Advertisement

A Compressive Spectral Collocation Method for the Diffusion Equation Under the Restricted Isometry Property

  • Simone BrugiapagliaEmail author
Chapter
  • 59 Downloads
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 137)

Abstract

We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in strong form at randomized points, by taking advantage of the compressive sensing principle. The proposed approach makes use of a number of collocation points substantially less than the number of basis functions when the solution to recover is sparse or compressible. Focusing on the case of the diffusion equation, we prove that, under suitable assumptions on the diffusion coefficient, the matrix associated with the compressive spectral collocation approach satisfies the restricted isometry property of compressive sensing with high probability. Moreover, we demonstrate the ability of the proposed method to reduce the computational cost associated with the corresponding full spectral collocation approach while preserving good accuracy through numerical illustrations.

Keywords

Partial differential equations Compressive spectral collocation method Compressive sensing Isometry property Spectral Sturm-Liouville approximation Diffusion equation 

Notes

Acknowledgements

The author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada through grant number 611675 and the Pacific Institute for the Mathematical Sciences (PIMS) through the program “PIMS Postdoctoral Training Centre in Stochastics”. Moreover, the author gratefully acknowledge Ben Adcock and the anonymous reviewer for providing helpful comments on the first version of this manuscript.

References

  1. 1.
    Adcock, B.: Univariate modified fourier methods for second order boundary value problems. BIT Numer. Math. 49(2), 49–280 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Adcock, B.: Multivariate modified Fourier series and application to boundary value problems. Numer. Math. 115(4), 511–552 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Adcock, B., Brugiapaglia, S., Webster, C.G.: Compressed sensing approaches for polynomial approximation of high-dimensional functions. In: Compressed Sensing and its Applications, pp. 93–124. Springer, Cham (2017)Google Scholar
  4. 4.
    Alberti, G.S., Santacesaria, M.: Infinite dimensional compressed sensing from anisotropic measurements (2017). Preprint. arXiv:1710.11093Google Scholar
  5. 5.
    Bouchot, J.-L., Rauhut, H., Schwab, C.: Multi-level compressed sensing Petrov-Galerkin discretization of high-dimensional parametric PDEs (2017). Preprint. arXiv:1701.01671Google Scholar
  6. 6.
    Brugiapaglia, S.: COmpRessed SolvING: sparse approximation of PDEs based on compressed sensing. PhD thesis, Politecnico di Milano, Italy (2016)Google Scholar
  7. 7.
    Brugiapaglia, S., Micheletti, S., Perotto, S.: Compressed solving: a numerical approximation technique for elliptic PDEs based on compressed sensing. Comput. Math. Appl. 70(6), 1306–1335 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brugiapaglia, S., Nobile, F., Micheletti, S., Perotto, S.: A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems. Math. Comput. 87(309), 1–38 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Canuto, C., Hussaini, M.Y., Quarteroni, A.M., Zhang, T.A.: Spectral Methods in Fluid Dynamics. Springer Science & Business Media, New York (2012)Google Scholar
  11. 11.
    Chkifa, A., Dexter, N., Tran, H., Webster, C.G.: Polynomial approximation via compressed sensing of high-dimensional functions on lower sets. Math. Comput. 87(311), 1415 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Daubechies, I., Runborg, O., Zou, J.: A sparse spectral method for homogenization multiscale problems. Multiscale Model. Simul. 6(3), 711–740 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Birkhäuser, Basel (2013)Google Scholar
  16. 16.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, vol. 26. SIAM, Philadelphia (1977)zbMATHCrossRefGoogle Scholar
  17. 17.
    Guermond, J.-L.: A finite element technique for solving first-order PDEs in L p. SIAM J. Numer. Anal. 42(2), 714–737 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Guermond, J.-L., Popov, B.: An optimal L 1-minimization algorithm for stationary Hamilton-Jacobi equations. Commun. Math. Sci. 7(1), 211–238 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Iserles, A., Nørsett, S.P.: From high oscillation to rapid approximation I: modified Fourier expansions. IMA J. Numer. Anal. 28(4), 862–887 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Iserles, A., Nørsett, S.P.: From high oscillation to rapid approximation III: multivariate expansions. IMA J. Numer. Anal. 29(4), 882–916 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jokar, S., Mehrmann, V., Pfetsch, M.E., Yserentant, H.: Sparse approximate solution of partial differential equations. Appl. Numer. Math. 60(4), 452–472 (2010). Special Issue: NUMAN 2008Google Scholar
  22. 22.
    Krahmer, F., Ward, R.: Stable and robust sampling strategies for compressive imaging. IEEE Trans. Image Process. 23(2), 612–622 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lavery, J.E.: Nonoscillatory solution of the steady-state inviscid Burgers’ equation by mathematical programming. J. Comput. Phys. 79(2), 436–448 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Lavery, J.E.: Solution of steady-state one-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 26(5), 1081–1089 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mackey, A., Schaeffer, H., Osher, S.: On the compressive spectral method. Multiscale Model. Simul. 12(4), 1800–1827 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mathelin, L., Gallivan, K.A.: A compressed sensing approach for partial differential equations with random input data. Commun. Comput. Phys. 12(4), 919–954 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Peng, J., Hampton, J., Doostan, A.: A weighted 1-minimization approach for sparse polynomial chaos expansions. J. Comput. Phys. 267, 92–111 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Rauhut, H., Schwab, C.: Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations. Math. Comput. 86(304), 661–700 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Rubinstein, R., Zibulevsky, M., Elad, M.: Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit. Technical report, Computer Science Department, Technion (2008)Google Scholar
  30. 30.
    Schaeffer, H., Caflisch, R., Hauck, C.D., Osher, S.: Sparse dynamics for partial differential equations. Proc. Natl. Acad. Sci. 110(17), 6634–6639 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Temlyakov, V.N.: Nonlinear methods of approximation. Found. Comput. Math. 3(1), 33 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Tran, G., Ward, R.: Exact recovery of chaotic systems from highly corrupted data. Multiscale Model. Simul. 15(3), 1108–1129 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yang, X., Karniadakis, G.E.: Reweighted 1 minimization method for stochastic elliptic differential equations. J. Comput. Phys. 248, 87–108 (2013)zbMATHCrossRefGoogle Scholar
  34. 34.
    Zhang, T.: Sparse recovery with orthogonal matching pursuit under RIP. IEEE Trans. Inf. Theory 57(9), 6215–6221 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© National Technology & Engineering Solutions of Sandia, and The Editor(s), under exclusive license to Springer Nature Switzerland AG  2020

Authors and Affiliations

  1. 1.Simon Fraser UniversityBurnabyCanada

Personalised recommendations