The inverse scattering problem for a partially coated cavity with interior measurements

We consider the interior inverse scattering problem of recovering the shape and the surface impedance of an impenetrable partially coated cavity from a knowledge of measured scatter waves due to point sources located on a closed curve inside the cavity. First, we prove uniqueness of the inverse problem, namely, we show that both the shape of the cavity and the impedance function on the coated part are uniquely determined from exact data. Then, based on the linear sampling method, we propose an inversion scheme for determining both the shape and the boundary impedance. Finally, we present some numerical examples showing the validity of our method.     

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.     

Recovery of interior eigenvalues from reduced near field data

We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The domain can be arbitrarily small but fixed.The scattered waves are measured on the same surface. An effective procedure is suggested for recovery of interior eigenvalues by these data.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

Mathematical basis of the linear sampling method for partially coated obstacles

We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method.     

Reconstruction of a penetrable cavity and the external obstacle

The inverse acoustic scattering of point sources by a penetrable cavity and the external obstacle is considered. Making use of the internal measurements of scattered field on a closed curve inside the cavity, we first derive a factorization method which provides a rigorous characterization of the support of the cavity without knowing the external object. Then under the condition of the cavity is known in advance, we show that the linear sampling method can be applied to recover the outside obstacle with the help of some Green function.     

Huygens’ principle and iterative methods in inverse obstacle scattering

The inverse problem we consider in this paper 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, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.     

New method for proving uniqueness theorems for obstacle inverse scattering problems

A new method is given to prove uniqueness of the solution to basic obstacle inverse scattering problems.     

Uniqueness in inverse obstacle scattering with conductive boundary conditions

We consider the uniqueness of the inverse obstacle scattering with conductive boundary conditions. This work is based on the original idea of Isakov for transmission boundary conditions, which utilize the solvability of the direct problem, orthogonality relations, approximations to solution of the direct problem and singular solutions. The methodology used is constructive and allows an extension to more general conditions and numerical methods.     

An improved time domain linear sampling method for Robin and Neumann obstacles

We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyse this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation.     

On the far-field operator in elastic obstacle scattering

We investigate the far-field operator for the scattering oftime-harmonic elastic plane waves by either a rigid body, acavity, or an absorbing obstacle. Extending results of Colton& Kress for acoustic obstacle scattering, for the spectrumof the far-field operator we show that there exist an infinitenumber of eigenvalues and determine disks in the complex planewhere these eigenvalues lie. In addition, as counterpart ofan identity in acoustic scattering due to Kress & Päivärinta,we will establish a factorization for the difference of thefar-field operators for two different scatterers. Finally, extendinga sampling method for the approximate solution of the acousticinverse obstacle scattering problem suggested by Kirsch to elasticity,this factorization is used for a characterization of a rigidscatterer in terms of the eigenvalues and eigenelements of thefar-field operator.     

电磁散射问题手性障碍边界重构的线性探测法的理论分析

In this paper,we consider the inverse scattering by chiral obstacle inelectromagnetic fields,and prove that the linear sampling method is also effective todetermine the support of a chiral obstacle from the noisy far field data.     

A uniqueness result for the inverse electromagnetic scattering problem in a piecewise homogeneous medium

The scattering of time-harmonic electromagnetic plane waves by an impenetrable obstacle in a piecewise homogeneous medium is considered. The well-posedness of the direct problem is proved by the variational method. Under the condition that the wave numbers in the innermost and outermost homogeneous layers coincide, we then establish a uniqueness result for the inverse problem, that is, the unique determination of the obstacle and its boundary condition from a knowledge of the electric far field pattern for incident plane waves. The proof is based on a generalization of the mixed reciprocity relation.     

局部扰动半平面上的散射问题

该文讨论半平面上有局部扰动情况下的散射问题．通过位势理论,应用边界积分方程的方法研究了该问题解的存在与唯一性．主要方法是运用对称反射,使该无界区域上的散射问题变成一个有界区域上的散射问题,只是这一有界区域的边界不光滑．通过仔细分析相应的边界积分算子,作者得到了其解的存在与唯一性．     

On the solvability of inverse Sturm–Liouville problems with self-adjoint boundary conditions

Theorems on the existence and uniqueness of a solution of the inverse Sturm–Liouville problem with self-adjoint nonseparated boundary conditions are proved. As spectral data two spectra and two eigenvalues are used. The theorems generalize the Levitan–Gasymov solvability theorem and Borg’s uniqueness theorem to the case of general boundary conditions.     

Inverse Sturm–Liouville problems with the potential known on an interior subinterval

The uniqueness problem of inverse Sturm–Liouville problems with the potential known on an interior subinterval is considered. We prove that the potential on the entire interval and boundary conditions are uniquely determined in terms of the known eigenvalues and some information on the eigenfunctions at some interior point (interior spectral data). Moreover, we also concern with the situation where the potential q is C^2k-smoothness at some given points.     

Uniqueness of reconstruction of the Sturm–Liouville problem with spectral polynomials in nonseparated boundary conditions

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.     

Uniqueness For An Inverse Boundary Value Problem For Dirac Operators

The uniqueness of both the inverse boundary value problem and inverse scattering problem for Dirac equation with a magnetic potential and an electrical potential are proved. Also, a relation between the Dirichlet to Dirichlet map for the inverse boundary value problem and the scattering amplitude for the inverse scattering problem is given     

An uniqueness result with some density theorems with interior transmission eigenvalues

Given a set of transmission eigenvalues, we apply Cartwright’s theory to show the density function inversely determines the indicator function. This indicator function gives a Weyl’s type of asymptotics on the transmission eigenvalues. The inverse uniqueness problem on the refraction index is reduced to identifying a parameter of an entire function. We use a Carlson’s type of theorem to prove the uniqueness as in entire function theory. Taking advantage of the uniqueness of rod density problem, we prove an uniqueness result with interior transmission eigenvalues.     

Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions

Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.