Tue, 21 July, 2009, 17:15-18:15, Foyer
We study the wave equation on a bounded domain or on a compact Riemannian manifold with boundary. Assume that we do not know the coefficients of the wave equation but only know the Robin-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. We show that for a fixed time a wave can be cut off outside a suitable set.
We consider how to focus waves, that is, how to find Robin boundary values so that at a given time the corresponding waves converges to a delta distribution
$\delta_x$ while the time derivative of the wave converges to zero. Such boundary values are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step.
A key feature of this result is that it does not require knowledge of the
coefficients in the wave equation, that is, the material parameters inside the media. However, we assume that the point $x$ where the wave focuses is known in travel time coordinates, and $x$ satisfies a certain geometrical conditions.
This work was done in collaboration with Matti Lassas and Matias Dahl (Helsinki University of Technology)
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