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Original Articles

On fast multipole methods for Fredholm integral equations of the second kind with singular and highly oscillatory kernels

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Pages 1391-1411 | Received 17 Jun 2018, Accepted 16 Apr 2019, Published online: 20 May 2019

References

  • M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964.
  • A. Amritkar, E. de Sturler, K. Świrydowicz, D. Tafti, and K. Ahuja, Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver, J. Comput. Phys. 303 (2015), pp. 222–237. doi: 10.1016/j.jcp.2015.09.040
  • G. Biros, L. Ying, and D. Zorin, A fast solver for the Stokes equations with distributed forces in complex geometries, J. Comput. Phys. 193 (2003), pp. 317–348. doi: 10.1016/j.jcp.2003.08.011
  • H. Brunner, A. Iserles, and S.P. Nørsett, The spectral probelm for a class of highly oscillatory Fredholm integral operators, IMA J. Numer. Anal. 30 (2010), pp. 108–130. doi: 10.1093/imanum/drn060
  • R.J. Burkholder and D.H. Kwon, High-frequency asymptotic acceleration of the fast multipole method, Radio Sci. 31 (2016), pp. 1199–1206. doi: 10.1029/96RS01785
  • M.R. Capobianco and G. Criscuolo, On quadrature for Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math. 156 (2003), pp. 471–486. doi: 10.1016/S0377-0427(03)00388-1
  • J. Carrier, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Statist. Comput. 9 (1988), pp. 669–686. doi: 10.1137/0909044
  • C. Cecka and E. Darve, Fourier-based fast multipole method for the Helmholtz equation, SIAM J. Sci. Comput 35 (2013), pp. A79–A103. doi: 10.1137/11085774X
  • R. Chen, Fast integration for Cauchy principal value integrals of oscillatory kind, Acta Appl. Math. 123 (2013), pp. 21–30. doi: 10.1007/s10440-012-9709-z
  • H. Cheng, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys. 155 (1999), pp. 468–498. doi: 10.1006/jcph.1999.6355
  • H. Cheng, W.Y. Crutchfield, Z. Gimbutas, L.F. Greengard, J.F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, A wideband fast multipole method for the Helmholtz equation in three dimensions, J. Comput. Phys. 216 (2006), pp. 300–325. doi: 10.1016/j.jcp.2005.12.001
  • W.C. Chew, J.M. Jin, E. Michielssen, and J. Song (eds.), Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, 2001.
  • M. Cho and W. Cai, A wideband fast multipole method for the two-dimensional complex Helmholtz equation, Comput. Phys. Commun. 181 (2010), pp. 2086–2090. doi: 10.1016/j.cpc.2010.09.010
  • G. Criscuolo, A new algorithm for Cauchy principle value and Hadamard finite-part integrals, J. Comput. Appl. Math. 78 (1997), pp. 255–275. doi: 10.1016/S0377-0427(96)00142-2
  • E. Darve, The fast multipole method: A numerical implementation, J. Comput. Phys. 160 (2000), pp. 195–240. doi: 10.1006/jcph.2000.6451
  • P. David, C. Cecka, and E. Darve, Cauchy fast multipole method for general analytic kernels, SIAM J. Sci. Comput 36 (2014), pp. A396-A426.
  • V. Domínguez, Filon-Clenshaw-Curtis rules for a class of highly oscillatory integrals with logarithmic singularities, J. Comput. Appl. Math. 261 (2014), pp. 299–319. doi: 10.1016/j.cam.2013.11.012
  • V. Domínguez, I.G. 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. doi: 10.1137/120884146
  • A. Dutt, M. Gu, and V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM J. Numer. Anal. 33 (1996), pp. 1689–1711. doi: 10.1137/0733082
  • B. Engquist and L. Ying, Fast directional multilevel algorithms for the oscillatory kernels, SIAM J. Sci. Comput. 29 (2007), pp. 1710–1737. doi: 10.1137/07068583X
  • W. Fong and E. Darve, The black-box fast multipole method, J. Comput. Phys. 228 (2009), pp. 8712–8725. doi: 10.1016/j.jcp.2009.08.031
  • Z. Gimbutas and V. Rokhlin, A generalized fast multipole method for nonoscillatory kernels, SIAM J. Sci. Comput. 24 (2002), pp. 796–817. doi: 10.1137/S1064827500381148
  • L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge, MA, 1988.
  • L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), pp. 325–348. doi: 10.1016/0021-9991(87)90140-9
  • L. Greengard and J. Strain, A fast algorithm for the evaluation of heat potentials, Comm. Pure. Appl. Math. 43 (1990), pp. 949–963. doi: 10.1002/cpa.3160430802
  • L.F. Greengard and J. Strain, The fast Gauss transform, SIAM J. Sci. Stat. Comput. 12 (1991), pp. 79–94. doi: 10.1137/0912004
  • G. He and S. Xiang, An improved algorithm for the evaluation of Cauchy principal value integrals of oscillatory functions and its application, J. Comput. Appl. Math. 280 (2015), pp. 1–13. doi: 10.1016/j.cam.2014.11.023
