generalized topological derivative for the navier equation and inverse scattering in the time domain abstract in this work the concept of generalized topological sensitivity developed recently to deal with inverse elastodynamic scattering by penetrable obstacles in the frequency domain is extended to permit non invasive defect reconstruction and material characterization by means of transient elastic waves this quantity which signifies the first order perturbation of a given cost functional due to nucleation of an infinitesimal elastic defect in the reference defect free solid is intended to be used as an obstacle indicator though an assembly of interior sampling points where it attains pronounced negative values from an asymptotic analysis of the transient scattered field emanating from a defect with vanishing size in the reference elastic solid the expression for topological sensitivity explicit in terms of i the green s function and ii the free i e incident field for the reference domain is obtained by employing the so called direct approach and a convolution based boundary integral representation of the featured elastodynamic wavefields for generality the proposed formula is recast in terms of its adjoint field counterpart that caters for general reference domains where the green s function may not be available through numerical simulations it is shown that the generalized topological sensitivity in the time domain provides an effective preliminary tool for the 3d reconstruction and material identification of discrete internal heterogeneities the results also demonstrate that the defect illumination by transient elastic waveforms results in a higher quality of reconstruction relative to its time harmonic monotonal counterpart

inverse scattering from an orthotropic medium abstract we consider the inverse scattering problem of determining the shape of an orthotropic homogeneous medium from a knowledge of the far field pattern corresponding to incident time harmonic plane waves we solve the direct problem by potential theory show the uniqueness of the inverse problem and then use a simple method recently developed by colton and kirsch 1996 and colton and monk 1997 to solve the inverse problem numerical examples are given showing the practicality of our method for solving the inverse problem

properties of the far field operator for a multilayered scatterer in acoustic scattering theory in this paper we consider the inverse scattering problem which is to determine the shape of a multilayered scatterer the adjoint of the far field operator is found in scattering theory the boundary data are given by the traces of plane waves the closure of the span of the images of these plane waves under the far field operator is characterized furthermore taking the affine hull instead of the span of the plane waves on the boundary some results are obtained

linear and nonlinear inverse scattering in this paper we discuss one dimensional scattering and inverse scattering for the helmholtz equation on the half line from the point of view of the layer stripping by full or nonlinear scattering we mean the transformation between the index of refraction actually half of its logarithmic derivative and the reflection coefficient we refer to this mapping as nonlinear scattering because the mapping itself is nonlinear another appropriate name is multiple scattering as this model includes the effects of multiple reflections by linear scattering we mean the born or single scattering approximation this is the frechet derivative of the full scattering transform at the constant index of refraction which can be calculated to be exactly the fourier transform in j sylvester d p winebrenner and f gylys colwell siam j appl math 56 1996 pp 736 754 we introduced a variant of layer stripping based on causality and the riesz transform rather than on trace formulas see a brickstein and

a regularized sampling method for solving three dimensional inverse scattering problems the inverse scattering problem under consideration is to determine the shape of an obstacle in re 3 from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region a method for solving this nonlinear and improperly posed problem is presented which is based on solving a linear integral equation of the first kind and avoids the use of nonlinear optimization methods numerical examples are given showing the practicality of this new approach

a modified gradient method for two dimensional problems in tomography abstract a method for reconstructing the complex index of refraction of a bounded two dimensional inhomogeneous object of known geometric configuration from measured scattered field data is presented this work is an extension of recent results on the direct scattering problem wherein the governing domain integral equation was solved iteratively by a successive over relaxation technique the relaxation parameter was chosen to minimize the residual error at each step convergence of this process was established for indices of refraction much larger than required for convergence of the born approximation for the inverse problem the same technique is applied except in this case both the index of refraction and the field are unknown iterative solutions for both unknowns are postulated with two relaxation parameters at each step they are determined by simultaneously minimizing the residual errors in satisfying the domain integral equation and matching the measured data this procedure retains the nonlinear relation between the two unknowns numerical results are presented for a number of representative two dimensional objects the algorithm is shown to be effective in cases where the iterative solution of the direct problem is rapidly convergent

