Stefan Ritter

A linear sampling method for inverse scattering from an open arc

In this paper, we develop a linear sampling method for the inverse scattering of time-harmonic plane waves by open arcs. We derive a characterization of the scatterer in terms of the spectral data of the scattering matrix analogously to the case of the scattering by bounded open domains. Numerical examples show that this theoretical result also leads to a very fast visualization technique for the unknown arc.

Verified computation of Lame´ functions with high accuracy

The Lamé functions are of considerable interest for boundary-value problems in electromagnetics and mechanics that involve ellipsoid geometry. Up to now efficient algorithms for the verified computation of these functions do not exist. In this article we provide a new effective scheme for the numerical treatment of Lamé functions of the first and second kind. It involves computing the Lamé functions with high accuracy combined with safe error estimates.

On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators

Eigenvalue problems for the double-layer potential operator and the electrostatic integral operator are of considerable interest in some models for permanent magnetization of compact bodies. For the case that the underlying surface is either a sphere, a spheroid or a triazial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lame functions and surface ellipsoidal harmonics. Since only few Lame functions can be written in closed form numerical methods are required. We provide an effective scheme for the computation these functions.

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lame functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.

The Nyström method for solving a class of singular integral equations and applications in 3D-plate elasticity

The paper consists of two parts. In the first part we investigate a Nystrom- 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 Nystrom 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 x 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 Nystrom method to solve the integral equations numerically.

On a class of Robin boundary value problems in physical geodesy

In physical geodesy one considers several (exterior) Robin boundary value problems for the Laplace equation in three dimensions. The ellipsoidal Stokes problem, which occurs in context of gravimetric determination of the geoid, belongs to this class. Up to now, this and related problems have been treated with high order series expansions of spherical and spheroidal harmonics. In this article we investigate the nullfield method for this class of boundary value problems. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. For the case that the underlying surface is a triaxial ellipsoid or an oblate spheroid, explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided.

Attacking a Conjecture in Mathematical Physics by Combining Methods of Computational Analysis and Scientific Computing

We consider a conjecture on the sum of eigenvalues of two integral operators arising in potential and scattering theory for the case that the underlying surface is a triaxial ellipsoid. This concerns computation of Lamé functions which are anyway of great interest in electromagnetics and mechanics. We provide a new effective scheme for the numerical treatment of these special functions. It involves computing the Lamé functions with high accuracy combined with safe error estimates.

The Integral Equation Method for Dipole Equilibrium

The equations of dipole distributions being in equilibrium in a compact three-dimensional domain are treated by the integral equation method. It is shown that electric or magnetic dipole equilibria are governed by the eigenfunctions of the electrostatic integral operator or its adjoint operator, respectively.

Matrizen und lineare Gleichungssysteme

Wir haben in Kap. 11 bereits gesehen, dass das Studium von Beziehungen von Vektoren in der Regel zu einem System von Gleichungen fuhrt. Solche Systeme von linearen Gleichungen treten auch in vielen anderen Bereichen der Mathematik und der Naturwissenschaften auf. Daher werden wir in diesem Kapitel lineare Gleichungssysteme und ihre Eigenschaften studieren und mithilfe von Matrizen Wege zur Untersuchung ihrer Losbarkeit und zum Finden ihrer Losungsmengen herleiten.

Induktive Statistik – Rückschlüsse von einer Stichprobe auf die Allgemeinheit ziehen

Wie bereits angesprochen geht es bei der induktiven Statistik darum, aus einer Stichprobe Ruckschlusse auf die zugrunde liegende Grundgesamtheit zu ziehen bzw. Vermutungen uber die Grundgesamtheit zu stutzen oder zu verwerfen.