On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

Consider the determination of Dirichlet-to-Neumann (D-to-N) map from the far-field pattern in inverse scattering problems, which is the key step in some recently developed inversion schemes such as probe method. Essentially, this problem is related to the reconstruction of the scattered wave from its far-field data. We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle. The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map, avoiding the numerical differentiation for the scattered wave. Then combining with the classical potential theory, a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed. Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed.     

二维逆散射问题探测方法的数值实现

探测方法是最近发展起来的逆散射问题的一种重要的求解方法,其主要思想是由散射波测量数据构造一个带有散射体外面参数点的指示函数,当参数点靠近散射体的边界时,指示函数爆破,由此重建散射体的边界．本文对具有Sound-soft边界的二维散射体给出了探测方法的数值实现．在给出标志函数的构造的基础上,进一步提出了利用模拟数据实现探测法的一个改进的逼近方法．为了更清楚地检验所提出的方法的数值结果,我们直接从Ω边界上的 D-to-N映射来研究探测方法的数值解．     

Inverse scattering from an open arc

A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a sound-soft or perfectly conducting arc from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. Fréchet differentiability with respect to the boundary is shown for the far-field operator, which for a fixed incident wave maps the boundary arc onto the far-field pattern of the scattered wave. For the sake of completeness, the first part of the paper gives a short outline on the corresponding direct problem via an integral equation method including the numerical solution.     

Reciprocity relations and completeness of far-field pattern vectors for obstacle scattering of acoustic wave in a stratified medium

Under a generalized Sommerfeld radiation condition, we proved the uniqueness and existence of the direct obstacle scattering problem of time-harmonic acoustic waves in a stratified medium [8]. In this paper, we study the asymptotic behaviour of the scattered waves and prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far-field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse obstacle scattering problems.     

On the far-field operator for electromagnetic scattering in chiral media

We consider time-harmonic electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium where a perfect conductor has been immersed. Assuming that the incident electric field is a superposition of plane incident electric waves, the corresponding scattered field and the far-field pattern are expressed as the superposition of the scattered fields and the far-field patterns respectively. It is also proved that the sets of far-field patterns are complete if and only if there does not exist an eigenfunction to the interior perfect conductor problem that vanishes on the boundary of the scatterer which is an electric Herglotz field. The Left-Circularly Polarized and the Right-Circularly Polarized far-field operators are defined and studied and using them the electric far-field operator is defined too. The properties of the above operators and Herglotz functions are related to the solution of the interior perfect conductor boundary value problem.     

The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.     

一类阻尼边界条件下的逆散射问题

１．引言 考虑声波在均匀非吸收介质中的传播,该声波被一个无限长的柱体所散射．在声波线性化理论中,众所周知,时间调和声波的振幅ｕ（ｘ）满足 Ｈｅｌｍｈｏｌｔｚ方程其中波数ｋ＝＞０时间调和声波的散射可以用方程（１．１）的适当的边界条件（例如Ｄｉｒｃｈｌｅｔ条件或Ｎｅｕｍａｎｎ条件）来描述．特别,ｕ｜Ｄ＝０对应于软边界（ｓｏｕｎｄ－ｓｏｆｔ）散射,而Ｄ＝ ０则是所谓的硬边界（ｓｏｕｎｄ－ｈａｒｄ）散射·这里 γ表示边界 Ｄ的外法向． 除了上述两类条件外,还有一类更能描述现实物理现象的阻尼边界条件,即物理上,…     

A spectrally accurate algorithm for electromagnetic scattering in three dimensions

In this work we develop, implement and analyze a high-order spectrally accurate algorithm for computation of the echo area, and monostatic and bistatic radar cross-section (RCS) of a three dimensional perfectly conducting obstacle through simulation of the time-harmonic electromagnetic waves scattered by the conductor. Our scheme is based on a modified boundary integral formulation (of the Maxwell equations) that is tolerant to basis functions that are not tangential on the conductor surface. We test our algorithm with extensive computational experiments using a variety of three dimensional perfect conductors described in spherical coordinates, including benchmark radar targets such as the metallic NASA almond and ogive. The monostatic RCS measurements for non-convex conductors require hundreds of incident waves (boundary conditions). We demonstrate that the monostatic RCS of small (to medium) sized conductors can be computed using over one thousand incident waves within a few minutes (to a few hours) of CPU time. We compare our results with those obtained using method of moments based industrial standard three dimensional electromagnetic codes CARLOS, CICERO, FE-IE, FERM, and FISC. Finally, we prove the spectrally accurate convergence of our algorithm for computing the surface current, far-field, and RCS values of a class of conductors described globally in spherical coordinates.     