an adaptive hybrid fem fdm method for an inverse scattering problem in scanning acoustic microscopy scanning acoustic microscopy based on focused ultrasound waves is a promising new tool in medical imaging in this work we apply an adaptive hybrid fem fdm finite element methods finite difference methods method to an inverse scattering problem for the time dependent acoustic wave equation where one seeks to reconstruct an unknown sound velocity c x from a single measurement of wave reflection data on a small part of the boundary e g to detect pathological defects in bone typically this corresponds to identifying an unknown object scatterer in a surrounding homogeneous medium the inverse problem is formulated as an optimal control problem where we use an adjoint method to solve the equations of optimality expressing stationarity of an associated augmented lagrangian by a quasi newton method to treat the problem of multiple minima of the objective function the optimization procedure is first performed on a coarse grid to smooth the high frequency error generating a starting point for optimization steps on successively refined meshes local refinement based on the results of previous steps will improve computational efficiency of the method as the main result then an a posteriori error estimate is proved for the error in the lagrangian and a corresponding adaptive method is formulated where the finite element mesh is refined from residual feedback the performance of the adaptive hybrid method and the usefulness of the a posteriori error estimator for problems with limited boundary data are illustrated in three dimensional numerical examples

why the high frequency inverse scattering by topological sensitivity may work this study deciphers the topological sensitivity ts as a tool for the reconstruction and characterization of impenetrable anomalies in the high frequency regime it is assumed that the anomaly is simply connected and convex and that the measurements of the scattered field are of the far field type in this setting the formula for ts which quantifies the perturbation of a cost functional due to a point like impenetrable scatterer is expressed as a pair of nested surface integrals one taken over the boundary of a hidden obstacle and the other over the measurement surface using multipole expansion the latter integral is reduced to a set of antilinear forms featuring green s function and its gradient the remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via kirchhoff approximation and pursuing an asymptotic expansion of the resulting fourier integral in this way the ts is found to survive upon three asymptotic lynchpins namely i the near boundary approximation for sampling points close to the exposed surface of an obstacle ii uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces lines and points and iii stationary phase approximation within the framework of catastrophe theory it is shown that in the case of the full source aperture the ts is asymptotically dominated by the explicit near boundary term which explains the previously reported reconstruction capabilities of this class of indicator functionals the analysis further shows that when the dirichlet or neumann character of an anomaly is unknown beforehand the latter can be effectively exposed by assuming point like dirichlet perturbation and considering the sign of the leading term inside the reconstruction

uniqueness for the determination of sound soft defects in an inhomogeneous planar medium by acoustic boundary measurements we consider the inverse problem of determining shape and location of sound soft defects inside a known planar inhomogeneous and anisotropic medium through acoustic imaging at low frequency in order to determine the defects we perform acoustic boundary measurements with prescribed boundary conditions of different types we prove that at most two suitably chosen measurements allow us to uniquely determine multiple defects under minimal regularity assumptions on the defects and the medium containing them finally we treat applications of these results to the case of inverse scattering

uniqueness of the inverse scattering problem of a locally perturbed half plane in this paper we consider the scattering problem by a locally perturbed half plane which can be reformulated as an exterior obstacle problem however the new obstacle is a bounded domain with nonsmooth boundary by using the generalization of green s theorem to nonsmooth domain we obtain that the obstacle is uniquely determined by the far field pattern

on the stress wave imaging of cavities in a semi infinite solid the problem of mapping underground cavities from surface seismic measurements is investigated within the framework of a regularized boundary integral equation bie method with the ground modeled as a uniform elastic half space the inverse analysis of elastic waves scattered by a three dimensional void is formulated as a task of minimizing the misfit between experimental observations and theoretical predictions for an assumed void geometry for an accurate treatment of the gradient search technique employed to solve the inverse problem sensitivities of the predictive bie model with respect to cavity parameters are evaluated semi analytically using an adjoint problem approach and a continuum kinematics description several key features of the formulation including the rigorous treatment of the radiation condition for semi infinite solids modeling of an illuminating seismic wave field and treatment of the prior information are highlighted a set of numerical examples with spherical and ellipsoidal cavity geometries is included to illustrate the performance of the method it is shown that the featured adjoint problem approach reduces the computational requirements by an order of magnitude relative to conventional finite difference estimates thus rendering the three dimensional elastic wave imaging of solids tractable for engineering applications

numerical resolution of an electromagnetic inverse medium problem at fixed frequency the aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency the governing equations are the time harmonic maxwell equations formulated in electric field in a two dimensional bounded domain we express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields our numerical reconstruction algorithm combines the bfgs method and an iterative process called the adaptive eigenspace inversion the unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator both the eigenspace and the mesh are iteratively adapted during the minimization procedure numerical experiments illustrate the performance of the reconstruction for various configurations

