The investigation of the far field operator and the Factorization Method in inverse scattering theory leads naturally to the study of corresponding interior transmission eigenvalue problems. In contrast to the classical Dirichlet- or Neumann eigenvalue problem for -Delta in bounded domains these interior transmiision eigenvalue problem fail to be selfadjoint. In general, existence of eigenvalues is an open problem. In this paper we prove existence of eigenvalues for the scalar Helmholtz equation (isotropic and anisotropic cases) and Maxwell's equations under the condition that the contrast of the scattering medium is large enough.

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L 2 affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L 2 does not suffer from our L 2 setting, compared to schemes in higher order Sobolev spaces.

In this paper the factorization method from inverse scattering theory and impedance tomography is extended to a class of general elliptic differential equations in divergence form. The inverse problem is to determine the interface ∂Ω of an interior change of the material parameters from the Neumann‐Dirichlet map. Since absorption is allowed a suitable combination of the real and imaginary part of the Neumann‐Dirichlet map is needed to explicitely characterize Ω by the data. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider a simple (but fully three-dimensional) mathematical model for the electromagnetic exploration of buried, perfect electrically conducting objects within the soil underground. Moving an electric device parallel to the ground at constant height in order to generate a magnetic field, we measure the induced magnetic field within the device, and factor the underlying mathematics into a product of three operations which correspond to the primary excitation, some kind of reflection on the surface of the buried object(s) and the corresponding secondary excitation, respectively. Using this factorization we are able to give a justification of the so-called sampling method from inverse scattering theory for this particular set-up.

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.

Wenn die Dichteverteilung in einem Körper bekannt ist, lässt sich ausrechnen, wie stark ein Röntgenstrahl abgeschwächt wird, wenn er den Körper in einer bestimmten Richtung durchläuft. Das umgekehrte Problem lautet: Kann man die Dichteverteilung im Körper bestimmen, wenn man weiß, wie stark Röntgenstrahlen in jeder vorgegebenen Richtung abgeschwächt werden? Auf der Lösung dieser Umkehraufgabe beruht ein wichtiges medizinisches Diagnoseverfahren, die so genannte Computertomografie (CT). Paare von zueinander inversen Problemen kommen häufig vor. Aus einem Modell für einen Ursache-Wirkungszusammenhang kann entweder aus bekannten Ursachen die Wirkung berechnet werden, oder aber umgekehrt, aus der (beobachteten) Wirkung lässt sich auf die Ursache(n) zurückschließen. Mathematisch geht es darum, Gleichungen zu lösen. Für die Praxis ist Gleichung aber nicht gleich Gleichung! Gleichungen hängen (fast immer) von Daten (Parametern) ab. Diese kennt man in der Regel meist nur ungenau, weil sie durch Messung bestimmt werden müssen und daher fehlerhaft sind. Das ist nicht weiter schlimm, falls sich die Lösung nur wenig ändert, wenn sich die Daten wenig ändern. Was aber, wenn eine geringfügige Veränderung der Daten eine massive Veränderung der Lösung bewirkt? Solche Probleme nennt man schlecht gestellt, englisch ill-posed. Die Pointe ist, dass man schlecht gestellte Probleme unter gewissen Bedingungen doch (einigermaßen) zufriedenstellend lösen kann. Das ist allerdings nicht offensichtlich. Aber es ist gut so: andernfalls würden verschiedene so genannte nicht-invasive medizinische Diagnoseverfahren nicht funktionieren. Warum soll man sich als Mathematiklehrerin, als Mathematiklehrer für schlecht gestellte inverse Probleme interessieren?

Two types of rod antennas of mobile phones are optimized so that the radiated energy absorbed by the head or body of the user is reduced and the radiation intensity to other areas especially to the receiver is increased. The mathematical modelling of this problem leads to an infinite dimensional bicriterial optimization problem. It is shown that this optimization problem and a discretized version of this problem are solvable. The relationship between the infinite and finite dimensional optimization problem is investigated. Numerical results are presented for mobile phones working with the GSM standards 900 and 1800.

The factorization method can be applied for certain classes of inverse problems, where the shape of a domain has to be determined. It constructs a binary criterion which determines whether a given point is inside or outside the domain. The present paper develops the theory of the factorization method for the time harmonic Maxwell system where the support of the contrast of the index of refraction has to be determined from the knowledge of the far-field patterns of the scattered fields for plane incident waves. The cases of an absorbing medium and a non-absorbing medium (where in the latter case the dielectricity varies smoothly) are considered.

First, we will briefly recall the physical models for the (linearised) acoustic and electromagnetic wave propagation and how they reduce to boundary value problems for the Helmholtz equation. For more details we refer to, e.g. [17, 18, 25, 50].

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.

We propose a Nyström/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces.Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel.The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f , and superalgebraic convergence is established in the case when f is infinitely smooth.Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case).The Nyström method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

The paper consists of two parts. In the first part we investigate a Nyström- or product integration method for second kind singular integral equations. We prove an asymptotically optimal error estimate in the scale of Sobolev Hilbert spaces. Although the result can also be obtained as a special case of a discrete iterated collocation method our proof is more direct and uses the Nyström interpolation. In the second part of this paper we consider the Dirichlet problem for thin elastic plates with transverse shear deformation. The boundary value problem is transformed into a 3 × 3 system of singular Fredholm integral equations of second kind. After discussing existence and uniqueness of the solution to the integral equations in a Sobolev space setting, we apply the Nyström method to solve the integral equations numerically. Copyright © 1999 John Wiley & Sons, Ltd.

This paper is concerned with the scattering of time harmonic plane waves by an inhomogeneous medium. We prove a factorization of the far-field operator F and characterize the range of (F*F)(1/4) by the range of an operator G which maps more general incident fields than plane waves into the far-field pattern. As an application we characterize the support of the index of refraction using only the spectral data of the far-field operator F.

In our paper [SIAM J. Appl. Math., 55 (1995), pp. 1324-1344] Theorem 4.2 b) is incorrect. This note shows how to avoid the use of Theorem 4.2 b) in the remainder of the paper. We emphasize that the main results of the paper (Theorems 4.6 and 5.3) are correct but that the proofs must be modified as shown in this note.

This paper is concerned with the inverse obstacle scattering problem for time harmonic plane waves. We derive a factorization of the far field operator F in the form and prove that the ranges of and G coincide. Then we give an explicit characterization of the scattering obstacle which uses only the spectral data of the far field operator F. This result is used to prove a convergence result for a recent numerical method proposed by Colton, Kirsch, Monk, Piana and Potthast. We illustrate this method by some numerical examples.

In this paper we consider a special optimization problem with two objectives which arises in antenna theory. It is shown that this abstract bicriterial optimization problem has at least one solution. Discretized versions of this problem are also discussed, and the relationships between these finite dimensional problems and the infinite dimensional problem are investigated. Moreover, we present numerical results for special parameters using a multiobjective optimization method.

This paper is devoted to the inverse scattering problem to recover a periodic structure by scattered waves measured above the structure. It is shown that a finite number of incident plane waves is sufficient to identify the structure. Additionally by a monotonicity principle for the eigenvalues of the Laplacian some upper bounds of the required number of wavenumbers are presented if a priori information on the height of the structure is available.