  • G. He, S. Xiang, and Z. Xu, A Chebyshev collocation method for a class of Fredholm integral equations with highly oscillatory kernels, J. Comput. Appl. Math. 300 (2016), pp. 354–368. doi: 10.1016/j.cam.2015.12.027
  • T. Hrycak and V. Rokhlin, An improved fast multipole algorithm for potential fields, SIAM J. Sci. Comput. 19 (1998), pp. 1804–1826. doi: 10.1137/S106482759630989X
  • D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (2006), pp. 1026–1048. doi: 10.1137/050636814
  • A. Iserles, On the numerical quadrature of highly-oscillating integrals. I. Fourier transforms, IMA J. Numer. Anal. 24 (2004), pp. 365–391. doi: 10.1093/imanum/24.3.365
  • A. Iserles, On the numerical quadrature of highly-oscillating integrals II: Fourier transforms, IMA J. Numer. Anal. 25 (2005), pp. 25–44. doi: 10.1093/imanum/drh022
  • P.K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Birkhauser, Boston, 2002.
  • J. Li, X. Wang, and T. Wang, Evaluation of Cauchy principal value integrals of oscillatory kind, Appl. Math. Comput. 217 (2010), pp. 2390–2396.
  • J. Li, X. Wang, S. Xiao, and T. Wang, A rapid solution of a kind of 1D Fredholm oscillatory integral equation, J. Comput. Appl. Math. 236 (2012), pp. 2696–2705. doi: 10.1016/j.cam.2012.01.007
  • Y. Liu, Fast Multipole Boundary Elementmethod: Theory and Applications in Engineering, Cambridge University Press, Cambridge, 2009.
  • G. Liu and S. Xiang, Fast multipole methods for approximating a function from sampling values, Numer. Algor. 76 (2017), pp. 727–743. doi: 10.1007/s11075-017-0279-z
  • M. Messner, M. Shanz, and E. Darve, Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation, J. Comput. Phys. 231 (2012), pp. 1175–1196. doi: 10.1016/j.jcp.2011.09.027
  • G. Monegato, Definitions, properties and applications of finite-part integrals, J. Comput. Appl. Math. 229 (2009), pp. 425–439. doi: 10.1016/j.cam.2008.04.006
  • N. Nishimura, Fast multipole accelerated boundary integral equation methods, Appl. Mech. Rev. 55 (2002), pp. 299–324. doi: 10.1115/1.1482087
  • G.E. Okecha, Quadrature formulae for Cauchy principal value integrals of oscillatory kind, Math. Comp. 49 (1987), pp. 259–268. doi: 10.1090/S0025-5718-1987-0890267-X
  • G.E. Okecha, Hermite interpolation and a method for evaluating Cauchy principal value integrals of oscillatory kind, Kragujevac J. Math. 29 (2006), pp. 91–98.
  • V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60 (1985), pp. 187–207. doi: 10.1016/0021-9991(85)90002-6
  • V. Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions, J. Comput. Phys. 86 (1990), pp. 414–439. doi: 10.1016/0021-9991(90)90107-C
  • V. Rokhlin, Diagonal forms of translation operators for the helmholtz equation in three dimensions, Tech. Rep., Yale University, New Haven, CT, 1992.
  • V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal. 1 (1993), pp. 82–93. doi: 10.1006/acha.1993.1006
  • E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993.
  • L.N. Trefethen, Chebfun Version 4.2, The Chebfun Development Team, 2011; software available at http://www.maths.ox.ac.uk/chebfun/.
  • J.L. Tsalamengas, Gauss-Jacobi quadratures for weakly, strongly, hyper-and nearly-singular integrals in boundary integral equation methods for domains with sharp edges and corners, J. Comput. Phys. 325 (2016), pp. 338–357. doi: 10.1016/j.jcp.2016.07.041
  • F. Ursell, Integral equations with a rapidly oscillating kernel, J. Lond. Math. Soc. 44 (1969), pp. 449–459. doi: 10.1112/jlms/s1-44.1.449
  • H. Wang and S. Xiang, Uniform approximations to Cauchy principal value integrals of oscillatory functions, Appl. Math. Comput. 215 (2009), pp. 1886–1894.
  • H. Wang and S. Xiang, On the evaluation of Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math. 234 (2010), pp. 95–100. doi: 10.1016/j.cam.2009.12.007
  • S. Xiang, Efficient Filon-type methods for ∫abf(x)eiwg(x)dx, Numer. Math. 105 (2007), pp. 633–658. doi: 10.1007/s00211-006-0051-0
  • S. Xiang and Q. Wu, Numerical solutions to Volterra integral equations of the second kind with oscillatory trigonometric kernels, Appl. Math. Comput. 223 (2013), pp. 34–44.
  • 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. doi: 10.1007/s10444-014-9377-9
  • S. Xiang, C. Fang, and Z. Xu, On uniform approximations to hypersingular finite-part integrals, J. Math. Anal. Appl. 435 (2016), pp. 1210–1228. doi: 10.1016/j.jmaa.2015.11.002
  • N. Yarvin and V. Rokhlin, An improved fast multipole algorithm for potential fields on the line, SIAM J. Numer. Anal. 36 (1999), pp. 629–666. doi: 10.1137/S0036142997329232
  • L. Ying, A kernel-independent fast multipole algorithm for radial basis functions, J. Comput. Phys. 213 (2006), pp. 451–457. doi: 10.1016/j.jcp.2005.09.010

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