Drossos Gintides

The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation

The recovery of a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, it is shown that v is uniquely determined by the data consisting of such transmission eigenvalues and their ‘multiplicities’, where the ‘multiplicity’ is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to b, the unique recovery is obtained when the data contain one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with ‘multiplicities’ for a related Schrodinger equation.

The linear sampling method for the transmission problem in three-dimensional linear elasticity

In this paper the sampling method for the shape reconstruction of a penetrable scatterer in three-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. We establish the interior transmission problem in the weak sense and consider the case where the nonhomogeneous boundary data are generated by a dyadic source point located in the interior domain. Assuming that the inclusion has absorbing behaviour, we prove the existence and uniqueness of the weak solution of the interior transmission problem. In this framework the main theorem for the shape reconstruction for the transmission case is established. As for the cases of the rigid body and the cavity an approximate far-field equation is derived with the known dyadic Green function term with the source point an interior point of the inclusion. The inversion scheme which is proposed is based on the unboundedness for the solution of an equation of the first kind. More precisely, the support of the body can be found by noting that the solution of the integral equation is not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interior points.

The factorization method in inverse elastic scattering from penetrable bodies

The present work is concerned with the extension of the factorization method to the inverse elastic scattering problem by penetrable isotropic bodies embedded in an isotropic host environment for time-harmonic plane wave incidence. Although the former method has been successfully employed for the shape reconstruction problem in the field of elastodynamic scattering by rigid bodies or cavities, no corresponding results have been recorded, so far, for the very interesting (both from a theoretical and a practical point of view) case of isotropic elastic inclusions. This paper aims at closing this gap by developing the theoretical framework which is necessary for the application of the factorization method to the inverse transmission problem in elastodynamics. As in the previous works referring to the particular reconstruction method, the main outcome is the construction of a binary criterion which determines whether a given point is inside or outside the scattering obstacle by using only the spectral data of the far-field operator.

The far-field equations in linear elasticity : An inversion scheme

In this paper the far-field equations in linear elasticity for the rigid body and the cavity are considered. The direct scattering problem is formulated as a dyadic one. This imbedding of the vector problem for the displacement field into a dyadic field is enforced by the dyadic nature of the free space Greens function. Assuming that the incident field is produced by a superposition of plane dyadic incident waves it is proved 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 in the far-field region. The properties of the Herglotz functions are used to derive solvability conditions and to build approximate far-field equations. Having this theoretical framework, approximate far-field equations are derived for a specific incidence which generates as far-field patterns simple known functions. An inversion scheme is proposed based on the unboundedness for the solutions of these approximate “far-field equations” and the support of the body is found by noting that the solutions of the integral equations are not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interioir points. It is also pointed that it is sufficient to recover the support of the body if only one approximate “far-field equation” is used. The case of the rigid sphere is considered to illuminate the unboundedness property on the boundary.

The linear sampling method for non-absorbing penetrable elastic bodies

In this paper the linear sampling method for the shape reconstruction of a penetrable non-dissipative scatterer in two-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. Assuming that the inclusion has non-absorbing behaviour, we consider this problem as a degenerate case of a non-dissipative anisotropic inclusion. Results for the existence and uniqueness of the weak solution of the interior transmission problem are obtained. In this framework the main theorem for the shape reconstruction for the transmission case is established. As in the previous works referring to the linear sampling method in acoustics and linear elasticity, the inversion scheme which is proposed is based on the unboundedness of the solution of an equation of the first kind having as the known term the far-field of the free-space Green dyadic, generated by a source inside the inclusion approaching the boundary. Numerical results are presented for different inclusion geometries, assuring the simple and efficient implementation of the algorithm, using synthetic data derived from the boundary element method.

Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide

Time-harmonic acoustic waves in an ocean of finite height are modeled by the Helmholtz equation inside a layer with suitable boundary conditions. Scattering in this geometry features phenomena unknown in free space: resonances might occur at special frequencies, and wave fields consist of partly evanescent modes. Inverse scattering in waveguides hence needs to cope with energy loss and limited aperture data due to the planar geometry. In this paper, we analyze direct wave scattering in a three-dimensional planar waveguide and show that resonance frequencies do not exist for a certain class of bounded penetrable scatterers. More important, we propose the factorization method for solving inverse scattering problems in the three-dimensional waveguide. This fast inversion method requires near-field data for special incident fields, and we rigorously show how to generate this data from standard point sources. Finally, we discuss our theoretical results in light of numerical examples.

Computation of light scattering by axisymmetric nonspherical particles and comparison with experimental results

A laboratory prototype of a novel experimental apparatus for the analysis of spherical and axisymmetric nonspherical particles in liquid suspensions has been developed. This apparatus determines shape, volume, and refractive index, and this is the main difference of this apparatus from commercially available particle analyzers. Characterization is based on the scattering of a monochromatic laser beam by particles [which can be inorganic, organic, or biological (such as red blood cells and bacteria)] and on the strong relation between the light-scattering pattern and the morphology and the volume, shape, and refractive index of the particles. To keep things relatively simple, first we focus attention on axisymmetrical particles, in which case hydrodynamic alignment can be used to simplify signal gathering and processing. Fast and reliable characterization is achieved by comparison of certain properly selected characteristics of the scattered-light pattern with the corresponding theoretical values, which are readily derived from theoretical data and are stored in a look-up table. The data in this table were generated with a powerful boundary-element method, which can solve the direct scattering problem for virtually arbitrary shapes. A specially developed fast pattern-recognition technique makes possible the on-line characterization of axisymmetric particles. Successful results with red blood cells and bacteria are presented.

Detection of point-like scatterers using one type of scattered elastic waves

In this paper, we are concerned with the detection of point-like obstacles using elastic waves. We show that one type of waves, either the P or the S scattered waves, is enough for localizing the points. We also show how the use of S incident waves gives better resolution than the P waves. These affirmations are demonstrated by several numerical examples using a MUSIC type algorithm.

A computational method for the inverse transmission eigenvalue problem

In this work, we consider the inverse transmission eigenvalue problem to determine the refractive index from transmission eigenvalues. We adopt a weak formulation of the problem and provide a Galerkin scheme in to compute transmission eigenvalues. Using a proper operator representation of the problem, we show convergence of the method. Next, we define the inverse transmission problem and show that numerically the problem can be considered as a discrete inverse quadratic eigenvalue problem. First, we investigate the case of a spherically symmetric piecewise constant refractive index and confirm our results with analytic computations. Then, we show that a relative small number of eigenvalues are sufficient for simple cases of a few layers by just minimizing the total error between measured and computed eigenvalues to reconstruct the refractive index. Finally, we propose a computational method based on a Newton-type algorithm for reconstructions of a general piecewise constant refractive index for any domain from transmission eigenvalues. The algorithm can be performed without having knowledge of the exact position of the eigenvalues in the spectrum.

On the uniqueness of the inverse elastic scattering problem for periodic structures

In this paper, Schiffers theorem of inverse elastic scattering theory for periodic rigid structures is formulated. The examination of this uniqueness problem requires the study of the spectral properties of an infinite periodic layer, defined by a pair of candidate scattering curves. It is proved that, knowledge of the scattered field on a straight line above the corrugations for P-type one-direction incidence and for an interval of wavenumbers, is sufficient for the unique determination of the scatterers curve. Moreover, Schiffers result holds for a finite set of wavenumbers, if a priori information about the height of the curve is available.