norm attaining compact operators abstract we show examples of compact linear operators between banach spaces which cannot be approximated by norm attaining operators this is the negative answer to an open question posed in the 1970s actually any strictly convex banach space failing the approximation property serves as the range space on the other hand there are examples in which the domain space has a schauder basis

morphological representation of discrete and binary images a general theory for the morphological representation of discrete and binary images is presented the basis of this theory relies on the generation of a set of nonoverlapping segments of an image via repeated erosions and set transformations which in turn produces a decomposition that guarantees exact reconstruction the relationship between the proposed representation and some existing shape analysis tools e g discrete size transform pattern spectrum skeletons is investigated thus introducing the representation as the basis of a unified theory for geometrical image analysis particular cases of the general representation scheme are shown to yield a number of useful image decompositions which are directly related to various forms of morphological skeletons the relationship between the representation and the various forms of morphological skeletons is studied as a result of this study a unified theory is developed for the mathematical description of the morphological skeleton decomposition of discrete and binary images

focusing and imaging using eigenfunctions of the scattering operator an inverse scattering method that uses eigenfunctions of the scattering operator is presented this approach provides a unified framework that encompasses eigenfunction methods of focusing and quantitative image reconstruction in arbitrary media scattered acoustic fields are described using a compact normal operator the eigenfunctions of this operator are shown to correspond to the far field patterns of source distributions that are directly proportional to the position dependent contrast of a scattering object conversely the eigenfunctions of the scattering operator specify incident wave patterns that focus on these effective source distributions these focusing properties are employed in a new inverse scattering method that represents unknown scattering media using products of numerically calculated fields of eigenfunctions a regularized solution to the nonlinear inverse scattering problem is shown to result from combinations of these products so that the products comprise a natural basis for efficient and accurate reconstructions of unknown inhomogeneities the corresponding linearized problem is solved analytically resulting in a simple formula for the low pass filtered scattering potential the linear formula is analytically equivalent to known filtered backpropagation formulas for born inversion and at least in the case of small scattering objects has advantages of computational simplicity and efficiency a similarly efficient and simple formula is derived for the nonlinear problem in which the total acoustic pressure can be determined based on an estimate of the medium computational results illustrate focusing of eigenfunctions on discrete and distributed scattering media quantitative imaging of inhomogeneous media using products of retransmitted eigenfunctions inverse scattering in an inhomogeneous background medium and reconstructions for data corrupted by noise

two numerical methods for recovering small inclusions from the scattering amplitude at a fixed frequency in this paper two noniterative algorithms for locating small electromagnetic inclusions from the scattering amplitude at a fixed frequency are developed in particular a variety of numerical results is presented to highlight their potential and their limitations

time reversal with partial information for wave refocusing and scatterer identification time reversal is a well known procedure in application fields involving wave propagation among other uses it can be applied as a computational tool for solving certain inverse problems the procedure is based on advancing the solution of the relevant wave problem backward in time one important use of numerical time reversal is that of refocusing where a reverse run is performed to recover the location of a source applied at an initial time based on measurements at a later time usually only partial noisy information is available at certain measurement locations on the field values that serve as data for the reverse run in this paper the question concerning the amount and characterization of the available data needed for a successful refocusing is studied for the scalar wave equation in particular a simple procedure is proposed which exploits multiple measurement times and is shown to be very beneficial for refocusing a tradeoff between availability of spatial and temporal information is discussed the effect of measurement noise is studied and the technique is shown to be quite robust sometimes even in the presence of very high noise levels the use of the technique as a basis for scatterer identification is also discussed a numerical study of these effects is presented employing finite elements in space and a standard explicit marching scheme in time in contrast to some previous studies the propagation medium is taken to be homogeneous

