A Geometric Multigrid Preconditioner for the Solution of the Helmholtz Equation in Three-Dimensional Heterogeneous Media on Massively Parallel Computers
We consider the numerical simulation of acoustic wave propagation in three-dimensional heterogeneous media as occurring in seismic exploration. We focus on forward Helmholtz problems written in the frequency domain, since this setting is known to be particularly challenging for modern iterative methods. The geometric multigrid preconditioner proposed by Calandra et al. (Numer Linear Algebra Appl 20:663–688, 2013) is considered for the approximate solution of the Helmholtz equation at high frequencies in combination with dispersion minimizing finite difference methods. We present both a strong scalability study and a complexity analysis performed on a massively parallel distributed memory computer. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application at high frequency.
KeywordsFinite Difference Scheme Multigrid Method Perfectly Match Layer Acoustic Imaging Krylov Subspace Method
The authors would like to thank TOTAL for the financial support over the past years. They also would like to acknowledge GENCI (Grand Equipement National de Calcul Intensif) for the dotation of computing hours on the IBM BG/Q computer at IDRIS, France. This work was granted access to the HPC resources of IDRIS under allocation 2015065068 and 2016065068 made by GENCI.
- 2.F. Aminzadeh, J. Brac, and T. Kunz. 3D Salt and Overthrust models. SEG/EAGE modeling series I, Society of Exploration Geophysicists, 1997.Google Scholar
- 8.H. Calandra, S. Gratton, R. Lago, X. Vasseur, and L. M. Carvalho. A modified block flexible GMRES method with deflation at each iteration for the solution of non-hermitian linear systems with multiple right-hand sides. SIAM J. Sci. Comput., 35(5):S345–S367, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Y. A. Erlangga. A robust and efficient iterative method for the numerical solution of the Helmholtz equation. PhD thesis, TU Delft, 2005.Google Scholar
- 24.O. Ernst and M. J. Gander. Why it is difficult to solve Helmholtz problems with classical iterative methods. In O. Lakkis I. Graham, T. Hou and R. Scheichl, editors, Numerical Analysis of Multiscale Problems. Springer, 2011.Google Scholar
- 28.W. Hackbusch and U. Trottenberg. Multigrid methods. Springer, 1982. Lecture Notes in Mathematics, vol. 960, Proceedings of the conference held at Köln-Porz, November 23–27 1981.Google Scholar
- 30.M. Hoemmen. Communication-avoiding Krylov subspace methods. PhD thesis, University of California, Berkeley, Department of Computer Science, 2010.Google Scholar
- 31.F. Liu and L. Ying Additive sweeping preconditioner for the Helmholtz equation ArXiv e-prints, 2015. http://arxiv.org/abs/1504.04058.
- 32.F. Liu and L. Ying Recursive sweeping preconditioner for the 3D Helmholtz equation ArXiv e-prints, 2015. http://arxiv.org/abs/1502.07266.
- 34.S. Operto, J. Virieux, P. R. Amestoy, J.-Y. L’Excellent, L. Giraud, and H. Ben Hadj Ali. 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study. Geophysics, 72–5:195–211, 2007.Google Scholar
- 36.X. Pinel. A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics. PhD thesis, CERFACS, 2010. TH/PA/10/55.Google Scholar
- 37.B. Reps, W. Vanroose, and H. bin Zubair. On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning. J. Comp. Phys., 229:8384–8405, 2010.Google Scholar
- 40.Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003. Second edition.Google Scholar
- 45.F. Sourbier, S. Operto, J. Virieux, P. Amestoy, and J. Y. L’ Excellent. FWT2D: a massively parallel program for frequency-domain full-waveform tomography of wide-aperture seismic data - part 1: algorithm. Computer & Geosciences, 35:487–495, 2009.Google Scholar
- 46.F. Sourbier, S. Operto, J. Virieux, P. Amestoy, and J. Y. L’ Excellent. FWT2D: a massively parallel program for frequency-domain full-waveform tomography of wide-aperture seismic data - part 2: numerical examples and scalability analysis. Computer & Geosciences, 35:496–514, 2009.Google Scholar
- 49.K. Stüben and U. Trottenberg. Multigrid methods: fundamental algorithms, model problem analysis and applications. In W. Hackbusch and U. Trottenberg, editors, Multigrid methods, Koeln-Porz, 1981, Lecture Notes in Mathematics, volume 960. Springer, 1982.Google Scholar
- 52.A. Toselli and O. Widlund. Domain Decomposition methods - Algorithms and Theory. Springer Series on Computational Mathematics, Springer, 34, 2005.Google Scholar
- 56.W. Vanroose, B. Reps, and H. bin Zubair. A polynomial multigrid smoother for the iterative solution of the heterogeneous Helmholtz problem. Technical Report, University of Antwerp, Belgium, 2010. http://arxiv.org/abs/1012.5379.