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An Adaptive Filon Algorithm for Highly Oscillatory Integrals

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Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Abstract

Based on the error analysis of Extended Filon Method (EFM), we present an adaptive Filon method to calculate highly oscillatory integrals. The main idea is to allow interpolation points depend upon underlying frequency in order to minimize the error. Typically, quadrature error need be examined in two regimes. Once frequency is large, asymptotic behaviour dominates and we need to choose interpolation points accordingly, while for small frequencies good choice of interpolation points is similar to classical, non-oscillatory quadrature. In this paper we choose frequency-dependent interpolation points according to a smooth homotopy function and the accuracy is superior to other EFMs. The basic algorithm is presented in the absence of stationary points but we extend it to cater for highly oscillatory integrals with stationary points. The presentation is accompanied by numerical experiments which demonstrate the power of our approach.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday.

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References

  1. Asheim, A., Huybrechs, D.: Complex Gaussian quadrature for oscillatory integral transforms. IMA J. Numer. Anal. 33(4), 1322–1341 (2013)

    Article  MathSciNet  Google Scholar 

  2. Chandler-Wilde, S.N., Graham, I.G., Langdon S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21, 89–305 (2012)

    Article  MathSciNet  Google Scholar 

  3. Deaño, A., Huybrechs, D., Iserles, A.: The kissing polynomials and their Hankel determinants. Technical Report, DAMTP, University of Cambridge (2015)

    Google Scholar 

  4. Domínguez, V., Ganesh, M.: Interpolation and cubature approximations and analysis for a class of wideband integrals on the sphere. Adv. Comput. Math. 39(3–4), 547–584 (2013)

    Article  MathSciNet  Google Scholar 

  5. Domínguez, V., Graham, I.G., Smyshlyaev, V.P.: Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal. 31(4), 1253–1280 (2011)

    Article  MathSciNet  Google Scholar 

  6. Ganesh, M., Langdon, S., Sloan, I.H.: Efficient evaluation of highly oscillatory acoustic scattering surface integrals. J. Comput. Appl. Math. 204(2), 363–374 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Gao, J., Iserles, A.: Error analysis of the extended Filon-type method for highly oscillatory integrals. Res. Math. Sci. 4(21/24) (2017). https://doi.org/10.1186/s40687-017-0110-4

  9. Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44(3), 1026–1048 (2006)

    Article  MathSciNet  Google Scholar 

  10. Iserles, A., Nørsett, S.P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math. 44(4), 755–772 (2004)

    Article  MathSciNet  Google Scholar 

  11. Iserles, A., Nørsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 461(2057), 1383–1399 (2005)

    Article  MathSciNet  Google Scholar 

  12. Levin, D.: Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67(1), 95–101 (1996)

    Article  MathSciNet  Google Scholar 

  13. Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA J. Numer. Anal. 26(2), 213–227 (2006)

    Article  MathSciNet  Google Scholar 

  14. Trevelyan, J., Honnor, M.E.: A numerical coordinate transformation for efficient evaluation of oscillatory integrals over wave boundary elements. J. Integral Equ. Appl. 21(3), 447–468 (2009)

    Article  MathSciNet  Google Scholar 

  15. Van’t Wout, E., Gélat, P., Timo, B., Simon, A.: A fast boundary element method for the scattering analysis of high-intensity focused ultrasound. J. Acoust. Soc. Am. 138(5), 2726–2737 (2015)

    Article  Google Scholar 

  16. Zhao, L.B., Huang, C.M.: An adaptive Filon-type method for oscillatory integrals without stationary points. Numer. Algorithms 75(3), 753–775 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of the first author has been supported by the Projects of International Cooperation and Exchanges NSFC-RS (Grant No. 11511130052), the Key Science and Technology Program of Shaanxi Province of China (Grant No. 2016GY-080) and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, People’s Republic of China

    Jing Gao

  2. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK

    Arieh Iserles

Authors
  1. Jing Gao
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  2. Arieh Iserles
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Correspondence to Arieh Iserles .

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Editors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Josef Dick

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Frances Y. Kuo

  3. Department of Computer Science, Columbia University, New York, USA

    Henryk Woźniakowski

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Gao, J., Iserles, A. (2018). An Adaptive Filon Algorithm for Highly Oscillatory Integrals. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_19

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