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

stochastic inversion in acoustic scattering a novel approach is presented for the probabilistic characterization of the solution in inverse acoustic scattering problems the approach consists of representing the unknown parameters of the scatterer as stochastic processes with spatially random fluctuations the stochastic processes are represented by their polynomial chaos decomposition the deterministic coefficients in this parametrization are then computed by solving a deterministic optimization algorithm with an appropriate objective function that captures the probabilistic significance of these coefficients the general procedure is demonstrated by its application to a two dimensional problem with the hidden object a hard obstacle being immersed in a heterogeneous medium the formulation provides as well a measure of validity of a particular predictive model at matching experimental evidence

a survey of finite element methods for time harmonic acoustics many of the current issues and methodologies related to finite element methods for time harmonic acoustics are reviewed the effective treatment of unbounded domains is a major challenge most prominent among the approaches that have been developed for this purpose are absorbing boundary conditions infinite elements and absorbing layers standard computational methods are unable to cope with wave phenomena at short wave lengths due to resolutions required to control dispersion and pollution errors leading to prohibitive computational demands since computation naturally separates the scales of a problem according to the mesh size multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them other issues addressed are related to the efficient solution of systems of specialized algebraic equations and inverse problems of acoustics the tremendous progress that has been made in all of the above areas in recent years will surely continue leading to many more exciting developments

properties of the far field operator for time harmonic acoustic waves abstract in this paper the eigenvalues of the far field operator according to the inverse scattering problem for time harmonic acoustic waves in the nonhomogeneous medium are characterized with an inequality in addition to we prove the far field operator is one to one and its range is dense in this paper we consider 0 where c 1 ir 3 is the density of the absorbing medium

a high order algorithm for obstacle scattering in three dimensions in this work we describe implement and analyse in detail a high order fully discrete spectral algorithm for solving the helmholtz equation exterior to a bounded sound soft sound hard or absorbing obstacle in three space dimensions with dirichlet neumann or robin impedance boundary conditions our algorithm may be thought of as a discrete glerkin method but it is also equivalent to a nystrom method after a simple transformation we test our algorithm with extensive computational experiments on a variety of three dimensional smooth and non smooth obstacles with conical singularities our tests include the computation of scattered and far fields induced by incident plane waves our method is shown to be very accurate for scattering from surfaces which are globally parameterised by spherical coordinates and tests show that it performs very much better than several of the well established fast algorithms for obstacle sattering on a range of such surfaces even some which are non smooth further we prove superalgebraic convergence of the scattered and far fields obtained using our algorithm in the case of smooth scatterers

a new non iterative singular sources method for the reconstruction of piecewise constant media we describe a novel non iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering which we call the singular sources method the basic idea of the method is to use the behaviour of the scattered field for singular incident fields multipoles to calculate the size of the refractive index n at some point z0 on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium the paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium also we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions using techniques that differ from those used to prove previous well known results

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

quality mesh generation in higher dimensions we consider the problem of triangulating a d dimensional region our mesh generation algorithm called qmg is a quadtree based algorithm that can triangulate any polyhedral region including nonconvex regions with holes furthermore our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles finally our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by qmg is bounded above by a factor times the number produced by any competing algorithm where the factor depends on the aspect ratio bound satisfied by the competing algorithm the qmg algorithm has been implemented in c and is used as a mesh generator for the finite element method

on the far field in obstacle scattering we will establish far field identities for the scattering of time harmonic acoustic waves from sound soft and sound hard obstacles and will illustrate two applications of these identities for the inverse obstacle scattering problem first we will provide an alternate proof for the frechet differentiability of the far field pattern with respect to the boundary second we will connect the identities with a new method for the approximate solution of the inverse obstacle scattering problem recently suggested by potthast

stability estimates and reconstructions in inverse acoustic scattering using singular sources abstract the problem of stability for the reconstruction of the scattered field us in the exterior of a scatterer d from far field data u and for the reconstruction of a sound soft or sound hard scatterer d from the far field pattern u x d x d for all incident plane waves is investigated in particular we show how stability estimates are linked to the behaviour of the scattered field s for incident point sources and to special minimum norm solutions for the herglotz wave operator the results can also be used to formulate a method of singular sources for the reconstruction of d which is new in inverse scattering the reconstruction method is described and numerical examples are provided

an improved imaging method for extended targets when we use direct imaging methods for solving inverse scattering problems we observe artificial lines which make it hard to determine the shape of targets we propose a signal space test to study the cause of the artificial lines and we use multiple frequency data to reduce the effect from them finally we use the active contour without edges acwe method to further improve the imaging results the final reconstruction is accurate and robust the computational cost is lower than iterative imaging methods which need to solve the forward problem in each iteration

