Skip to main content
Springer Nature Link
Log in

A fast solver for integral equations with convolution-type Kernel

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

This paper studies the data redundancy of the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms of solving the integral equation whose kernel includes a convolution function factor. We develop lossless matrix compression strategies, which reduce the cost of integral evaluations and the storage to linear complexity, i.e., the same order of the approximation space dimensions. We establish that this algorithm preserves the convergence order of the approximate solution. We also propose a hardware-aware parallel algorithm for these strategies.

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

Similar content being viewed by others

References

  1. Alpert, B.: A class of bases in L 2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24, 246–262 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  3. Atkinson, K.E., Graham, I., Sloan, I.: Piecewise continuous collocation for integral equations. SIAM J. Numer. Anal. 20, 172–186 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertero, M., Brianzi, P., Pike, E.: Super-resolution in confocal scanning microscopy. Inverse Problems 3, 195–212 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44, 141–183 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. de Boor, C.: A practical guide to spline. Applied Mathematical Science, vol. 27. Springer (1978)

  7. Bose, N., Boo, K.: High-resolution image reconstruction with multisensors. Int. J. Imaging Syst. Technol. 9, 294–304 (1998)

    Article  Google Scholar 

  8. Chen, H., Peng, S.: A quesi-wavelet algorithm for second kind boundary integral equations. Adv. Comput. Math. 11, 355–375 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, M., Chen, Z., Chen, G.: Approximate Solutions of Operator Equations. World Scientific Publishing Co. (1997)

  10. Chen, Z., Micchelli, C.A., Xu, Y.: The Petrov–Galerkin methods for second kind integral equations II: multiwavelets scheme. Adv. Comput. Math. 7, 199–233 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, Z., Micchelli, C.A., Xu, Y.: A construction of interpolating wavelets on invariant sets. Math. Comput. 68, 1569–1587 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Z., Micchelli, C.A., Xu, Y.: Discrete wavelet Petrov–Galerkin methods. Adv. Comput. Math. 16, 1–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, Z., Micchelli, C.A., Xu, Y.: A multilevel method for solving operator equations. J. Math. Anal. Appl. 262, 688–699 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, Z., Micchelli, C.A., Xu, Y.: Fast collocation methods for second kind integral equations. SIAM J. Numer. Anal. 40, 344–375 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, Z., Wu, B., Xu, Y.: Multilevel augmentation methods for solving operator equations. Num. Math. J. China. Univ. 14, 31–55 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Chen, Z., Wu, B., Xu, Y.: Error control strategies for numerical integrations in fast collocation methods. Northeast Math. J. 21, 233–252 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Chen, Z., Wu, B., Xu, Y.: Fast numerical collocation solutions of integral equations. Commun. Pure App. Anal. 6, 643–666 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, Z., Wu, B., Xu, Y.: Fast collocation methods for high-dimensional weakly singular integral equations. J. Integ. Appl. 20, 49–92 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, Z., Xu, Y.: The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal. 35, 406–434 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chui, C.K.: Multivariate splines. CBMS-NSF Series in Applied Math., no. 54. SIAM Publ., Philadelphia (1988)

    Book  Google Scholar 

  21. Cope, D.K.: Recursive generation of the Galerkin–Chebyshev matrix for convolution kernels. Numerical treatment of boundary integral equations. Adv. Comput. Math. 9(1–2), 21–35 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)

    Article  MathSciNet  Google Scholar 

  23. Ford, N.J., Edwards, J.T., Frischmuth, K.: Qualitative Behaviour of Approximate Solutions of Some Volterra Integral Equations with Non-Lipschitz Nonlinearity. Numerical Analysis Report 310. Manchester Centre for Computational Mathematics, Manchester (1997)

  24. Gonzalez, R., Woods, R.: Digital Image Processing. Addison-Wesley, Boston (1993)

    Google Scholar 

  25. Hansen, P.C.: Deconvolution and regularization with Topelitz matrices. Numer. Algorithms 29, 323–328 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lu, Y., Shen, L., Xu, Y.: Integral equation models for image restoration: high accuracy methods and fast algorithms. Inverse Problems 26, 1–32 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kaneko, H., Xu, Y.: Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math. Comput. 62, 739–753 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kress, R.: Linear Integral Equations. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  29. Malekejad, K., Aghazaden, N.: Numerical solution of Volterra integral equation of second kind with convolution kernel by using Taylor-series expension method. Appl. Math. Comput. 161, 15–24 (2005)

    Article  Google Scholar 

  30. Micchelli, C.A., Xu, Y.: Using the matrix refinement equation for the construction of wavelets on invariant sets. Appl. Comput. Harmon. Anal. 1, 391–401 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Micchelli, C.A., Xu, Y., Zhao, Y.: Wavelet Galerkin methods for second kind integral equations. J. Comput. Appl. Math. 86, 251–270 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. NG Michael, K.: Circulant preconditioners for convolution-like integral equations with higher-order quadrature rules. Electron. Trans. Numer. Anal. 5, 18–28 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Sehwab, C.: Variable order composite quadrature of singular and nearly singular integrals. Computing 53, 173–194 (1994)

    Article  MathSciNet  Google Scholar 

  34. Sezer, M.: Taylor polymial solution of Volterra integral equation. Math. Educ. Sci. Technol. 25, 625–629 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. von Petersdorff, T., Schwab, C.: Wavelet approximation of first kind integral equations in a polygon, Numer. Math. 74, 479–516 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. von Petersdorff, T., Schwab, C., Schneider, R.: Multiwavelets for second-kind integral equations. SIAM J. Numer. Anal. 34, 2212–2227 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China

    Weicai Ye & Yongdong Zhang

Authors
  1. Weicai Ye
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yongdong Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yongdong Zhang.

Additional information

Communicated by Charles A. Micchelli.

This research is supported in part by the Natural Science Foundation of China under grants 11071264, 11171359 and supported in part by the Guangdong Province Key Laboratory of Computational Science and the Guangdong Province Computational Science Innovative Research Team.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ye, W., Zhang, Y. A fast solver for integral equations with convolution-type Kernel. Adv Comput Math 39, 45–67 (2013). https://doi.org/10.1007/s10444-011-9268-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-011-9268-2

Keywords

Mathematics Subject Classifications (2010)