Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Using \(\ell _1\) Regularization to Improve Numerical Partial Differential Equation Solvers

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Sparse regularization plays a central role in many recent developments in imaging and other related fields. However, it is still of limited use in numerical solvers for partial differential equations (PDEs). In this paper we investigate the use of \(\ell _1\) regularization to promote sparsity in the shock locations of hyperbolic PDEs. We develop an algorithm that uses a high order sparsifying transform which enables us to effectively resolve shocks while still maintaining stability. Our method does not require a shock tracking procedure nor any prior information about the number of shock locations. It is efficiently implemented using the alternating direction method of multipliers. We present our results on one and two dimensional examples using both finite difference and spectral methods as underlying PDE solvers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. Even orders were not used in [2] because the post-processing techniques used for pinpointing the edges assumed that maximum (minimum) values occurred at the edge, which is true only for odd orders.

  2. We also applied \(\ell _1\) enhancement to the spectral viscosity method for the Fourier and Chebyshev cases. Both resulted in improved accuracy that essentially mirrored the approximations displayed in Figs. 4 and 10. Hence they are not reported here.

  3. With a small decrease in accuracy, the mapped Chebyshev method allows the time step to increase to \(\mathcal {O}(\frac{1}{N})\), [19].

  4. The explicit matrix entries for (30) for (24) can be found in [5, 16]. In our examples, we use the mapped Chebyshev points, [19], so the derivative matrix depends on the chosen grid points \(x_j\).

  5. MATLAB code is available at [32].

  6. We did not separately analyze our method for Euler’s equations, (47).

  7. More information about these images can be found at [14, 29], and [34].

References

  1. Archibald, R., Gelb, A., Platte, R.B.: Image reconstruction from undersampled Fourier data using the polynomial annihilation transform. J. Sci. Comput. 67(2), 432–452 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Archibald, R., Gelb, A., Yoon, J.: Polynomial fitting for edge detection in irregularly sampled signals and images. SIAM J. Numer. Anal. 43(1), 259–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Argenti, F., Lapini, A., Bianchi, T., Alparone, L.: A tutorial on speckle reduction in synthetic aperture radar images. IEEE Geosci. Remote Sens. Mag. 1(3), 6–35 (2013)

    Article  Google Scholar 

  4. Aubert, G., Aujol, J.F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quarteroni, A.M., Thomas Jr., A., et al.: Spectral Methods in Fluid Dynamics. Springer, New York (2012)

    MATH  Google Scholar 

  6. Danaila, I., Joly, P., Kaber, S.M., Postel, M.: An Introduction to Scientific Computing: Twelve Computational Projects Solved with MATLAB. Springer, New York (2007)

    Book  MATH  Google Scholar 

  7. Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun guide (2014)

  9. Durand, S., Froment, J.: Reconstruction of wavelet coefficients using total variation minimization. SIAM J. Sci. Comput. 24(5), 1754–1767 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

  11. Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goodman, J.W.: Speckle Phenomena in Optics: Theory and Applications. Roberts and Company Publishers, Greenwood Village (2007)

    Google Scholar 

  13. Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guermond, J.L., Marpeau, F., Popov, B., et al.: A fast algorithm for solving first-order pdes by l1-minimization. Commun. Math. Sci. 6(1), 199–216 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. GOTCHA volumetric SAR data set overview. https://www.sdms.afrl.af.mil/index.php?collection=gotcha. Accessed 19 Aug 2016

  16. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-dependent Problems, vol. 21. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  17. Hou, T.Y., Li, Q., Schaeffer, H.: Sparse + low-energy decomposition for viscous conservation laws. J. Comput. Phys. 288, 150–166 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jameson, A.: Energy estimates for nonlinear conservation law with applications to solutions of the burgurs equation and one-dimensional viscous flow in a shock tube by central difference schemes. In: 18th Computational Fluid Dynamics Conference by the AIAA, Miami, vol. 28 (2007)

  19. Kosloff, D., Tal-Ezer, H.: Modified chebyshev pseudospectral method with \({O}({N}^{-1})\) time step restriction. J. Comput. Phys. 104(2), 457–469 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lavery, J.E.: Solution of steady-state one-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 26(5), 1081–1089 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lavery, J.E.: Solution of steady-state, two-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 28(1), 141–155 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. LeVeque, R.J.: Numerical Methods for Conservation Laws. Springer, New York (1992)

