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Institute for Applied and Numerical Mathematics 4: Numerical Simulation, Optimization and High Performance Computing

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Kollegiengebäude Mathematik (20.30)
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Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
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76131 Karlsruhe

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Modellansatz: Modell062 - Splitting Waves

modellansatz.de/splitting-waves

On closer inspection, we find science and especially mathematics throughout our everyday lives, from the tap to automatic speed regulation on motorways, in medical technology or on our mobile phone. What the researchers, graduates and academic teachers in Karlsruhe puzzle about, you experience firsthand in our Modellansatz Podcast: "The modeling approach“.

Der Modellansatz: Splitting Waves. Simulation: Kray

To separate one single instrument from the acoustic sound of a whole orchestra- just by knowing its exact position- gives a good idea of the concept of wave splitting, the research topic of Marie Kray. Interestingly, an approach for solving this problem was found by the investigation of side-effects of absorbing boundary conditions (ABC) for time-dependent wave problems- the perfectly matched layers are an important example for ABCs.

Marie Kray works in the Numerical Analysis group of Prof. Grote in Mathematical Department of the University of Basel. She did her PhD 2012 in the Laboratoire Jacques-Louis Lions in Paris and got her professional education in Strasbourg and Orsay.

Since boundaries occur at the surface of volumes, the boundary manifold has one spatial dimension less than the actual regarded physical domain. Therefore, the treatment of normal derivatives as in the Neumann boundary condition needs special care.

The implicit Crank-Nicolson method turned out to be a good numerical scheme for integrating the time derivative, and an upwinding scheme solved the discretized hyperbolic problem for the space dimension.

An alternative approach to separate the signals from several point sources or scatterers is to apply global integral boundary conditions and to assume a time-harmonic representation.

The presented methods have important applications in medical imaging: A wide range of methods work well for single scatterers, but Tumors often tend to spread to several places. This serverely impedes inverse problem reconstruction methods such as the TRAC method, but the separation of waves enhances the use of these methods on problems with several scatterers.

Literature and additional material