American Institute of Mathematical Sciences

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

Efficient representation and accurate evaluation of oscillatory integrals and functions

Gregory Beylkin 1, and  Lucas Monzón 1,

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.
Keywords: functional approximation, Oscillatory integrals, bandlimited functions., numerical integration, quadratures.
Mathematics Subject Classification: Primary: 65D15, 65D30; Secondary: 41A5.
Citation: Gregory Beylkin, Lucas Monzón. Efficient representation and accurate evaluation of oscillatory integrals and functions. Discrete & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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