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.


















Similar content being viewed by others
References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1965)
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)
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)
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)
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)
Alpert, B., Greengard, L., Hagstrom, T.: Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys. 180, 270–296 (2002)
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)
Beylkin, G., Monzón, L.: On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19, 17–48 (2005)
Beylkin, G., Monzón, L.: On generalized Gaussian quadratures for exponentials and their applications. Appl. Comput. Harmon. Anal. 12, 332–373 (2002)
Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J. Sci. Comput. 32, 1761–1788 (2010)
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)
Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, San Diego (1972)
Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L ∞-error bounds. Int. J. Control 39, 1115–1193 (1984)
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)
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)
Gutknecht, M.H.: Rational Carathéodory-Fejér approximation on a disk, a circle, and an interval. J. Approx. Theory 41, 257–278 (1984)
Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47–106 (1999)
Hammarling, S.: Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Numer. Anal. 2, 303–323 (1982)
Hardy, G.H.: On the mean value of the modulus of an analytic function. Proc. Lond. Math. Soc. s2_14, 269–277 (1915)
Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1998)
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)
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)
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)
Lee, J., Greengard, L.: A fast adaptive numerical method for stiff two-point boundary value problems. SIAM J. Sci. Comput. 18, 403–429 (1997)
Li, J.R.: Low order approximation of the spherical nonreflecting boundary kernel for the wave equation. Linear Algebra Appl. 415, 455–468 (2006)
Li, J.R.: A fast time stepping method for evaluating fractional integrals. SIAM J. Sci. Comput. 31, 4696–4714 (2010)
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)
López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004)
López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44, 1332–1350 (2006)
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)
Lubich, C.: Convolution quadrature revisited. BIT Numer. Math. 44, 503–514 (2004)
Lubich, C., Schädle, A.: Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput. 24, 161–182 (2002)
Ma, J., Rokhlin, V., Wandzura, S.: Generalized Gaussian quadrature rules for systems of arbitrary functions. SIAM J. Numer. Anal. 33, 971–996 (1996)
Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26, 17–32 (1981)
Muller, D.: A method for solving algebraic equations using an automatic computer. Math. Tables Other Aids Comput. 10, 208–215 (1956)
Peller, V.V.: Hankel Operators and Their Applications. Springer Monographs in Mathematics. Springer, New York (2003)
Penzl, T.: Algorithms for model reduction of large dynamical systems. Linear Algebra Appl. 415, 322–343 (2006)
Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math. 8, 33–48 (1998)
Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)
Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)
Safonov, M.G., Chiang, R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control 34, 729–733 (1989)
Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28, 421–438 (2006)
Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, New York (1998)
Trefethen, L.N.: Rational Chebyshev approximation on the unit disk. Numer. Math. 37, 297–320 (1981)
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)
Trefethen, L.N., Gutknecht, M.H.: The Carathéodory-Fejér method for real rational approximation. SIAM J. Numer. Anal. 20, 420–436 (1983)
Trefethen, L.N., Weideman, J., Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006)
Yarvin, N., Rokhlin, V.: Generalized Gaussian quadratures and singular value decompositions of integral operators. SIAM J. Sci. Comput. 20, 699–718 (1998)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, K., Jiang, S. A Bootstrap Method for Sum-of-Poles Approximations. J Sci Comput 55, 16–39 (2013). https://doi.org/10.1007/s10915-012-9620-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-012-9620-9