# 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.