Direct Scattering Problems

Acoustic, electromagnetic, or elastic waves are scattered when they encounter obstacles. The propagation of these waves is described by (partial) differential equations, and the scattering problems are modeled by boundary value problems for these differential equations. My research in this area has focused on the investigation of integral equation methods for these boundary value problems (existence, uniqueness, stability) and the analysis and implementation of numerical algorithms.

Inverse Scattering Problems

Inverse scattering problems occur, for example, in engineering (non-destructive material testing), geophysics (deep geomagnetic sounding), and medicine (localization of cancer cells). Unlike direct scattering problems, in which the primary wave and the parameters of the scattering medium are known and the scattered wave must be calculated, inverse scattering problems involve determining the scattering object from measurements of the scattered field.

The two images show the reconstruction of the scattering objects from 32 primary waves each, whose scattered components were measured in 32 directions—- without knowing the nature of the scattering object (open bounded set consisting of one or two components, each with Dirichlet boundary conditions).

Mathematical Antenna Theory

In general, an antenna is defined as any structure capable of carrying a current distribution and thereby generating an electromagnetic field. The department investigates optimization problems in antenna theory: The current distribution on the antenna is determined such that certain performance characteristics, such as the radiated energy in a given sector, are maximized. The two plots show the amplitude of the optimal far field for a circular antenna of radius 1 when the radiated energy is maximized in the angular range between 0° and 45°. The wavelength is 1 (left) and (right).

Publications

My publications can be found on this page.

Monographs

Andreas Kirsch, Wolfgang Warth, and Jochen Werner: Necessary Optimality Conditions and Their Application. Volume 152 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin-Heidelberg-New York, 1978.

Thomas Angell and Andreas Kirsch. Optimization Methods in Electromagnetic Radiation. Springer Monographs in Mathematics. Springer, New York, 2004.

Andreas Kirsch and Natalia Grinberg. The Factorization Method for Inverse Problems. Oxford University Press, Oxford, 2008. Corrections and plots can be found here: The Factorization Method for Inverse Problems

Andreas Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Volume 120 of Applied Mathematical Sciences. Springer, New York, third edition, 2021.

Andreas Kirsch and Frank Hettlich. The Mathematical Theory of Time–Harmonic Maxwell’s Equations. Expansion–, Integral–, and Variational Methods. Applied Mathematical Sciences. Springer, New York, 2015.