There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Finite element methods for Maxwell's equations.

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.

https://iopscience.iop.org/article/10.1088/0266-5611/14/6/009/pdf

Characterization of the shape of a scattering obstacle using the spectral data of the far field operator

We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.

Finite Element Methods for Maxwell's Equations

We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using different popular models, such as for example Maxwell's equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion approximation. These models are also representative of a broader class of inverse problems. Starting out from the original binary approach of Santosa for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, such as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in two dimensions and in three dimensions for demonstrating the performance of level set techniques in realistic applications.

https://iopscience.iop.org/article/10.1088/0266-5611/22/4/R01/pdf

Level set methods for inverse 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.

https://iopscience.iop.org/article/10.1088/0266-5611/11/2/007/pdf

Frechet derivatives in inverse obstacle scattering

The Mathematical Theory of Time-Harmonic Maxwell's Equations

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.

http://iopscience.iop.org/article/10.1088/0266-5611/14/1/008/pdf

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

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.

Vibration parameter extraction from endoscopic image series of the vocal folds

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.

https://iopscience.iop.org/article/10.1088/0266-5611/12/3/006/pdf

Frank Hettlich

Papers

Frechet derivatives in inverse obstacle scattering

The Mathematical Theory of Time-Harmonic Maxwell's Equations

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

Vibration parameter extraction from endoscopic image series of the vocal folds

Iterative methods for the reconstruction of an inverse potential problem