August  2016, 36(8): 4077-4100. doi: 10.3934/dcds.2016.36.4077

Efficient representation and accurate evaluation of oscillatory integrals and functions

1. 

Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO 80309-0526, United States, United States

Received  May 2015 Revised  October 2015 Published  March 2016

We introduce a new method for functional representation of oscillatory integrals within any user-supplied accuracy. Our approach is based on robust methods for nonlinear approximation of functions via exponentials. The complexity of evaluation of the resulting representations of the oscillatory integrals does not depend or depends only mildly on the size of the parameter responsible for the oscillatory behavior.
Citation: Gregory Beylkin, Lucas Monzón. Efficient representation and accurate evaluation of oscillatory integrals and functions. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4077-4100. doi: 10.3934/dcds.2016.36.4077
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th edition, Dover Publications, 1970.

[2]

G. E. Andrews, R. Askey and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937.

[3]

G. Beylkin and T. S. Haut, Nonlinear approximations for electronic structure calculations, Proc. R. Soc. A, 469 (2013), 20130408. doi: 10.1098/rspa.2013.0231.

[4]

G. Beylkin and L. Monzón, On generalized Gaussian quadratures for exponentials and their applications, Appl. Comput. Harmon. Anal., 12 (2002), 332-373. doi: 10.1006/acha.2002.0380.

[5]

G. Beylkin and L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal., 19 (2005), 17-48. doi: 10.1016/j.acha.2005.01.003.

[6]

G. Beylkin and L. Monzón, Nonlinear inversion of a band-limited Fourier transform, Appl. Comput. Harmon. Anal., 27 (2009), 351-366. doi: 10.1016/j.acha.2009.04.003.

[7]

G. Beylkin and L. Monzón, Approximation of functions by exponential sums revisited, Appl. Comput. Harmon. Anal., 28 (2010), 131-149. doi: 10.1016/j.acha.2009.08.011.

[8]

G. Beylkin and K. Sandberg, Wave propagation using bases for bandlimited functions, Wave Motion, 41 (2005), 263-291. doi: 10.1016/j.wavemoti.2004.05.008.

[9]

G. Beylkin and K. Sandberg, O{DE} solvers using bandlimited approximations, J. Comp. Phys., 265 (2014), 156-171. doi: 10.1016/j.jcp.2014.02.001.

[10]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 2nd edition, Dover Publications Inc., New York, 1986.

[11]

A. Bonami and A. Karoui, Uniform bounds of prolate spheroidal wave functions and eigenvalues decay, Comptes Rendus Mathematique, 352 (2014), 229-234. doi: 10.1016/j.crma.2014.01.004.

[12]

E. Candès and L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris, 336 (2003), 395-398, doi: 10.1016/S1631-073X(03)00095-5.

[13]

E. Candès, L. Demanet and L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput., 29 (2007), 2464-2493. doi: 10.1137/060671139.

[14]

E. J. Candès and L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58 (2005), 1472-1528. doi: 10.1002/cpa.20078.

[15]

M. Condon, A. Deaño and A. Iserles, On highly oscillatory problems arising in electronic engineering, ESAIM Mathematical Modelling and Numerical Analysis, 43 (2009), 785-804. doi: 10.1051/m2an/2009024.

[16]

M. Condon, A. Deaño and A. Iserles, On second-order differential equations with highly oscillatory forcing terms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1809-1828. doi: 10.1098/rspa.2009.0481.

[17]

M. Condon, A. Deaño and A. Iserles, Asymptotic solvers for oscillatory systems of differential equations, SeMA J., 53 (2011), 79-101.

[18]

L. Demanet and L. Ying, Fast wave computation via fourier integral operators, Math. Comp., 81 (2012), 1455-1486. doi: 10.1090/S0025-5718-2012-02557-9.

[19]

B. Engquist, A. Fokas, E. Hairer and A. Iserles, Highly Oscillatory Problems, vol. 366 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9781139107136.

[20]

G. B. Folland, Real Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1984, Modern techniques and their applications, A Wiley-Interscience Publication.

[21]

F. G. Friedlander and J. B. Keller, Asymptotic expansions of solutions of $(\nabla^2+k^2)u=0$, Comm. Pure Appl. Math., 8 (1955), 387-394. doi: 10.1002/cpa.3160080306.

[22]

A. Gil, J. Segura and N. M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898717822.

[23]

T. S. Haut and G. Beylkin, Fast and accurate con-eigenvalue algorithm for optimal rational approximations, SIAM J. Matrix Anal. Appl., 33 (2012), 1101-1125. doi: 10.1137/110821901.

[24]

T. S. Haut, G. Beylkin and L. Monzón, Solving Burgers' equation using optimal rational approximations, Appl. Comput. Harmon. Anal., 34 (2013), 83-95. doi: 10.1016/j.acha.2012.03.004.

