Fast and accurate simulation of wave propagation through an elastic medium is a key component in geophysical exploration. By assuming a time-harmonic wave field and after discretizing the governing equation, the wave can be computed by solving a large complex and indefinite system of linear equations. This time-harmonic elastic wave equation has the same structure as the Helmholtz equation for acoustic wave propagation in the sense that both can be written as $(K + if C -f^2 M)\mathbf{x} = \mathbf{b}$. Here $f$ is the (angular) frequency, $i = \sqrt{-1}$ is the imaginary unit number, $K$, $C$ and $M$ are symmetric and positive definite matrices and the right-hand side vector $\mathbf{b}$ corresponds to a source term. Such systems typically have to be solved at several frequencies and for dozens of right-hand side vectors $\mathbf{b}$. The grid size must be small enough to describe a wave, which results in very large systems for high frequencies (many millions of linear equations) that are hard to solve due to the indefiniteness of the system matrix. The time-harmonic elastic wave equation is even more difficult to solve than the Helmholtz equation since the number of unknowns per grid points is three instead of one, resulting in an even larger system, with a system matrix that is more dense (more non-zeros per row) and more ill-conditioned. My project aims to develop an efficient iterative mehtod for solving the time-harmonic elastic wave equation at multiple frequencies and for multiple source terms. In order to develop this method, algorithms that have been successfully applied for the Helmholtz equation will be generalized for the time-harmonic elastic wave equation. We aim for (nearly) frequency and grid independent convergence so that the solver can be successfully applied to problems at the relevant frequencies for geophysical applications.

- M. Baumann (2017). Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple Frequencies.
*PhD Thesis*. Also available online from the TU Delft repository and as eBook.

- M. Baumann and M. B. van Gijzen (2015). Nested
Krylov methods for shifted linear systems.
*SIAM J. Sci. Comput.,***37**(5), S90-S112.
Preprint
bibtex
- M. Baumann, R. Astudillo, Y. Qiu, E. Ang, M. B. van Gijzen, and R.-E. Plessix (2017). An MSSS-Preconditioned Matrix Equation Approach for the Time-Harmonic Elastic Wave Equation at Multiple Frequencies.
*Springer Computat. Geosci.,***22**(1):43-61.
bibtex
- M. Baumann and M. B. van Gijzen (2017). Efficient iterative methods for multi-frequency wave propagation problems: A comparison study.
*Procedia Comput. Sci., Vol.***108**, pp. 645-654. Preprint bibtex

- M. Baumann and M.B. van Gijzen (2016).
A Fast Iterative Solution of the Time-harmonic Wave Equation
with MSSS-preconditioned IDR(s).
*Appeared in: Extended abstract for 78th EAGE Conference & Exhibition 2016 in Vienna, Austria, EarthDoc.*Preprint
bibtex
- Baumann, M., Heiland, J., and Schmidt, M. (2015). Discrete Input/Output Maps and their Relation to Proper Orthogonal Decomposition. In:
*Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory. Festschrift in Honor of Volker Mehrmann.*Springer International Publishing. 585-608.
Preprint
bibtex

- M. Baumann and M.B. van Gijzen (2017). An Efficient Two-Level Preconditioner for Multi-Frequency Wave Propagation Problems. DIAM Technical Report 17-03, TU Delft pdf Code Applet
- M. Baumann, P. Benner, J. Heiland (2016). Space-time Galerkin POD with application in optimal control of semi-linear parabolic partial differential equations. arXiv:1611.04050. pdf Code
- M. Baumann, R. Astudillo, Y. Qiu, E. Ang, M.B. Van Gijzen and R.-E. Plessix (2016). An MSSS-Preconditioned Matrix Equation Approach for the Time-Harmonic Elastic Wave Equation at Multiple Frequencies. DIAM Technical Report 16-04, TU Delft. pdf Code
- M. Baumann, M. B. van Gijzen (2014). Nested Krylov methods for shifted linear systems. DIAM Technical Report 14-01, TU Delft. pdf Code

- Preconditioning the Elastic Wave Equation in 2D and 3D based on Inexact MSSS Matrix Computations, presented at International Conference On Preconditioning Techniques For Scientific And Industrial Applications, Jul. 31 - Aug. 2, 2017, Vancouver, CAN (poster, abstract)
- Preconditioning the Time-Harmonic Elastic Wave Equation at Multiple Frequencies, presented at COSSE alumni workshop, Nov. 10-12, 2016, Stockholm, Sweden and at DCSE Kick-Off Meeting, May 23, 2017, TU Delft.
- A generalized POD space-time Galerkin scheme for parameter dependent dynamical systems, presented at MoRePas - Model Reduction of Parametrized Systems, Oct. 13-16, 2015, Trieste, Italy (presenter: Jan Heiland)
- A Set of Fortran 90 and Python Routines for Solving Linear Equation with IDR(s), presented at SIAM Conference on Applied Linear Algebra, Oct. 26-30, 2015, Atlanta, US (poster, blitz)
- A generalization of the Proper Orthogonal Decomposition method for nonlinear model-order reduction, presented at 40th Woudschoten Conference, Oct. 7-9, 2015, Zeist, The Netherlands (poster)
- Nested Krylov methods for shifted linear systems, presented at Numerical Algebra, Matrix Theory, Differential-Algebraic Equations, and Control Theory, May 6-9, 2015, TU Berlin, Berlin, Germany (published at scienceopen)
- Nested Krylov methods for shifted linear systems arising in frequency-domain wave simulations, presented at 39th Woudschoten Conference, Oct. 8-10, 2014, Zeist, The Netherlands (poster, my_prize)
- Fast iterative solution of the time-harmonic elastic wave equation at multiple frequencies, presented at 38th Woudschoten Conference, Oct. 2-4, 2013, Zeist, The Netherlands (poster)

- 01-08-2017: Talk at Preconditioning Conference 2017, Vancouver, CAN (abstract, slides, applet)
- 14-06-2017: Talk at International Conference on Computational Science, ETH Zurich, CH (slides)
- 19-05-2017: Talk at WSC Spring Meeting, U Antwerp, BEL (slides)
- 31-01-2017: Talk at AG ModNumDiff, TU Berlin, GER (slides)
- 17-07-2016: Talk at SIAM Annual Meeting 2016, Boston, USA (slides)
- 03-06-2016: Talk at EAGE16 - WS13, Vienna, AU (slides, movie)
- 26-10-2015: Talk at SIAM Conference on Applied Linear Algebra, Atlanta, US (slides)
- 19-06-2015: Talk at Preconditioning Conference 2015, TU Eindhoven, NL (slides)
- 25-09-2014: Talk at the mini-workshop on shifted linear systems at the University of Wuppertal, Wuppertal, GER (slides, movie)
- 03-09-2014: Talk at the 4th IMA Conference on Numerical Linear Algebra and Optimisation, Birmingham, UK (abstract, slides)

- We currently offer the possibility of a master thesis in the framework of my PhD project.