a periodic faddeev type solution operator abstract we construct periodic solution operators for the equation u 2i u fin a bounded domain with the help of fourier series we prove that thel2 norms of these operators converge to zero if the parameter im goes to infinity then we apply these operators to show that functionsu c20 rd satisfying an inequality u x m u x inrdmust vanish everywhere we extend this result to other second order elliptic differential operators with constant coefficients replacing the laplacian finally we use the solution operators to derive that the span of products of solutions to differential equations is dense inl1

the reconstruction of shape and impedance exploiting space mapping with inverse difference method the reconstruction of shape and impedance obtained from measurements of scattered electromagnetic fields is considered for the two dimensional case space mapping with inverse difference technique which is newly developed to solve inverse problem is used for this purpose one of the advantages of this technique over other space mapping techniques is that it does not need parameter extraction during optimization process as the contour of the cylinder is periodic it can be represented by fourier series measurements of scattered field and fourier coefficients of the contour are used as inputs and outputs respectively the results are compared with exact shapes and good agreement is obtained it is also observed that reconstructions obtained by space mapping based technique are better than those obtained by artificial neural network for exact and noisy data sharp cornered tetragonal contour is also investigated for both noisy and noiseless case in order to show efficiency of the technique

on the uniqueness of the shape of a penetrable anisotropic obstacle we consider the scattering of time harmonic waves in an isotropic homogeneous medium which contains a penetrable anisotropic inhomogeneous obstacle after reviewing existence uniqueness and regularity of a variational solution we prove that a knowledge of the scattered near fields for all incident plane waves at a fixed wave number uniquely determines the obstacle provided the matrix characterizing the anisotropic medium satisfies a suitable condition to this end we examine the corresponding interior transmission problem and prove that it is a compact perturbation of an interior transmission problem which is well posed

the linear sampling method for anisotropic media we consider the inverse scattering problem of determining the support of an anisotropic inhomogeneous medium from a knowledge of the incident and scattered time harmonic acoustic wave at fixed frequency to this end we extend the linear sampling method from the isotropic case to the case of anisotropic medium in the case when the coefficients are real we also show that the set of transmission eigenvalues forms a discrete set

locating an obstacle in a 3d finite depth ocean using the convex scattering support we consider an inverse scattering problem in a 3d homogeneous shallow ocean specifically we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle this reconstruction is obtained from the far field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency the technique is based on an implementation of the theory of the convex scattering support s kusiak j sylvester the scattering support commun pure appl math 2003 1525 1548 a few examples are presented to show the feasibility of the method

eigenvalues of the far field operator for time harmonic acoustic waves by an inhomogeneous medium and inverse scattering theory in this paper the far field pattern for the acoustic scattering waves in the nonhomogeneous medium is studied both the eigenvalue for the far field operator and the identity operator for the scattering operator are investigated also this paper the inverse scattering problem is studied all this researches density of the absorbing medium email protected c 1 r 3 is assumed that email protected 0

the linear sampling method for solving the electromagnetic inverse scattering problem we consider the inverse scattering problem of determining the shape of an obstacle in mathbb r from knowledge of the time harmonic incident electromagnetic wave and the far field pattern of the scattered wave with frequency in the resonance region the approach used is the linear sampling method which does not require a priori knowledge of either the boundary condition or the connectivity of the scattering obstacle numerical examples are given for the case of both a simply connected perfect conductor and multiply connected obstacles satisfying an impedance boundary condition on the boundary of one component and satisfying a perfect conducting boundary condition on the boundary of another component

elastic scatterer reconstruction via the adjoint sampling method an inverse problem dealing with the reconstruction of voids in a uniform semi infinite solid from near field elastodynamic waveforms is investigated via the linear sampling method to cater to active imaging applications that are characterized by a limited density of illuminating sources existing formulation of the linear sampling method is advanced in terms of its adjoint statement that features integration over the receiver surface rather than its source counterpart to deal with an ill posedness of the integral equation that is used to reconstruct the obstacle the problem is solved by alternative means of tikhonov regularization and a preconditioned conjugate gradient method through a set of numerical examples it is shown i that the adjoint statement elevates the performance of the linear sampling method when dealing with scarce illuminating sources and ii that a combined use of the existing formulation together with its adjoint counterpart represents an effective tool for exposing an undersam