[25]

A. Iserles and D. Levin, Asymptotic expansion and quadrature of composite highly oscillatory integrals, Mathematics of Computation, 80 (2011), 279-296. doi: 10.1090/S0025-5718-2010-02386-5.

[26]

A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772. doi: 10.1007/s10543-004-5243-3.

[27]

A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1383-1399. doi: 10.1098/rspa.2004.1401.

[28]

A. Iserles and S. P. Nørsett, From high oscillation to rapid approximation. III. Multivariate expansions, IMA J. Numer. Anal., 29 (2009), 882-916. doi: 10.1093/imanum/drn020.

[29]

A. Iserles, S. P. Nørsett and S. Olver, Highly oscillatory quadrature: The story so far, in Numerical mathematics and advanced applications, Springer, Berlin, 2006, 97-118. doi: 10.1007/978-3-540-34288-5_6.

[30]

H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell System Tech. J., 40 (1961), 65-84. doi: 10.1002/j.1538-7305.1961.tb03977.x.

[31]

H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty III, Bell System Tech. J., 41 (1962), 1295-1336. doi: 10.1002/j.1538-7305.1962.tb03279.x.

[32]

P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646. doi: 10.1215/S0012-7094-57-02471-7.

[33]

R. D. Lewis, G. Beylkin and L. Monzón, Fast and accurate propagation of coherent light, Proc. R. Soc. A, 469 (2013), 20130458. doi: 10.1098/rspa.2013.0323.

[34]

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974, Computer Science and Applied Mathematics.

[35]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010.

[36]

A. Osipov, Certain inequalities involving prolate spheroidal wave functions and associated quantities, Applied and Computational Harmonic Analysis, 35 (2013), 359-393. doi: 10.1016/j.acha.2012.10.002.

[37]

A. Osipov, V. Rokhlin and H. Xiao, Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation, vol. 187, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-8259-8.

[38]

M. Reynolds, G. Beylkin and L. Monzón, On generalized Gaussian quadratures for bandlimited exponentials, Appl. Comput. Harmon. Anal., 34 (2013), 352-365. doi: 10.1016/j.acha.2012.07.002.

[39]

D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J., 43 (1964), 3009-3057. doi: 10.1002/j.1538-7305.1964.tb01037.x.

[40]

D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty V. The discrete case, Bell System Tech. J., 57 (1978), 1371-1430. doi: 10.1002/j.1538-7305.1978.tb02104.x.

[41]

D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Review, 25 (1983), 379-393. doi: 10.1137/1025078.

[42]

D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell System Tech. J., 40 (1961), 43-63. doi: 10.1002/j.1538-7305.1961.tb03976.x.

[43]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.

[44]

H. Xiao, V. Rokhlin and N. Yarvin, Prolate spheroidal wavefunctions, quadrature and interpolation, Inverse Problems, 17 (2001), 805-838. doi: 10.1088/0266-5611/17/4/315.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th edition, Dover Publications, 1970.

[2]

G. E. Andrews, R. Askey and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937.

[3]

G. Beylkin and T. S. Haut, Nonlinear approximations for electronic structure calculations, Proc. R. Soc. A, 469 (2013), 20130408. doi: 10.1098/rspa.2013.0231.

[4]

G. Beylkin and L. Monzón, On generalized Gaussian quadratures for exponentials and their applications, Appl. Comput. Harmon. Anal., 12 (2002), 332-373. doi: 10.1006/acha.2002.0380.

[5]

G. Beylkin and L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal., 19 (2005), 17-48. doi: 10.1016/j.acha.2005.01.003.

[6]

G. Beylkin and L. Monzón, Nonlinear inversion of a band-limited Fourier transform, Appl. Comput. Harmon. Anal., 27 (2009), 351-366. doi: 10.1016/j.acha.2009.04.003.

[7]

G. Beylkin and L. Monzón, Approximation of functions by exponential sums revisited, Appl. Comput. Harmon. Anal., 28 (2010), 131-149. doi: 10.1016/j.acha.2009.08.011.

[8]

G. Beylkin and K. Sandberg, Wave propagation using bases for bandlimited functions, Wave Motion, 41 (2005), 263-291. doi: 10.1016/j.wavemoti.2004.05.008.

[9]

G. Beylkin and K. Sandberg, O{DE} solvers using bandlimited approximations, J. Comp. Phys., 265 (2014), 156-171. doi: 10.1016/j.jcp.2014.02.001.

[10]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 2nd edition, Dover Publications Inc., New York, 1986.

[11]

A. Bonami and A. Karoui, Uniform bounds of prolate spheroidal wave functions and eigenvalues decay, Comptes Rendus Mathematique, 352 (2014), 229-234. doi: 10.1016/j.crma.2014.01.004.

[12]

E. Candès and L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris, 336 (2003), 395-398, doi: 10.1016/S1631-073X(03)00095-5.

[13]

E. Candès, L. Demanet and L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput., 29 (2007), 2464-2493. doi: 10.1137/060671139.

