Skip to main content
SpringerLink
Log in

Efficient computation of oscillatory integrals by exponential transformations

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The modified Filon–Clenshaw–Curtis rules, proposed earlier by the author, are combined with the (double) exponential transformations in such a way that (1) the necessity of computing the inverse of the oscillator function is released, (2) possible inaccuracy due to rounding error, when the amplitude function has endpoint singularities, is treated, (3) difficulties raised by the stationary points of the oscillator function are treated, and (4) all the benefits of the original Filon–Clenshaw–Curtis rules are preserved. We also carry out some numerical experiments, which illustrate the efficiency of the proposed algorithms while earlier methods based on Filon–Clenshaw–Curtis rules result in poor approximations or even fail.

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

Access this article

Log in via an institution

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. Intel Core i7-8550 CPU with a clock speed between 1.80 GHz and 1.99 GHz with 8 GB of RAM.

References

  1. Asheim, A., Huybrechs, D.: Extraction of uniformly accurate phase functions across smooth shadow boundaries in high frequency scattering problems. SIAM J. Appl. Math. 74, 454–476 (2014)

    Article  MathSciNet  Google Scholar 

  2. Berrut, J.P., Trefethen, L.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)

    Article  MathSciNet  Google Scholar 

  3. Chandler-Wilde, S., Graham, I., Langdon, S., Spence, E.: Numerical-asymptotic boundary integral methods in high-frequency scattering. Acta Numerica 21, 89–305 (2012)

    Article  MathSciNet  Google Scholar 

  4. Chandler-Wilde, S., Langdon, S.: A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45, 610–640 (2007)

    Article  MathSciNet  Google Scholar 

  5. Deaño, A., Huybrechs, D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112, 197–219 (2009)

    Article  MathSciNet  Google Scholar 

  6. Domínguez, V., Graham, I., Kim, T.: Filon–Clenshaw–Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points. SIAM J. Numer. Anal. 51, 1542–1566 (2013)

    Article  MathSciNet  Google Scholar 

  7. Domínguez, V., Graham, I., Smyshlyaev, V.: Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal. 31, 1253–1280 (2011)

    Article  MathSciNet  Google Scholar 

  8. Gao, J., Iserles, A.: A generalization of Filon–Clenshaw–Curtis quadrature for highly oscillatory integrals. BIT 57, 943–961 (2017)

    Article  MathSciNet  Google Scholar 

  9. Higham, N.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)

    Article  MathSciNet  Google Scholar 

  10. Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026–1048 (2006)

    Article  MathSciNet  Google Scholar 

  11. Iserles, A., Nørsett, S.: On quadrature methods for highly oscillatory integrals and their implementation. BIT 44, 755–772 (2004)

    Article  MathSciNet  Google Scholar 

  12. Iserles, A., Nørsett, S.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. Lond. A 461, 1383–1399 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Kim, T.: Asymptotic and numerical methods for high-frequency scattering problems. Ph.D. thesis, University of Bath (2012)

  14. Levin, D.: Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67, 95–101 (1996)

    Article  MathSciNet  Google Scholar 

  15. Majidian, H.: Modified Filon–Clenshaw–Curtis rules for oscillatory integrals with a nonlinear oscillator. Preprint available at arXiv:1604.05074

  16. Majidian, H.: Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations. BIT 59, 155–181 (2019)

    Article  MathSciNet  Google Scholar 

  17. Mori, M.: An IMT-type double exponential formula for numerical integration. Publ. Res. Inst. Math. Sci. 14, 713–729 (1978)

    Article  MathSciNet  Google Scholar 

  18. Mori, M.: Discovery of the double exponential transformation and its developments. Publ. Res. Inst. Math. Sci. 41, 897–935 (2005)

    Article  MathSciNet  Google Scholar 

  19. Mori, M., Sugihara, M.: The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127, 287–296 (2001)

    Article  MathSciNet  Google Scholar 

  20. Rutishauser, H.: Vorlesungen über numerische Mathematik, vol. 1. Birkhäuser, Basel, Stuttgart, 1976; English translation, Lectures on Numerical Mathematics, Walter Gautschi, ed. Birkhäuser, Boston (1990)

  21. Spence, E., Chandler-Wilde, S., Graham, I., Smyshlyaev, V.: A new frequency-uniform coercive boundary integral equation for acoustic scattering. Commun. Pure Appl. Math. 64, 1384–1415 (2011)

    Article  MathSciNet  Google Scholar 

  22. Takahasi, H., Mori, M.: Quadrature formulas obtained by variable transformation. Numer. Math. 21, 206–219 (1973)

    Article  MathSciNet  Google Scholar 

  23. Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci. 9, 721–741 (1974)

    Article  MathSciNet  Google Scholar 

  24. Xiang, S.: Efficient Filon-type methods for \(\int ^b_af(x)e^{{\rm i}\omega g(x)}\,{\rm d} x\). Numer. Math. 105, 633–658 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Group of Multidisciplinary Studies, Faculty of Encyclopedia Studies, Institute for Humanities and Cultural Studies, PO Box 14155-6419, Tehran, Iran

    Hassan Majidian

Authors
  1. Hassan Majidian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hassan Majidian.

Additional information

Communicated by Christian Lubich.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Majidian, H. Efficient computation of oscillatory integrals by exponential transformations. Bit Numer Math 61, 1337–1365 (2021). https://doi.org/10.1007/s10543-021-00855-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-021-00855-2

Keywords

Mathematics Subject Classification

Search

Navigation