A comparative study of Filon-type rules for oscillatory integrals

Hassan Majidian

doi:10.33993/jnaat531-1380

Authors

Hassan Majidian Department of Multidisciplinary Studies, Faculty of Encyclopedia Studies, Institute for Humanities and Cultural Studies, Tehran, Iran

DOI:

https://doi.org/10.33993/jnaat531-1380

Keywords:

oscillatory integral, Filon-Clenshaw-Curtis rule, extended FCC rule, adaptive FCC rule

Abstract views: 128

Abstract

Our aim is to answer the following question: "Among the Filon-type methods for computing oscillatory integrals, which one is the most efficient in practice?". We first discuss why we should seek the answer among the family of Filon-Clenshaw-Curtis rules. A theoretical analysis accompanied by a set of numerical experiments reveals that the plain Filon-Clenshaw-Curtis rules reach a given accuracy faster than the (adaptive) extended Filon-Clenshaw-Curtis rules. The comparison is based on the CPU run-time for certain wave numbers (medium and large).

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References

F. Bornemann, Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals, Found. Comput. Math., 11 (2011), pp. 1–63, https://doi.org/10.1007/s10208-010-9075-z. DOI: https://doi.org/10.1007/s10208-010-9075-z

F. Bornemann and G. Wechslberger, Optimal contours for high-order derivatives, IMA J. Numer. Anal., 33 (2013), pp. 403–412, https://doi.org/10.1093/imanum/drs030. DOI: https://doi.org/10.1093/imanum/drs030

A. Deano, D. Huybrechs, and A. Iserles, Computing Highly Oscillatory Integrals, SIAM, 2017, https://doi.org/10.1137/1.9781611975123. DOI: https://doi.org/10.1137/1.9781611975123

V. Dominguez, I. Graham, and T. Kim, Filon-Clenshaw-Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points, SIAM J. Numer. Anal., 51 (2013), pp. 1542–1566, https://doi.org/10.1137/120884146. DOI: https://doi.org/10.1137/120884146

V. Dominguez, I. Graham, and V. Smyshlyaev, Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals, IMA J. Numer. Anal., 31 (2011), pp. 1253–1280, https://doi.org/10.1093/imanum/drq036. DOI: https://doi.org/10.1093/imanum/drq036

A. Dutt, M. Gu, and V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM J. Numer. Anal., 33 (1996), pp. 1689–1711, https://doi.org/10.1137/0733082. DOI: https://doi.org/10.1137/0733082

J. Gao and A. Iserles, A generalization of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals, BIT Numer. Math., 57 (2017), pp. 943–961, https://doi.org/10.1007/s10543-017-0682-9. DOI: https://doi.org/10.1007/s10543-017-0682-9

L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325–348, https://doi.org/10.1016/0021-9991(87)90140-9. DOI: https://doi.org/10.1016/0021-9991(87)90140-9

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, 2008, https://doi.org/10.5555/1455489. DOI: https://doi.org/10.1137/1.9780898717761

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

A. Iserles and S. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. A, 461 (2005), pp. 1383–1399, https://doi.org/10.1098/rspa.2004.1401. DOI: https://doi.org/10.1098/rspa.2004.1401

G. Lantoine, R. Russell, and T. Dargent, Using multicomplex variables for automatic computation of high-order derivatives, ACM Trans. Math. Softw., 38 (2012), pp. 1–21, https://doi.org/10.1145/2168773.2168774. DOI: https://doi.org/10.1145/2168773.2168774

H. Majidian, Adaptive FCC+ rules for oscillatory integrals, J. Comput. Appl. Math., 424 (2023), p. 115012, https://doi.org/10.1016/j.cam.2022.115012. DOI: https://doi.org/10.1016/j.cam.2022.115012

H. Majidian, Efficient construction of FCC+ rules, J. Comput. Appl. Math., 417 (2023), p. 114592, https://doi.org/10.1016/j.cam.2022.114592. DOI: https://doi.org/10.1016/j.cam.2022.114592

H. Majidian, M. Firouzi, and A. Alipanah, On the stability of Filon-Clenshaw-Curtis rules, Bull. Iran. Math. Soc., 48 (2022), pp. 2943–2964, https://doi.org/10.1007/s41980-022-00681-4. DOI: https://doi.org/10.1007/s41980-022-00681-4

J. Martins, P. Sturdza, and J. Alonso, The complex-step derivative approximation, ACM Trans. Math. Softw., 29 (2003), pp. 245–262, https://doi.org/10.1145/838250.838251. DOI: https://doi.org/10.1145/838250.838251

H. Millwater and S. Shirinkam, Multicomplex Taylor series expansion for computing high order derivatives, Int. J. Appl. Math., 27 (2014), pp. 311–334, https://doi.org/10.12732/ijam.v27i4.2. DOI: https://doi.org/10.12732/ijam.v27i4.2

R. Neidinger, Introduction to automatic differentiation and MATLAB object-oriented programming, SIAM Rev., 52 (2010), pp. 545–563, https://doi.org/10.1137/080743627. DOI: https://doi.org/10.1137/080743627

M. Patterson, M. Weinstein, and A. Rao, An efficient overloaded method for computing derivatives of mathematical functions in MATLAB, ACM Trans. Math. Softw., 39 (2013), pp. 1–36, https://doi.org/10.1145/2450153.2450155. DOI: https://doi.org/10.1145/2450153.2450155

S. Xiang, X. Chen, and H. Wang, Error bounds for approximation in Chebyshev points, Numer. Math., 116 (2010), pp. 463–491, https://doi.org/10.1007/s00211-010-0309-4. DOI: https://doi.org/10.1007/s00211-010-0309-4

S. Xiang, G. He, and Y. Cho, On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals, Adv. Comput. Math., 41 (2015), pp. 573–597, https://doi.org/10.1007/s10444-014-9377-9. DOI: https://doi.org/10.1007/s10444-014-9377-9