[14]

E. J. Candès and L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58 (2005), 1472-1528. doi: 10.1002/cpa.20078.

[15]

M. Condon, A. Deaño and A. Iserles, On highly oscillatory problems arising in electronic engineering, ESAIM Mathematical Modelling and Numerical Analysis, 43 (2009), 785-804. doi: 10.1051/m2an/2009024.

[16]

M. Condon, A. Deaño and A. Iserles, On second-order differential equations with highly oscillatory forcing terms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1809-1828. doi: 10.1098/rspa.2009.0481.

[17]

M. Condon, A. Deaño and A. Iserles, Asymptotic solvers for oscillatory systems of differential equations, SeMA J., 53 (2011), 79-101.

[18]

L. Demanet and L. Ying, Fast wave computation via fourier integral operators, Math. Comp., 81 (2012), 1455-1486. doi: 10.1090/S0025-5718-2012-02557-9.

[19]

B. Engquist, A. Fokas, E. Hairer and A. Iserles, Highly Oscillatory Problems, vol. 366 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9781139107136.

[20]

G. B. Folland, Real Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1984, Modern techniques and their applications, A Wiley-Interscience Publication.

[21]

F. G. Friedlander and J. B. Keller, Asymptotic expansions of solutions of $(\nabla^2+k^2)u=0$, Comm. Pure Appl. Math., 8 (1955), 387-394. doi: 10.1002/cpa.3160080306.

[22]

A. Gil, J. Segura and N. M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898717822.

[23]

T. S. Haut and G. Beylkin, Fast and accurate con-eigenvalue algorithm for optimal rational approximations, SIAM J. Matrix Anal. Appl., 33 (2012), 1101-1125. doi: 10.1137/110821901.

[24]

T. S. Haut, G. Beylkin and L. Monzón, Solving Burgers' equation using optimal rational approximations, Appl. Comput. Harmon. Anal., 34 (2013), 83-95. doi: 10.1016/j.acha.2012.03.004.

[25]

A. Iserles and D. Levin, Asymptotic expansion and quadrature of composite highly oscillatory integrals, Mathematics of Computation, 80 (2011), 279-296. doi: 10.1090/S0025-5718-2010-02386-5.

[26]

A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772. doi: 10.1007/s10543-004-5243-3.

[27]

A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1383-1399. doi: 10.1098/rspa.2004.1401.

[28]

A. Iserles and S. P. Nørsett, From high oscillation to rapid approximation. III. Multivariate expansions, IMA J. Numer. Anal., 29 (2009), 882-916. doi: 10.1093/imanum/drn020.

[29]

A. Iserles, S. P. Nørsett and S. Olver, Highly oscillatory quadrature: The story so far, in Numerical mathematics and advanced applications, Springer, Berlin, 2006, 97-118. doi: 10.1007/978-3-540-34288-5_6.

[30]

H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell System Tech. J., 40 (1961), 65-84. doi: 10.1002/j.1538-7305.1961.tb03977.x.

[31]

H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty III, Bell System Tech. J., 41 (1962), 1295-1336. doi: 10.1002/j.1538-7305.1962.tb03279.x.

[32]

P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646. doi: 10.1215/S0012-7094-57-02471-7.

[33]

R. D. Lewis, G. Beylkin and L. Monzón, Fast and accurate propagation of coherent light, Proc. R. Soc. A, 469 (2013), 20130458. doi: 10.1098/rspa.2013.0323.

[34]

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974, Computer Science and Applied Mathematics.

[35]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010.

[36]

A. Osipov, Certain inequalities involving prolate spheroidal wave functions and associated quantities, Applied and Computational Harmonic Analysis, 35 (2013), 359-393. doi: 10.1016/j.acha.2012.10.002.

[37]

A. Osipov, V. Rokhlin and H. Xiao, Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation, vol. 187, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-8259-8.

[38]

M. Reynolds, G. Beylkin and L. Monzón, On generalized Gaussian quadratures for bandlimited exponentials, Appl. Comput. Harmon. Anal., 34 (2013), 352-365. doi: 10.1016/j.acha.2012.07.002.

[39]

D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J., 43 (1964), 3009-3057. doi: 10.1002/j.1538-7305.1964.tb01037.x.

[40]

D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty V. The discrete case, Bell System Tech. J., 57 (1978), 1371-1430. doi: 10.1002/j.1538-7305.1978.tb02104.x.

[41]

D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Review, 25 (1983), 379-393. doi: 10.1137/1025078.

[42]

D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell System Tech. J., 40 (1961), 43-63. doi: 10.1002/j.1538-7305.1961.tb03976.x.

[43]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.

[44]

H. Xiao, V. Rokhlin and N. Yarvin, Prolate spheroidal wavefunctions, quadrature and interpolation, Inverse Problems, 17 (2001), 805-838. doi: 10.1088/0266-5611/17/4/315.

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