Frank Hettlich

Frechet derivatives in inverse obstacle scattering

We are concerned with inverse obstacle scattering problems. For a fixed incident wave we consider the operator mapping an obstacle onto the far-field pattern of the scattered wave. The existence and characterizations of the Frechet derivatives for the exterior Robin problem and the transmission problem are proved.

The determination of a discontinuity in a conductivity from a single boundary measurement

We consider the determination of the interior domain where D is characterized by a different conductivity from the surrounding medium. This amounts to solving the inverse problem of recovering the piecewise constant conductivity in from boundary data consisting of Cauchy data on the boundary of the exterior domain . We will compute the derivative of the map from the domain D to this data and use this to obtain both qualitative and quantitative measures of the solution of the inverse problem.

Vibration parameter extraction from endoscopic image series of the vocal folds

An approach is given to extract parameters affecting phonation based upon digital high-speed recordings of vocal fold vibrations and a biomechanical model. The main parameters which affect oscillation are vibrating masses, vocal fold tension, and subglottal air pressure. By combining digital high-speed observations with the two-mass-model by Ishizaka and Flanagan (1972) as modified by Steinecke and Herzel (1995), an inversion procedure has been developed which allows the identification and quantization of laryngeal asymmetries. The problem is regarded as an optimization procedure with a nonconvex objective function. For this kind of problem, the choice of appropriate initial values is important. This optimization procedure is based on spectral features of vocal fold movements. The applicability of the inversion procedure is first demonstrated in simulated vocal fold curves. Then, inversion results are presented for a healthy voice and a hoarse voice as a case of functional dysphonia caused by laryngeal asymmetry.

ITERATIVE METHODS FOR THE RECONSTRUCTION OF AN INVERSE POTENTIAL PROBLEM

This paper considers an inverse potential problem which seeks to recover the shape of an obstacle separating two different densities by measurements of the potential. A representation for the domain derivative of the corresponding operator is established and this allows the investigation of several iterative methods for the solution of this ill-posed problem.

Two methods for solving an inverse conductive scattering problem

The inverse conductive scattering problem we are concerned with is to determine the shape of an obstacle which is covered by an infinitely thin layer or high conductivity by measurements of the far-field patterns of scattered waves. We derive two methods proposed by Kirsch, Kress and Monk (1988) and Colton and Monk (1989) respectively on numerically solving this improperly posed problem. In particular the effects of such boundary conditions are illustrated in some examples.

Identification of a discontinuous source in the heat equation

In this paper the feasibility of identification of a discontinuous source term in a parabolic equation is investigated. It is shown that a minimal set of data according to uniqueness of the inverse problem is given by flux measurements in time at two distinct points on the boundary. Besides the uniqueness result iterative regularization schemes are developed. The methods are based on the domain derivative and a general existence theory and a representation of the domain derivative for parabolic equations is derived.

On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation

In this paper we consider the inverse scattering problem by penetrable obstacles with conductive boundary conditions. We prove that the obstacle is uniquely determined if the far-field patterns of the scattered waves for all incident plane waves with fixed wavenumber are known. Also we establish uniqueness results for the other parameters of the conductive scattering problem.

Schiffer's theorem in inverse scattering theory for periodic structures

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.

Iterative regularization schemes in inverse scattering by periodic structures

The paper is devoted to the reconstruction of periodic structures from measurements of a scattered field. Iterative regularization schemes are established based on derivatives with respect to variations of the boundary. Therefore, representations of the required derivatives are presented which can be applied numerically. Additionally an integral equation method is derived solving the direct scattering problem numerically. Through some numerical examples the performance of such regularization schemes is illustrated.

The resistive and conductive problems for the Exterior Helmholtz Equation

The existence, uniqueness, and continuous dependence of solutions of the Helmholtz equation, which are subject to either resistive or conductive conditions at the boundary of a closed smooth scatterer, are considered. A boundary integral equation of the second kind for each problem whose unique solution is the trace on the boundary of the unique solution of that problem is devised. Continuous dependence results are proven using the integral equations.