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Dive into the research topics where Natalia Grinberg is active.

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Featured researches published by Natalia Grinberg.


Archive | 2007

The factorization method for inverse problems

Andreas Kirsch; Natalia Grinberg

Preface 1. The Simplest Cases: Dirichlet and Neumann Boundary Conditions 2. The Factorization Method for Other Types of Inverse Obstacle Scattering Problems 3. The Mixed Boundary Value Problem 4. The MUSIC Algorithm and Scattering by an Inhomogenous Medium 5. The Factorization method for Maxwells Equations 6. The Factorization Method in Impedance Tomography 7. Alternative Sampling and Probe Methods Bibliography


Journal of Inverse and Ill-posed Problems | 2002

The linear sampling method in inverse obstacle scattering for impedance boundary conditions

Natalia Grinberg; Andreas Kirsch

Abstract - In this paper we study the inverse scattering problem to determine the shape of a scatterer from either far field data for plane wave incidence or near field data for point sources as incident fields. As the simplest case of an absorbing medium we consider an impedance boundary condition with complex valued impedance λ on the boundary of the obstacle. We extend a new approach which characterizes the domain by those points z ∈ℝ3 for which a certain function attains zero as its minimal value. This function is given as the cost functional of an optimization problem and depends explicitly on the data and the points z. An valuable feature of this approach is that it does not assume any a priori knowledge on the number of components of the obstacle or even the type of boundary condition. Some examples show the usefulness of this approach also from the numerical point of view.


Computing | 2005

A Complete Factorization Method for Scattering by Periodic Surfaces

Tilo Arens; Natalia Grinberg

The Factorization Method, a well established method in inverse scattering problems for bounded obstacles, is extended to the case of scattering by a periodic surface. The method is rigorously proved to provide accurate reconstructions for the cases of the total field satisfying a Dirichlet or an impedance boundary condition on the scattering surface. A number of computational examples are given with an emphasis on exploring the number of evanescent modes for which data has to be reliably measured to obtain satisfactory reconstructions.


Mathematics and Computers in Simulation | 2004

The factorization method for obstacles with a-priori separated sound-soft and sound-hard parts

Natalia Grinberg; Andreas Kirsch

We consider the direct and inverse scattering problem for obstacles with mixed Dirichlet and Robin boundary conditions. We derive an explicit and fast obstacle visualization by the factorization method in the case when sound-soft and sound-hard scatterers are a-priori geometrically separated.


Inverse Problems | 1991

Inverse scattering problem for an elastic layered medium

Natalia Grinberg

A new method of reconstructing the density and Lame-coefficients of an elastic horizontal-homogeneous medium is presented. It is based on a solution of the one-dimensional inverse problem for the wave equation. A connection between ruptures of elastic characteristics of a medium and the scattering data is obtained.


European Journal of Applied Mathematics | 2000

Local uniqueness for the inverse boundary problem for the two-dimensional diffusion equation

Natalia Grinberg

We study an inverse boundary problem for the diffusion equation in ℝ 2 . Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div( D grad u ) - ( c μ a + i ω 0 ) u = 0; ω 0 ≠ 0, where D and μ a are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and μ a inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency ω 0 , determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.


Inverse Problems | 2001

Obstacle localization in an homogeneous half-space

Natalia Grinberg

A method to reconstruct an obstacle embedded in a homogeneous half-space x3?0 from measurements of the scattered waves above the obstacle is proposed. The wave satisfies the Dirichlet or Neumann boundary condition on the plane x3 = 0 and the Dirichlet or Robin boundary condition on the reflecting surface. As scattering data we take the far-field operator or, alternatively, the near-field data from sources and observation points that cover the same plane x3 = h away from the obstacle. For an aribitrary point z we propose a minimization algorithm involving the far- or near-field operator to determine whether the point is inside or outside the obstacle.


Journal of Mathematical Physics | 1995

Scattering on small three‐dimensional, nonspherically symmetric potentials

Natalia Grinberg

Three‐dimensional anisotropic inhomogeneous scatterer is considered, which is modeled by the Schrodinger operator with a pseudodifferential potential. Definition of the scattering amplitude is given based on the scattering of the plane wave. The amplitude, as well as the potential, depends on five real parameters. The correspondence between the scatterer (potential) and the scattering amplitude is discussed. It is stated, in particular, that any small nontrivial potential produces a nonzero scattering amplitude.


Nonlinearity | 1993

On an uncertainty principle for a scattering transform

Natalia Grinberg

Nonlinear extension of the Heisenberg uncertainty relation is presented. It is proved that the deviation of the wave velocity c from one and its scattering transform r/k, r is a reflection coefficient, obey the following principle: a product of their dispersions exceeds some positive quantity depending only on Var log c mod - infinity + infinity and sup mod log c mod .


Inverse Problems | 1996

Spectral and scattering properties of three-dimensional anisotropic Schrödinger operators

Natalia Grinberg

A three-dimensional Schrodinger operator with a small, zero order, pseudodifferential potential is considered. Equivalence of H and in is shown. Eigenfunction expansion is given explicitly in terms of incoming and outgoing waves. Existence and completeness of wave operators are proven. Scattering operator and scattering matrix are studied.

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Andreas Kirsch

Karlsruhe Institute of Technology

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Tilo Arens

Karlsruhe Institute of Technology

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