Numerical methods for multivariate highly oscillatory integrals

References

• A. Asheim, Applying the numerical method of steepest descent on multivariate oscillatory integrals in scattering theory, arXiv preprint arXiv:1302.1019, 2013.
   Google Scholar
• A. Asheim and D. Huybrechs, Asymptotic analysis of numerical steepest descent with path approximations, Found. Comput. Math. 10 (2010), pp. 647–671. doi: 10.1007/s10208-010-9068-y
   Web of Science ®Google Scholar
• I. Aziz, Siraj-ul-Islam, and W. Khan, Quadrature rules for numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appl. 61 (2011), pp. 2770–2781. doi: 10.1016/j.camwa.2011.03.043
   Web of Science ®Google Scholar
• C. Bouwkamp, Difraction theory, Rep. Progr. Phys. 17 (1954), pp. 35–100. doi: 10.1088/0034-4885/17/1/302
   Web of Science ®Google Scholar
• P.J. Harris and Ke Chen, An efficient method for evaluating the integral of a class of highly oscillatory functions, J. Comput. Appl. Math. 230 (2009), pp. 433–442. doi: 10.1016/j.cam.2008.12.026
   Web of Science ®Google Scholar
• D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SAIM J. Numer. Anal. 44 (2006), pp. 1026–1048. doi: 10.1137/050636814
   Web of Science ®Google Scholar
• D. Huybrechs and S. Vandewalle, The construction of cubature rules for multivariate highly oscillatory integrals, Math. Comput. 76 (2007), pp. 1955–1980. doi: 10.1090/S0025-5718-07-01937-0
   Web of Science ®Google Scholar
• A. Iserles and S. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comput. 75(255) (2006), pp. 1233–1258. doi: 10.1090/S0025-5718-06-01854-0
   Web of Science ®Google Scholar
• A. Iserles and S.P. Nørsett, Efficient quadrature of highly oscillatory integrals, Proc. Roy. Soc. 461 (2005), pp. 1383–1399. doi: 10.1098/rspa.2004.1401
   Google Scholar
• A. Iserles and S.P. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comput. 75 (2006), pp. 1233–1258. doi: 10.1090/S0025-5718-06-01854-0
   Web of Science ®Google Scholar
• A. Iserles and S.P. Nørsett, On the computation of highly oscillatory multivariate integrals with critical points, BIT 46 (2006), pp. 549–566. doi: 10.1007/s10543-006-0071-2
   Web of Science ®Google Scholar
• S. Langdon, S.N. Chandler-Wilde, I.G. Graham, and E.A. Spence, Numerical asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer. 21 (2012), pp. 89–305. doi: 10.1017/S0962492912000037
   Web of Science ®Google Scholar
• D. Levin, Procedures for computing one and two-dimensional integrals of functions with rapid irregular oscillations, Math. Comput. 158 (1982), pp. 531–538. doi: 10.1090/S0025-5718-1982-0645668-7
   Google Scholar
• J. Li, X. Wang, T. Wang, and C. Shen, Delaminating quadrature method for multi-dimensional highly oscillatory integrals, J. Comput. Appl. Math. 209 (2009), pp. 327–338. doi: 10.1016/j.amc.2008.12.061
   Web of Science ®Google Scholar
• J. Li, X. Wang, T. Wang, and S. Xiao, An improved Levin quadrature method for highly oscillatory integrals, J. Appl. Numer. Math. 60 (2010), pp. 833–842. doi: 10.1016/j.apnum.2010.04.009
   Web of Science ®Google Scholar
• A. Miks, J. Novak, and P. Novak, Method for numerical integration of rapidly oscillating functions in diffraction theory, Internat. J. Numer. Methods Engrg. 82 (2010), pp. 525–536. doi: 10.1002/nme.2783
   Web of Science ®Google Scholar
• J. Nedelec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer, NY, 2001, 144.
   Google Scholar
• S. Olver, On the quadrature of multivariate highly oscillatory integrals over non-polytope domains, Numer. Math. 103(4) (2006), pp. 643–665. doi: 10.1007/s00211-006-0009-2
   Web of Science ®Google Scholar
• S. Olver, Numerical approximation of highly oscillatory integrals, Ph.D thesis, 2008.
   Google Scholar
• S. Olver, Fast, numerically stable computation of oscillatory integrals with stationary points, BIT 50 (2010), pp. 149–171. doi: 10.1007/s10543-010-0251-y
   Web of Science ®Google Scholar
• A. Pierc, Acoustics: An introduction to its physical principles and applications, McGraw-Hill Book Co, New York, 1981.
   Google Scholar
• Siraj-ul-Islam, A.S. Al-Fhaid, and S. Zaman, Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points, Eng. Anal. Bound. Elem. 37 (2013), pp. 1136–1144. doi: 10.1016/j.enganabound.2013.04.012
   Web of Science ®Google Scholar
• Siraj-ul-Islam, I. Aziz, and Fazal-e-Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appl. 59(6) (2010), pp. 2026–2036. doi: 10.1016/j.camwa.2009.12.005
   Web of Science ®Google Scholar
• Siraj-ul-Islam, I. Aziz, and W. Khan, Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions, Eng. Anal. Bound. Elem. 36 (2012), pp. 1284–1295. doi: 10.1016/j.enganabound.2012.01.008
   Web of Science ®Google Scholar
• Siraj-ul-Islam and S. Zaman, New quadrature rules for highly oscillatory integrals with stationary points, J. Comput. Appl. Math. 278 (2015), pp. 75–89. doi: 10.1016/j.cam.2014.09.019
   Web of Science ®Google Scholar
• E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscilitory Integrals, Princeton Unversity Press, Princeton NJ, 1993.
   Google Scholar
• R. Wong, Asymptotic approximation of integrals, SIAM, Philadelphia, 2001.
   Google Scholar
• S. Xiang, Efficient quadrature for highly oscillatory integrals involving critical points, J. Comput. Appl. Math. 206 (2007), pp. 688–698. doi: 10.1016/j.cam.2006.08.018
   Web of Science ®Google Scholar
• S. Zaman and Siraj-ul-Islam, Efficient numerical methods for bessel type of oscillatory integrals, J. Comput. Appl. Math. 315 (2017), pp. 161–174. doi: 10.1016/j.cam.2016.10.032
   Web of Science ®Google Scholar