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.

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.

Far field patterns and inverse scattering problems for imperfectly conducting obstacles

1989 ◽  
Vol 106 (3) ◽  
pp. 553-569 ◽  
Author(s):  
T. S. Angell ◽  
David Colton ◽  
Rainer Kress
Keyword(s):  
Boundary Condition ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Impedance Boundary Condition ◽  
Far Field ◽  
Scattering Problems ◽  
Impedance Boundary ◽  
Field Patterns

AbstractWe first examine the class of far field patterns for the scalar Helmholtz equation in ℝ2 corresponding to incident time harmonic plane waves subject to an impedance boundary condition where the impedance is piecewise constant with respect to the incident direction and continuous with respect to x ε ∂ D where ∂ D is the scattering obstacle. We then examine the class of far field patterns for Maxwell's equations in subject to an impedance boundary condition with constant impedance. The results obtained are used to derive optimization algorithms for solving the inverse scattering problem.

The inverse elastic scattering by an impenetrable obstacle and a crack

2020 ◽  
Author(s):  
Jianli Xiang ◽  
Guozheng Yan
Keyword(s):  
Mixed Type ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Operator ◽  
Integral Equation Method ◽  
Factorization Method ◽  
Boundary Integral ◽  
Far Field ◽  
Direct Scattering ◽  
Time Harmonic

Abstract This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator $F$ is related to the boundary integral matrix operator $A$, which is obtained in the study of the direct scattering problem. So, in the first part, we show the well posedness of the direct scattering problem by the boundary integral equation method. Some numerical examples are presented at the end of the paper to demonstrate the feasibility and effectiveness of the inverse algorithm.

Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Leonid L. Frumin
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Numerical Algorithms ◽  
Scattering Problems ◽  
Block Matrices ◽  
Nonlinear Schrödinger ◽  
Direct Scattering ◽  
Vector Case ◽  
Manakov Model ◽  
Vector Variant

AbstractWe introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

Dielectric Metrology VIA Microwave Tomography: Present and Future

1994 ◽  
Vol 347 ◽  
Author(s):  
J.Ch. Bolomey ◽  
N. Joachimowicz
Keyword(s):  
Inverse Scattering ◽  
A Priori ◽  
Scattering Problem ◽  
Measurement Techniques ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Dielectric Characterization ◽  
Sample Shape ◽  
Three Dimensional Configuration ◽  
Characterization Of Materials

ABSTRACTUntil now, the measurement techniques used for the dielectric characterization of materials require severe limitations in terms of sample shape, size and homogeneity. This paper considers the dielectric permittivity measurement as a non-linear inverse scattering problem. Such an approach allows to identify the quantities to be measured and suggests possible experimental arrangements. The problem is shown to be significantly simplified if the shape of the material is known and if some a priori knowledge of the averaged value of the permittivity in the material under test is available. Two test cases have been selected to illustrate the state of the art in solving such inverse problems. The first one consists of a two-dimensional configuration which is applicable to cylindrical objects, and the second one to a vector three-dimensional configuration applicable, for instance, to cubic samples. The main limitations of such an inverse scattering approach are discussed and expected improvements in the near future are analysed.

scholarly journals Fast Imaging of Thin, Curve-Like Electromagnetic Inhomogeneities without a Priori Information

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 799 ◽  
Author(s):  
Won-Kwang Park
Keyword(s):  
Imaging Technique ◽  
A Priori ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Mathematical Structure ◽  
Fast Imaging ◽  
A Priori Information ◽  
Expansion Formula ◽  
Priori Information ◽  
Thin Curve

It is well-known that subspace migration is a stable and effective non-iterative imaging technique in inverse scattering problem. However, for a proper application, a priori information of the shape of target must be estimated. Without this consideration, one cannot retrieve good results via subspace migration. In this paper, we identify the mathematical structure of single- and multi-frequency subspace migration without any a priori of unknown targets and explore its certain properties. This is based on the fact that elements of so-called multi-static response (MSR) matrix can be represented as an asymptotic expansion formula. Furthermore, based on the examined structure, we improve subspace migration and consider the multi-frequency subspace migration. Various results of numerical simulation with noisy data support our investigation.

Simultaneous Scatterer Shape Estimation and Partial Aperture Far-Field Pattern Denoising

2012 ◽  
Vol 11 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Yaakov Olshansky ◽  
Eli Turkel
Keyword(s):  
Scattering Problem ◽  
Least Square ◽  
Field Pattern ◽  
Error Estimator ◽  
Far Field ◽  
The Finite Element Method ◽  
Shape Estimation ◽  
Ill Posed ◽  
Filtering Technique ◽  
Far Field Pattern

AbstractWe study the inverse problem of recovering the scatterer shape from the far-field pattern(FFP) in the presence of noise. Furthermore, only a discrete partial aperture is usually known. This problem is ill-posed and is frequently addressed using regularization. Instead, we propose to use a direct approach denoising the FFP using a filtering technique. The effectiveness of the technique is studied on a scatterer with the shape of the ellipse with a tower. The forward scattering problem is solved using the finite element method (FEM). The numerical FFP is additionally corrupted by Gaussian noise. The shape parameters are found based on a least-square error estimator. If ũ∞ is a perturbation of the FFP then we attempt to find Γ, the scatterer shape, which minimizes ∣∣ũ∞ − ũ∞∣∣ using the conjugate gradient method for the denoised FFP

scholarly journals An injective far-field pattern operator and inverse scattering problem in a finite depth ocean

1991 ◽  
Vol 34 (2) ◽  
pp. 295-311 ◽  
Author(s):  
Yongzhi Xu
Keyword(s):  
Inverse Scattering ◽  
Acoustic Waves ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Finite Depth ◽  
Field Pattern ◽  
Far Field ◽  
Optimal Scheme ◽  
Loss Of Information ◽  
Far Field Pattern

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

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.

Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part II: Numerical Evaluations

1983 ◽  
Vol 5 (4) ◽  
pp. 376-392 ◽  
Author(s):  
Michael L. Tracy ◽  
Steven A. Johnson
Keyword(s):  
Inverse Problem ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Incident Radiation ◽  
Multiple Sources ◽  
Reconstruction Quality ◽  
Ill Posed ◽  
Band Limited ◽  
Pseudo Inverse

In part I, we presented a method for solving the inverse scattering problem using multiple sources and detectors. Allowance for multiple angles of incident radiation improves the ill-posed nature of the inverse problem by improving the quality and quantity of information gathered at detector points. This paper describes implementation and numerical evaluation of the method. An 11 by 11 image reconstructed from noisy scattered field data is shown to closely match the original scattering object, and the improvement possible by constraining the reconstruction to be spatially band limited is demonstrated. Furthermore, for a somewhat simpler “pseudo-inverse problem,” we give findings on the effects that detector radius, degree of overdetermination, noise, and object contrast have on reconstruction quality.

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