Reconstruction of Scattered Field from Far-Field by Regularization

In this paper, we consider an inverse scattering problem for an obstacle D R^2 with Robin boundary condition. By applying the point source, we give a regularizing method to recover the scattered field from the far-field pattern. Numerical implementations are also presented.     

海洋波导的可穿透障碍物散射场远场分布的若干性质

研究了海洋波导中可穿透目标时谐声波散射传播远场分布的性质,构造了透射问题解的集合,使得所构造解的集合在边界上的限制在某个Hilbert空间中是稠密的,确定了传播远场分布的集合在某个Hilbert空间中是完备的．这些性质对研究海洋波导中的逆透射问题有重要的应用．     

Continuous Frechet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problems

We consider direct acoustic scattering problems with eithera sound-soft or sound-hard obstacle, or lossy boundary conditions,and establish continuous Fréchet differentiability withrespect to the shape of the scatterer of the scattered fieldand its corresponding far-field pattern. Our proof is basedon the implicit function theorem, and assumes that the boundaryof the scatterer as well as the deformation are only Lipschitzcontinuous. From continuous Fréchet differentiability,we deduce a stability estimate governing the variation of thefar-field pattern with respect to the shape of the scatterer.We illustrate this estimate with numerical results obtainedfor a two-dimensional high-frequency acoustic scattering problem.     

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.     

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.     

Uniqueness Theorems for the Inverse Problem of Elastodynamic Boundary Scattering

A uniqueness theorem is established for the elastodynamic problemof direct scattering of harmonic small-amplitude waves by anobstacle embedded in an isotropic medium with spatially varyingparameters. Uniqueness theorems are then developed for the inverseproblem of boundary determination of the obstacle associatedwith this direct scattering problem. In particular, it is shownfor nondegenerate elastodynamic problems that knowledge of boththe lamellar and the solenoidal far-field patterns is necessaryto reconstruct the obstacle's boundary.     

Error analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems

In this paper we present an error analysis for a high-order accurate combined Dirichlet-to-Neumann (DtN) map/finite element (FE) algorithm for solving two-dimensional exterior scattering problems. We advocate the use of an exact DtN (or Steklov–Poincaré) map at an artificial boundary exterior to the scatterer to truncate the unbounded computational region. The advantage of using an exact DtN map is that it provides a transparent condition which does not reflect scattered waves unphysically. Our algorithm allows for the specification of quite general artificial boundaries which are perturbations of a circle. To compute the DtN map on such a geometry we utilize a boundary perturbation method based upon recent theoretical work concerning the analyticity of the DtN map. We also present some preliminary work concerning the preconditioning of the resulting system of linear equations, including numerical experiments.     

Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern

The problem considered is that of determining the shape of aplane acoustically sound-soft obstacle from the knowledge ofthe far-field pattern for one time-harmonic incident field.An iterative procedure is proposed based on two boundary integralsrepresenting the incident field and the far-field pattern, respectively.Numerical examples are included which show that the proceduregives accurate numerical approximations in relatively few iterations.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

A uniqueness theorem in inverse scattering of elastic waves

The scattering of plane elastic waves in an isotropic inhomogeneousmedium is considered. The existence and uniqueness of the directproblem is stated, and a reciprocity principle for the far fieldsof the scattered waves formulated. Finally, it is proved thatthe knowledge of the far-field patterns for a bounded sequenceof different frequencies and certain sets of incoming planewaves uniquely determines the density of the medium.     

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.     

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.