1887

Abstract

Summary

We propose a ‘learned’ iterative solver for the Helmholtz equation, by combining traditional Krylov-based solvers with machine learning. The method is, in principle, able to circumvent the shortcomings of classical iterative solvers, and has clear advantages over purely data-driven approaches. We demonstrate the effectiveness of this approach under a 1.5-D assumption, when adequate a priori information about the velocity distribution is known.

Loading

Article metrics loading...

/content/papers/10.3997/2214-4609.201901542
2019-06-03
2024-06-15
Loading full text...

Full text loading...

References

  1. Adler, J. and Öktem, O.
    [2017] Solving ill-posed inverse problems using iterative deep neural networks. Inverse Problems, 33(12), 4007.
    [Google Scholar]
  2. [2018] Learned Primal-dual Reconstruction. arXiv preprint.
    [Google Scholar]
  3. Erlangga, Y.A, Oosterlee, C.W and Vuik, C.
    [2006] A novel multigrid based preconditioner for heterogeneous Helmholtz problems. J. Sci.Comput., 27(4), 1471–1492.
    [Google Scholar]
  4. Ernst, O.G and Gander, M.J
    [2012] Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods. Numerical Analysis of Multiscale Problems, 83, 325–363.
    [Google Scholar]
  5. He, K., Zhang, X., Ren, S. and Sun, J.
    [2015] Deep Residual Learning for Image Recognition. arXiv preprint.
    [Google Scholar]
  6. Johnson, J., Alahi, A. and Fei-Fei, L.
    [2016] Perceptual Losses for Real-Time Style Transfer and Super-Resolution. arXiv preprint.
    [Google Scholar]
  7. Kingma, D.P and Ba, J.
    [2014] Adam: A Method for Stochastic Optimization. arXiv preprint.
    [Google Scholar]
  8. Krizhevsky, A., Sutskever, I. and Hinton, G.E
    [2012] Imagenet classification with deep convolutional neural networks. In: Advances in neural information processing systems. 1097–1105.
    [Google Scholar]
  9. Rizzuti, G. and Mulder, W.A
    [2016] Multigrid-based ‘shifted-Laplacian’ preconditioning for the time-harmonic elastic wave equation. J. Comput. Phys., 317(15), 47–65.
    [Google Scholar]
  10. Saad, Y.
    [2003] Iterative Methods for Sparse Linear Systems. SIAM, 2 edn.
    [Google Scholar]
  11. Siahkoohi, A., Louboutin, M., Kumar, R. and Herrmann, F.J
    [2018] Deep-convolutional neural networks in prestack seismic: Two exploratory examples. SEG Technical Program Expanded Abstracts 2018, 2196–2200.
    [Google Scholar]
  12. Trottenberg, U., Oosterlee, C.W and Schüller, A.
    [2001] Multigrid. Academic Press.
    [Google Scholar]
  13. Zhu, J.Y, Park, T., Isola, P. and Efros, A.A
    [2017] Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint.
    [Google Scholar]
/content/papers/10.3997/2214-4609.201901542
Loading
/content/papers/10.3997/2214-4609.201901542
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