morphological shape description using geometric spectrum on multidimensional binary images abstract a useful morphological shape description tool is presented called geometric spectrum or g spectrum for quantifying the geometric features on multidimensional binary images the basis of this tool relies upon the cardinality of a set of non overlapping segments in an image using morphological operations the g spectrum preserves the translation invariance property with a chosen set of isotropic structuring elements the g spectrum also preserves the rotation invariance after the procedure of normalization the g spectrum can also preserve the scaling invariance the properties and proofs of the g spectrum are discussed

eigenvalues of the far field operator for the helmholtz equation in an absorbing medium it is shown that there exist an infinite number of eigenvalues of the far field operator corresponding to the scattering of a time harmonic plane wave by either an imperfectly conducting obstacle or an absorbing inhomogeneous medium regions in the complex plane are found where these eigenvalues must lie and in the case of obstacle scattering numerical examples are given showing how precise these regions are

transmission eigenvalues and a problem of hans lewy abstract subject to certain assumptions on the matrix n n x we show that except for a discrete set of values of k called transmission eigenvalues the only solution of n u k 2 u 0 v k 2 v 0 in d u v u v on d where d r n is a domain with smooth boundary d having unit outward normal is the trivial solution u v 0 this problem has important applications to the inverse scattering problem for anisotropic media and is in a class of problems first considered by lewy bull amer math soc 65 1959 37 58

the scattering support we discuss inverse problems for the helmholtz equation at fixed energy specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle we introduce the convex scattering support of a far field a set which will be a subset of the convex hull of the support of any source which can produce it we give several theorems which explain how to compute the convex scattering support and how to relate it to the actual support of a source medium or obstacle 2003 wiley periodicals inc

scattering of time harmonic electromagnetic waves by anisotropic inhomogeneous scatterers or impenetrable obstacles we investigate an overlapping solution technique to compute the scattering of time harmonic electromagnetic waves in two dimensions the technique can be used to compute waves scattered by penetrable anisotropic inhomogeneous scatterers or impenetrable obstacles the major focus is on implementing the method using finite elements we prove existence of a unique solution to the discretized problem and derive an optimal convergence rate for the scheme which is verified by numerical examples

inverse scattering by a multilayered obstacle in this paper we consider the inverse scattering problem of a plane acoustic wave by a multilayered obstacle which is important in various areas of imaging and nondestructive testing when the core is penetrable with transmission boundary conditions we obtain that the core is determined uniquely by the corresponding far field pattern

three dimensional inverse obstacle scattering for time harmonic acoustic waves a numerical method a numerical method for a three dimensional inverse acoustic scattering problem is considered from the knowledge of several far fields patterns of the helmholtz equation a closed surface partial d representing the boundary of an unknown obstacle d is reconstructed the obstacle d is supposed to be acoustically soft or acoustically hard or characterized by a given acoustic impedance

numerical solution of an inverse medium scattering problem for maxwell s equations at fixed frequency consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in r 3 in this paper well posedness of the variational problem for the direct scattering is examined an energy estimate for the scattered field is obtained on which the born approximation is based a regularized recursive linearization method for the inverse medium scattering which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field is developed the algorithm requires only single frequency data using an initial guess from the born approximation each update is obtained via continuation on the spatial frequency of a two parameter family of plane waves by solving one direct problem and one adjoint problem of the maxwell equation

microwave biomedical data inversion using the finite difference contrast source inversion method we present a contrast source inversion csi technique which is based on a finite difference fd solver for use in microwave biomedical imaging the algorithm is capable of inverting complex permittivity biomedical data sets without the explicit use of a forward solver at each iteration the fd solver is based in the frequency domain utilizes perfectly matched layer pml boundary conditions and the stiffness matrix is solved via an lu decomposition and gaussian elimination an important feature of the fd csi algorithm is that the stiffness matrix associated with the fd solver depends only upon the background medium and frequency and thus the lu decomposition is only performed once before the iterative inversion process unlike the usual integral equation ie based inversion techniques the fd csi algorithm is readily capable of utilizing an arbitrary backarbitrary backgroundground medium for the inversion process

identification of small inhomogeneities asymptotic factorization we consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions we give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential the location of the inhomogeneities and their geometry as the size of the inhomogeneities tends to zero such asymptotic expansions have already been used to design direct i e noniterative reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary e g music type methods our derivation of the asymptotic formulas is based on integral equation methods it demonstrates the strong relation between factorization methods and music type methods for the solution of this inverse problem