# Manuel Baumann: Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple Frequencies

Time: Thu 2018-02-01 14.15 - 15.00

Lecturer: Manuel Baumann, TU Delft

Location: Room F11, Lindstedtsvägen 22, våningsplan 2, F-huset, KTH Campus.

Abstract:

The time-harmonic elastic wave equation after spatial discretization reads,

$$(K + i \omega_k C - \omega_k^2 M) \mathbf{x}_k = \mathbf{b}, \quad k > 1,$$  (1)

where each solution $$\mathbf{x}_k$$ corresponds to a different (angular) wave frequency $$\omega_k$$. The challenge in a seismic full-waveform inversion algorithm is the efficient simultaneous numerical solution of (1) when $$k$$ is of the order of tens to hundreds, i.e. at multiple frequencies.

During our research in the last years, we have developed several approaches for the efficient iterative solution of this multi-frequency problem. One way is to reformulate (1) as shifted linear systems where the shifts are equal to $$\omega_k$$. In order to apply a nested Krylov method to all shifted systems simultaneous, it is a particular mathematical difficulty to preserve the shifted structure of the corresponding Krylov subspaces, cf. [2].

Another aspect of our work that I would like to discuss is the efficient application of a single preconditioner at a so-called seed frequencies. Based on spectral analysis, I will demonstrate how an optimal seed shift $$\tau^\ast$$ can be chosen for a given set of frequencies $$\{\omega_1,...,\omega_N\}$$ in (1).

All results from the presentation can be found in my recent dissertation with the same title [1].

References

[1] M. Baumann. Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple Frequencies. PhD thesis, Delft University of Technology, 2018.

[2] M. Baumann and M. B. van Gijzen. Nested Krylov methods for shifted linear systems. SIAM J. Sci. Comput., 37(5):S90-S112, 2015.

2018-02-01T14:15 2018-02-01T15:00 Manuel Baumann: Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple Frequencies Manuel Baumann: Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple Frequencies
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