Advertisement

Journal of Scientific Computing

, Volume 55, Issue 1, pp 16–39 | Cite as

A Bootstrap Method for Sum-of-Poles Approximations

  • Kuan Xu
  • Shidong Jiang
Article

Abstract

A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138–1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples.

Keywords

Rational approximation Sum-of-poles approximation Model reduction Balanced truncation method Square root method 

Notes

Acknowledgements

S. Jiang was supported in part by National Science Foundation under grant CCF-0905395 and would like to thank Dr. Bradley Alpert at National Institute of Standards and Technology for many useful discussions on this project. Both authors would like to thank the anonymous referees for their careful reading and very useful suggestions which have greatly enhanced the presentation of the work.

References

  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1965) Google Scholar
  2. 2.
    Adamyan, V.M., Arov, D.Z., Krein, M.G.: Infinite Hankel matrices and generalized Carathéodory-Fejér and I. Schur problems. Funct. Anal. Appl. 2, 269–281 (1968) zbMATHCrossRefGoogle Scholar
  3. 3.
    Adamyan, V.M., Arov, D.Z., Krein, M.G.: Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and Riesz problems. Funct. Anal. Appl. 2(1), 1–18 (1968) zbMATHCrossRefGoogle Scholar
  4. 4.
    Adamyan, V.M., Arov, D.Z., Krein, M.G.: Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. Mat. Sb. 86, 34–75 (1971) MathSciNetGoogle Scholar
  5. 5.
    Alpert, B., Greengard, L., Hagstrom, T.: Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal. 37, 1138–1164 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Alpert, B., Greengard, L., Hagstrom, T.: Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys. 180, 270–296 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schädle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4, 729–796 (2008) MathSciNetGoogle Scholar
  8. 8.
    Beylkin, G., Monzón, L.: On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19, 17–48 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Beylkin, G., Monzón, L.: On generalized Gaussian quadratures for exponentials and their applications. Appl. Comput. Harmon. Anal. 12, 332–373 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J. Sci. Comput. 32, 1761–1788 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Causley, M., Petropolous, P., Jiang, S.: Incorporating the Havriliak-Negami dielectric model in numerical solutions of the time-domain Maxwell equations. J. Comput. Phys. 230, 3884–3899 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, San Diego (1972) zbMATHGoogle Scholar
  13. 13.
    Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds. Int. J. Control 39, 1115–1193 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gutknecht, M.H., Trefethen, L.N.: Real and complex Chebyshev approximation on the unit disk and interval. Bull., New Ser., Am. Math. Soc. 8, 455–458 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gutknecht, M.H., Smith, J.O., Trefethen, L.N.: The Carathéodory-Fejér (CF) method for recursive digital filter design. IEEE Trans. Acoust. Speech Signal Process. 31, 1417–1426 (1983) CrossRefGoogle Scholar
  16. 16.
    Gutknecht, M.H.: Rational Carathéodory-Fejér approximation on a disk, a circle, and an interval. J. Approx. Theory 41, 257–278 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47–106 (1999) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hammarling, S.: Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Numer. Anal. 2, 303–323 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hardy, G.H.: On the mean value of the modulus of an analytic function. Proc. Lond. Math. Soc. s2_14, 269–277 (1915) CrossRefGoogle Scholar
  20. 20.
    Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1998) Google Scholar
  21. 21.
    Jiang, S.: Fast evaluation of the nonreflecting boundary conditions for the Schrödinger equation. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, New York (2001) Google Scholar
  22. 22.
    Jiang, S., Greengard, L.: Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions. Commun. Pure Appl. Math. 61, 261–288 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Laub, A., Heath, M., Paige, C., Ward, R.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Autom. Control 32, 115–122 (1987) zbMATHCrossRefGoogle Scholar
  24. 24.
    Lee, J., Greengard, L.: A fast adaptive numerical method for stiff two-point boundary value problems. SIAM J. Sci. Comput. 18, 403–429 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Li, J.R.: Low order approximation of the spherical nonreflecting boundary kernel for the wave equation. Linear Algebra Appl. 415, 455–468 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Li, J.R.: A fast time stepping method for evaluating fractional integrals. SIAM J. Sci. Comput. 31, 4696–4714 (2010) zbMATHCrossRefGoogle Scholar
  27. 27.
    Lin, L., Lu, J., Ying, L., E, W.: Pole-based approximation of the Fermi-Dirac function. Chin. Ann. Math., Ser. B 30, 729–742 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44, 1332–1350 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    López-Fernández, M., Lubich, C., Schädle, A.: Adaptive, fast, and oblivious convolution in evolution equations with memory. SIAM J. Sci. Comput. 30, 1015–1037 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Lubich, C.: Convolution quadrature revisited. BIT Numer. Math. 44, 503–514 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Lubich, C., Schädle, A.: Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput. 24, 161–182 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Ma, J., Rokhlin, V., Wandzura, S.: Generalized Gaussian quadrature rules for systems of arbitrary functions. SIAM J. Numer. Anal. 33, 971–996 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26, 17–32 (1981) zbMATHCrossRefGoogle Scholar
  35. 35.
    Muller, D.: A method for solving algebraic equations using an automatic computer. Math. Tables Other Aids Comput. 10, 208–215 (1956) zbMATHCrossRefGoogle Scholar
  36. 36.
    Peller, V.V.: Hankel Operators and Their Applications. Springer Monographs in Mathematics. Springer, New York (2003) zbMATHCrossRefGoogle Scholar
  37. 37.
    Penzl, T.: Algorithms for model reduction of large dynamical systems. Linear Algebra Appl. 415, 322–343 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math. 8, 33–48 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981) zbMATHGoogle Scholar
  40. 40.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987) zbMATHGoogle Scholar
  41. 41.
    Safonov, M.G., Chiang, R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control 34, 729–733 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28, 421–438 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, New York (1998) zbMATHGoogle Scholar
  44. 44.
    Trefethen, L.N.: Rational Chebyshev approximation on the unit disk. Numer. Math. 37, 297–320 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Trefethen, L.N.: Chebyshev approximation on the unit disk. In: Werner, K.E., Wuytack, L., Ng, E. (eds.) Computational Aspects of Complex Analysis, pp. 309–323. D. Reidel Publishing, Dordrecht (1983) CrossRefGoogle Scholar
  46. 46.
    Trefethen, L.N., Gutknecht, M.H.: The Carathéodory-Fejér method for real rational approximation. SIAM J. Numer. Anal. 20, 420–436 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Trefethen, L.N., Weideman, J., Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Yarvin, N., Rokhlin, V.: Generalized Gaussian quadratures and singular value decompositions of integral operators. SIAM J. Sci. Comput. 20, 699–718 (1998) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.One Research CircleGE Global ResearchNiskayunaUSA
  2. 2.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA

Personalised recommendations