1887

Abstract

Summary

For the Full Waveform Inversion in frequency-domain, the fast numerical solution of the time-harmonic wave equation is required. For large three-dimensional problems, the problem size exceeds several million of unknowns, and a short-recurrence Krylov method such as IDR(s) is used to solve linear systems of this size. Especially for high-frequency simulations, an efficient preconditioner needs to be applied in order to speed-up convergence.

In our presentation, we introduce a new preconditioner for the time-harmonic wave equation that exploits the hierarchical structure of the discretized problem. We use multilevel sequentially semiseparable (MSSS) matrix computations for the approximate inversion of the preconditioner. For large three-dimensional problems, we present a memory-efficient modification of the MSSS preconditioner that resembles the approximate solution of a sequence of two-dimensional problems. We conclude our presentation with numerical examples for the time-harmonic wave equation in both acoustic and elastic media, and in two and three spatial dimensions.

Loading

Article metrics loading...

/content/papers/10.3997/2214-4609.201601667
2016-05-31
2019-12-11
Loading full text...

Full text loading...

References

  1. Astudillo, R., Baumann, M., Qiu, Y., Ang, E., Van Gijzen, M.B. and Plessix, R.E.
    [2016] A Preconditioned Matrix Equation Approach for the Time-Harmonic Elastic Wave Equation at Multiple Frequencies. Tech. Rep. in preparation, Delft University of Technology.
    [Google Scholar]
  2. Astudillo, R. and Van Gijzen, M.B.
    [2015] Induced Dimension Reduction method for solving linear matrix equations. Tech. Rep.15–05, Delft University of Technology.
    [Google Scholar]
  3. Baumann, M. and Van Gijzen, M.B.
    [2015] Nested Krylov methods for shifted linear systems. SIAM J. Sci. Comput., 37(5), S90–S112.
    [Google Scholar]
  4. Martin, G.S., Marfurt, K.J. and Larsen, S.
    [2002] Marmousi-2: an updated model for the investigation of AVO in structurally complex areas. 72nd Annual International Meeting, SEG, Expanded Abstract, 1979–1982.
    [Google Scholar]
  5. Plessix, R.E. and Mulder, W.A.
    [2004] Seperation-of-variables as a preconditioner for an iterative Helmholtz solver. Appl. Numer. Math., 44, 385–400.
    [Google Scholar]
  6. Plessix, R.E. and Pérez Solano, C.A.
    [2015] Modified surface boundary conditions for elastic waveform inversion of low-frequency wide-angle active land seismic data. Geophys. J. Int., 201, 1324–1334.
    [Google Scholar]
  7. Qiu, Y., Van Gijzen, M.B., Van Wingerden, J.W., Verhaegen, M. and Vuik, C.
    [2015] Efficient Preconditioners for PDE-Constrained Optimization Problems with a Multilevel Sequentially SemiSeparable Matrix Structure. Electron. Trans. Numer. Anal., 44, 367–400.
    [Google Scholar]
  8. Sonneveld, P. and Van Gijzen, M.B.
    [2008] IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. SIAM J. Sci. Comput., 31(2), 1035–1062.
    [Google Scholar]
  9. Virieux, J. and Operto, S.
    [2009] An overview of full-waveform inversion in exploration geophysics. Geophysics, 73(6), VE135–VE144.
    [Google Scholar]
  10. Van der Vorst, H.A.
    [1992] Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput., 13, 631–644.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/papers/10.3997/2214-4609.201601667
Loading
/content/papers/10.3997/2214-4609.201601667
Loading

Data & Media loading...

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error