    Book  MATH  Google Scholar 

  23. Le Veque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  24. LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  25. Li, C.: An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing. Ph.D. thesis, Rice University (2009)

  26. Li, C., Yin, W., Jiang, H., Zhang, Y.: An efficient augmented Lagrangian method with applications to total variation minimization. Comput. Optim. Appl. 56(3), 507–530 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lustig, M., Donoho, D., Pauly, J.M.: Sparse MRI: the application of compressed sensing for rapid mr imaging. Magn. Reson. Med. 58(6), 1182–1195 (2007)

    Article  Google Scholar 

  28. Moulin, P.: A wavelet regularization method for diffuse radar-target imaging and speckle-noise reduction. In: Wavelet Theory and Application, pp. 123–134. Springer, New York (1993)

  29. MSTAR overview. https://www.sdms.afrl.af.mil/index.php?collection=mstar. Accessed 19 Aug 2016

  30. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sanders, T.: Matlab imaging algorithms: Image reconstruction, restoration, and alignment, with a focus in tomography. http://www.toby-sanders.com/software, doi:10.13140/RG.2.2.33492.60801. Accessed 19 Aug 2016

  33. Sanders, T., Gelb, A., Platte, R.B.: Composite SAR imaging using sequential joint sparsity. J. Comput. Phys. 338, 357–370 (2017)

    Article  MathSciNet  Google Scholar 

  34. Scarnati, T., Zelnio, E., Paulson, C.: Exploiting the sparsity of edge information in synthetic aperture radar imagery for speckle reduction. In: SPIE Defense \(+\) Security, p. 102010C. International Society for Optics and Photonics (2017)

  35. Schaeffer, H., Caflisch, R., Hauck, C.D., Osher, S.: Sparse dynamics for partial differential equations. Proc. Natl. Acad. Sci. 110(17), 6634–6639 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shu, C.W., Wong, P.S.: A note on the accuracy of spectral method applied to nonlinear conservation laws. J. Sci. Comput. 10(3), 357–369 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  37. Solbo, S., Eltoft, T.: A stationary wavelet-domain wiener filter for correlated speckle. IEEE Trans. Geosci. Remote Sens. 46(4), 1219–1230 (2008)

    Article  Google Scholar 

  38. Stefan, W., Renaut, R.A., Gelb, A.: Improved total variation-type regularization using higher order edge detectors. SIAM J. Imaging Sci. 3(2), 232–251 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tadmor, E.: Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26(1), 30–44 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tadmor, E.: Shock capturing by the spectral viscosity method. Comput. Methods Appl. Mech. Eng. 80(1–3), 197–208 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tadmor, E.: Super viscosity and spectral approximations of nonlinear conservation laws. In: M. Baines, K. Morton (eds.) Proceedings of the 1992 Conference on Numerical Methods for Fluid Dynamics, pp. 69–82 (1993)

  42. Tadmor, E., Waagan, K.: Adaptive spectral viscosity for hyperbolic conservation laws. SIAM J. Sci. Comput. 34(2), A993–A1009 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17(1), 227–238 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wasserman, G., Archibald, R., Gelb, A.: Image reconstruction from fourier data using sparsity of edges. J. Sci. Comput. 65(2), 533–552 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Theresa Scarnati & Rodrigo B. Platte

  2. Department of Mathematics, Dartmouth College, 27 N. Main St., Hanover, NH, 03755, USA

    Anne Gelb

Authors
  1. Theresa Scarnati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anne Gelb
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rodrigo B. Platte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anne Gelb.

Additional information

This work is supported in part by the Grants NSF-DMS 1502640, NSF-DMS 1522639 and AFOSR FA9550-15-1-0152.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Scarnati, T., Gelb, A. & Platte, R.B. Using \(\ell _1\) Regularization to Improve Numerical Partial Differential Equation Solvers. J Sci Comput 75, 225–252 (2018). https://doi.org/10.1007/s10915-017-0530-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0530-8

Keywords

Mathematics Subject Classification

Search

Navigation