantseva
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Coefficient ◽  
Scattering Data ◽  
Scattering Problems ◽  
Scattering Coefficients ◽  
Spectral Spaces ◽  
The One ◽  
Physical Reasons

Abstract Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys 74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another.

scholarly journals On the Mathematical Basis of the Linear Sampling Method

2003 ◽  
Vol 10 (3) ◽  
pp. 411-425
Author(s):  
Fioralba Cakoni ◽  
David Colton
Keyword(s):  
Inverse Scattering ◽  
Electromagnetic Waves ◽  
Sampling Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Linear Integral Equation ◽  
Mathematical Basis ◽  
Linear Sampling Method ◽  
Mathematical Justification ◽  
Linear Sampling

Abstract The linear sampling method is an algorithm for solving the inverse scattering problem for acoustic and electromagnetic waves. The method is based on showing that a linear integral equation of first kind has a solution that becomes unbounded as a parameter 𝑧 approaches the boundary of the scatterer 𝐷 from inside 𝐷. However, except for the case of the transmission problem, the case where z is in the exterior of 𝐷 is unresolved. Since for the inverse scattering problem 𝐷 is unknown, this step is crucial for the mathematical justification of the linear sampling method. In this paper we give a mathematical justification of the linear sampling method for arbitrary 𝑧 by using the theory of integral equations of first kind with singular kernels.

Uniqueness and stability of the recovery of an absorbing obstacle from a knowledge of its scattering resonances

2000 ◽  
Vol 130 (1) ◽  
pp. 63-80
Author(s):  
C. Labreuche
Keyword(s):  
A Priori ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Large Error ◽  
Small Error ◽  
Compact Set ◽  
Far Field ◽  
Scattering Problems ◽  
Resonant Frequencies ◽  
Ill Posed

In a previous paper, I investigated the use (for the inverse scattering problem) of the resonant frequencies and the associated eigen far-fields. I showed that the shape of a sound soft obstacle is uniquely determined by a knowledge of one resonant frequency and one associated eigen far-field. Inverse obstacle scattering problems are ill-posed in the sense that a small error in the measurement may imply a large error in the reconstruction. This is contrary to the idea of continuity. I proved that, by adding some a priori information, the reconstruction becomes continuous. More precisely, continuity holds if we assume that the obstacle lies a fixed and known compact set.The goal of this paper is to extend these results to the case of absorbing obstacles.

Preliminary Results of Integrating Tikhonov’s Regularization in Forward-Backward Time-Stepping Technique for Object Detection

2016 ◽  
Vol 833 ◽  
pp. 170-175 ◽  
Author(s):  
Andrew Sia Chew Chie ◽  
Kismet Anak Hong Ping ◽  
Yong Guang ◽  
Ng Shi Wei ◽  
Nordiana Rajaee
Keyword(s):  
Image Reconstruction ◽  
Inverse Scattering ◽  
Time Domain ◽  
Regularization Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Time Stepping ◽  
Ill Posed ◽  
Tikhonov’S Regularization

The inverse scattering in time domain known as Forward-Backward Time-Stepping (FBTS) technique is applied to determine the sizes, shape and location of the embedded objects. Tikhonov’s regularization method has been proposed in order to improve or solve the ill-posed of FBTS inverse scattering problem. The reconstructed results showed that FBTS technique can detect the presence of embedded objects. The reconstructed results of FBTS technique utilizing with the Tikhonov’s regularization method shown better results than the results only applied FBTS technique. Tikhonov’s regularization combined with FBTS technique to improve the quality of image reconstruction.

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