Splitting Waves

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

F. Assous, M. Kray, F. Nataf, E. Turkel: Time-reversed
absorbing condition: application to inverse problems, Inverse
Problems, 27(6), 065003, 2011.

F. Assous, M. Kray, F. Nataf: Time reversal techniques for
multitarget identification, in Ultrasonics Symposium (IUS), IEEE
International (pp. 143-145). IEEE, 2013.

M. Grote, M. Kray, F. Nataf, F. Assous: Wave splitting for
time-dependent scattered field separation, Comptes Rendus
Mathematique, 353(6), 523-527, 2015.