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On the application of GMRES to oscillatory singular integral equations

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Abstract

We present a new method for the numerical solution of singular integral equations on the real axis. The method’s value stems from a new formula for the Cauchy integral of a rational function with an oscillatory exponential factor. The inner product of such functions is also computed explicitly. With these tools in hand, the GMRES algorithm is applied to both non-oscillatory and oscillatory singular integral equations. In specific cases, ideas from Fredholm theory and Riemann–Hilbert problems are used to motivate preconditioners for these singular integral equations. A significant acceleration in convergence is realized for these examples. This presents a useful link between the theory of singular integral equations and the numerical analysis of such equations. Furthermore, this method presents a first step towards a solver for the inverse scattering transform that does not require the deformation of a Riemann–Hilbert problem.

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References

  1. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  2. Ablowitz, M.J., Fokas, A.S.: Complex Variables: Introduction and Applications, 2nd edn. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  3. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC (1970)

    Google Scholar 

  4. Atkinson, K., Han, W.: Theoretical Numerical Analysis. Springer, Berlin (2009)

    MATH  Google Scholar 

  5. Beals, R., Coifman, R.R.: Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math. 37(iv), 39–90 (1984)

  6. Beals, R., Deift, P., Tomei, C.: Direct and Inverse Scattering on the Line. Mathematical Surveys and Monographs, vol. 28. American Mathematical Society, Providence (1988)

  7. Brešar, M., Šemrl, P.: Derivations mapping into the socle. Math. Proc. Camb. Philos. Soc. 120(2), 339–346 (1996)

    Article  MATH  Google Scholar 

  8. Deift, P.: Orthogonal polynomials and random matrices: a Riemann–Hilbert approach. American Mathematical Society, Providence, RI (2008)

  9. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Bull. Am. Math. Soc. 26, 119–124 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deift, P., Zhou, X.: Long-time Behavior of the Non-focusing Nonlinear Schrödinger Equation—A Case Study. Lectures in Mathematical Sciences, vol. 1. University of Tokyo, Tokyo (1994)

  11. Deift, P., Zhou, X.: Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Commun. Pure Appl. Math. 56, 1029–1077 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dienstfrey, A.: The Numerical Solution of a Riemann–Hilbert Problem Related to Random Matrices and the Painlevé V ODE. PhD thesis, Courant Institute of Mathematical Sciences (1998)

  13. Fokas, A.S.: A Unified Approach to Boundary Value Problems. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  14. Gasparo, M.G., Papini, A., Pasquali, A.: Some properties of GMRES in Hilbert spaces. Numer. Funct. Anal. Optim. 29(11–12), 1276–1285 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Keller, P.: A practical algorithm for computing Cauchy principal value integrals of oscillatory functions. Appl. Math. Comput. 218(9), 4988–5001 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Springer, New York (1980)

    Google Scholar 

  17. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  18. Olver, S.: Numerical solution of Riemann–Hilbert problems: Painlevé II. Found. Comput. Math. 11(2), 153–179 (2010)

    Article  MathSciNet  Google Scholar 

  19. Olver, S.: Computing the Hilbert transform and its inverse. Math. Comput. 80, 1745–1767 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Olver, S.: A general framework for solving Riemann–Hilbert problems numerically. Numer. Math. 122(2), 305–340 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Olver, S., Trogdon, T.: Nonlinear Steepest descent and numerical solution of Riemann-Hilbert problems. Commun. Pure Appl. Math., pp. 1–36 (2013, to appear)

  22. Olver, S., Trogdon, T.: Numerical solution of Riemann–Hilbert problems: random matrix theory and orthogonal polynomials. Constr. Approx. 39(1), 101–149 (2013)

    Article  MathSciNet  Google Scholar 

  23. Prösdorf, S., Silbermann, B.: Numerical Analysis for Integral and Related Operator Equations. Birkhäuser, Basel (1991)

    Google Scholar 

  24. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stein, E.M., Shakarchi, R.: Real Analysis. Princeton Lectures in Analysis, III. Princeton University Press, Princeton (2005)

    Google Scholar 

  26. Trogdon, T.: Riemann–Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions. PhD thesis, University of Washington (2013)

  27. Trogdon, T.: Rational approximation, oscillatory Cauchy integrals and Fourier transforms (2014). arXiv Prepr. arXiv1403.2378

  28. Trogdon, T., Olver, S.: Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations. Proc. R. Soc. A 469(2149), 20120330 (2013)

    Article  MathSciNet  Google Scholar 

  29. Trogdon, T., Olver, S., Deconinck, B.: Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations. Phys. D Nonlinear Phenom. 241(11), 1003–1025 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wang, H., Zhang, L., Huybrechs, D.: Asymptotic expansions and fast computation of oscillatory Hilbert transforms. Numer. Math. 123(4), 709–743 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zhou, X.: The Riemann–Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20(4), 966–986 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We acknowledge the National Science Foundation for its generous support through grants NSF-DMS-1008001 and NSF-DMS-1303018. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources. We also thank the anonymous referees for their input which improved this manuscript.

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  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA

    Thomas Trogdon

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  1. Thomas Trogdon
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Correspondence to Thomas Trogdon.

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Communicated by Anne Kværnø.

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Trogdon, T. On the application of GMRES to oscillatory singular integral equations. Bit Numer Math 55, 591–620 (2015). https://doi.org/10.1007/s10543-014-0502-4

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