The numerical solution of an inverse scattering problem for acoustic waves

In a previous paper, the authors presented a dual space methodfor the numerical solution of the two-dimensional inverse scatteringproblem for acoustic waves in an inhomogeneous medium. Here,by making major modifications to the dual space method, a dramaticimprovement in the numerical performance of this method is achievedfor solving the inverse scattering problem.     

On an inverse problem for acoustic and seismic waves

Two basic inverse problems for determining the constitutive relation in an attenuating medium from given absorption behavior of plane waves in the medium are dealt with in the frequency domain by methods of complex analysis. In particular the case of a prescribed constant internal friction function over the seismic frequencies is considered in detail.     

用正则化方法求解声波散射反问题

研究了从声波散射场的远场模式的信息来再现散射物边界形状的反问题.首先构造表达散射物特征的指示函数,然后利用该函数之特性,建立求解该类反问题的基本方程,从而确定散射物的边界形状.在这个算法中,不需预先知道散射物的边界类型和形状等知识,从T ikhonov正则化方法进行的数值计算结果表明了该方法是有效的和实用的.     

The inverse scattering problem for semi-infinite crystals

The inverse problem for the scattering by semi-infinite crystals, is studied in the one- and three-dimensional cases. The three-dimensional problem is reduced to a system of one-dimensional coupled ones. The inversion procedure is applied to one-dimensional differential equations to obtain the Fourier components of the potential describing the crystal, in terms of the scattering amplitudes and surface states information. The analytic properties of the scattering amplitudes are analyzed by using matching conditions.

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Sommaire Le problème inverse de la diffusion par cristaux semi-infinis est étudié dans les cas à une et trois dimensions. Le problème à trois dimensions est réduit à celui de systèmes couplés à une dimension. Le procédé d'inversion est appliqué aux équations différentialles à une dimension pour obtenir les composés de Fourier du potentiel décrivant le crystal, en fonctión des amplitudes de diffusion et des états de surface donnés. Les propriétés analytiques des amplitudes de diffusión sont analysées en utilisant des conditions de continuité.

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The financial support has been provided by Junta de Energia Nuclear (Madrid).     

The inverse scattering problem for partially penetrable obstacles

We consider the direct and inverse problems for the scattering of a partially penetrable obstacle. Here ‘partially penetrable obstacle’ means that the waves transmit into the obstacle just from partial boundary of the obstacle with the rest of the boundary touching a known perfect and thin scatterer. The solvability of the direct scattering problem is presented using the classical boundary integral equation method. An interesting interior transmission problem is investigated for the purpose of solving the inverse obstacle scattering problem. Then the linear sampling method is proposed to reconstruct the shape and location of the obstacle from near field measurements. We note that the inversion algorithm can be implemented by avoiding the use of background Green function as a test function due to a mixed reciprocal principle.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media

A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.     

The inverse scattering problem of quantum theory

The inverse phase-type scattering problem for the boundary-value problem?y″+q(x)y=k ² y (0?x<∞), (1)y′ (0)=hy (0) (2) is studied. It is shown that, for each function δ(k) satisfying the hypotheses of Levinson's theorem, there exists a problem (1)–(2) with h≠∞ and another problem (1)–(2) with h=∞ (i.e., with the boundary condition o (0)=0). The solvability condition for the Riemann-Hilbert problem is used more directly than has been done heretofore by others in deriving boundary condition (2).     

The inverse electromagnetic scattering problem for a partially coated dielectric

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric infinite cylinder from knowledge of the far field pattern of the scattered TM polarized electromagnetic wave at fixed frequency. A mathematical justification of the method is provided based on the use of a complete family of solutions. Numerical examples are given showing the efficiency of our method.     

The inverse scattering problem for a perturbed difference Hill equation

We consider the inverse scattering problem for the difference analog of a perturbed Hill equation. The perturbation coefficients are recovered from the periodic coefficients and from the scattering data.     

The inverse scattering problem for cavities with impedance boundary condition

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ?D\partial D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ?D\partial D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.     

An inverse medium problem for scanning acoustic microscopy

For nondestructive testing of materials, the scanning acoustic microscope is commonly used for the qualitative visualization of elastic properties. However, especially for medical applications, quantitative evaluation of the measured data is of considerable interest. Based on a mathematical model of the process of scanning acoustic microscopy, the problem of recovering elastomechanical parameters of human bone can be cast into the form of identifying the unknown space-dependent speed of sound in the three-dimensional acoustic wave equation. Specific methods have to be devised for this kind of inverse medium problem due to the reduced set of measured data available. Here, an optimal control approach is used. We discuss uniqueness and stability of the reconstruction and present a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.     

用Backus-Gilbert方法求解声波散射问题

利用位势理论将散射问题的外边界问题转化为第一类边界积分方程求解,再利用Backus-Gilbert方法给出了二维空间的数值结果,与Tikhonov正则化方法比较,虽然精度稍差一些,但是计算方法和计算机实现比较简单.     

三组谱的Sturm-Liouville反问题

考虑与三组谱关联的逆Sturm-Liouville问题,证明了若对于给定的两组数列,在一定条件下,可划分为三组数列,使其分别为区间[0,a]上三个Sturm-Liouville问题的部分特征值,则通过三组谱的部分特征值能唯一确定区间[0,a]上的势函数q(x).     

Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.