Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


Error estimates for Gaussian beam superpositions

Authors: Hailiang Liu, Olof Runborg and Nicolay M. Tanushev
Journal: Math. Comp. 82 (2013), 919-952
MSC (2010): Primary 35J10, 35L05, 35A35, 41A60, 35L30
Published electronically: November 6, 2012
MathSciNet review: 3008843
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrödinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength $ \varepsilon $. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrödinger equations subject to highly oscillatory initial data of the form $ Ae^{i\Phi /\varepsilon }$. Through a careful estimate of an oscillatory integral operator, we prove that the $ k$-th order Gaussian beam superposition converges to the original wave field at a rate proportional to $ \varepsilon ^{k/2}$ in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, $ \varepsilon $-scaled, energy norm and for the Schrödinger equation in the $ L^2$ norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in $ \mathbb{R}^2$ to analyze the sharpness of the theoretical results.

References [Enhancements On Off] (What's this?)

  • 1. Gil Ariel, Björn Engquist, Nicolay M. Tanushev, and Richard Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization, J. Comput. Phys. 230 (2011), no. 6, 2303–2321. MR 2764548, 10.1016/
  • 2. V. M. Babič and V. S. Buldyrev, Short-wavelength diffraction theory, Springer Series on Wave Phenomena, vol. 4, Springer-Verlag, Berlin, 1991. Asymptotic methods; Translated from the 1972 Russian original by E. F. Kuester. MR 1245488
  • 3. V. M. Babič and T. F. Pankratova, The discontinuities of the Green’s function for a mixed problem for the wave equation with a variable coefficient, Problems of mathematical physics, No. 6 (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 9–27 (Russian). MR 0355352
  • 4. V. M. Babich and M. M. Popov, The method of summing Gaussian beams (a survey), Izv. Vyssh. Uchebn. Zaved. Radiofiz. 32 (1989), no. 12, 1447–1466 (Russian); English transl., Radiophys. and Quantum Electronics 32 (1989), no. 12, 1063–1081 (1990). MR 1045524, 10.1007/BF01038632
  • 5. Salma Bougacha, Jean-Luc Akian, and Radjesvarane Alexandre, Gaussian beams summation for the wave equation in a convex domain, Commun. Math. Sci. 7 (2009), no. 4, 973–1008. MR 2604628
  • 6. V. Červený, M. M. Popov, and I. Pšenčík.
    Computation of wave fields in inhomogeneous media -- Gaussian beam approach.
    Geophys. J. R. Astr. Soc., 70:109-128, 1982.
  • 7. Björn Engquist and Olof Runborg, Computational high frequency wave propagation, Acta Numer. 12 (2003), 181–266. MR 2249156, 10.1017/S0962492902000119
  • 8. N. R. Hill.
    Gaussian beam migration.
    Geophysics, 55(11):1416-1428, 1990.
  • 9. N. R. Hill.
    Prestack Gaussian beam depth migration.
    Geophysics, 66(4):1240-1250, 2001.
  • 10. Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 0388463
  • 11. Lars Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Enseignement Math. (2) 17 (1971), 99–163. MR 0331124
  • 12. Lars Hörmander, The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773
  • 13. Lars Hörmander, The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007. Pseudo-differential operators; Reprint of the 1994 edition. MR 2304165
  • 14. Shi Jin, Hao Wu, and Xu Yang, Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Commun. Math. Sci. 6 (2008), no. 4, 995–1020. MR 2511703
  • 15. A. P. Katchalov and M. M. Popov.
    Application of the method of summation of Gaussian beams for calculation of high-frequency wave fields.
    Sov. Phys. Dokl., 26:604-606, 1981.
  • 16. Joseph B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems., Ann. Physics 4 (1958), 180–188. MR 0099207
  • 17. L. Klimeš.
    Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams.
    Geophys. J. R. Astr. Soc., 79:105-118, 1984.
  • 18. L. Klimeš.
    Discretization error for the superposition of Gaussian beams.
    Geophys. J. R. Astr. Soc., 86:531-551, 1986.
  • 19. Yu. A. Kravtsov.
    On a modification of the geometrical optics method.
    Izv. VUZ Radiofiz., 7(4):664-673, 1964.
  • 20. Shingyu Leung and Jianliang Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys. 228 (2009), no. 8, 2951–2977. MR 2509304, 10.1016/
  • 21. S. Leung, J. Qian, and R. Burridge.
    Eulerian Gaussian beams for high frequency wave propagation.
    Geophysics, 72:SM61-SM76, 2007.
  • 22. Hailiang Liu and James Ralston, Recovery of high frequency wave fields for the acoustic wave equation, Multiscale Model. Simul. 8 (2009/10), no. 2, 428–444. MR 2581028, 10.1137/090761598
  • 23. Hailiang Liu and James Ralston, Recovery of high frequency wave fields from phase space-based measurements, Multiscale Model. Simul. 8 (2009/10), no. 2, 622–644. MR 2600299, 10.1137/090756909
  • 24. Donald Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math. 19 (1966), 215–250. MR 0196254
  • 25. V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics, vol. 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. Translated from the Russian by J. Niederle and J. Tolar; Contemporary Mathematics, 5. MR 634377
  • 26. M. Motamed and O. Runborg.
    A wave front Gaussian beam method for high-frequency wave propagation.
    In Proceedings of WAVES 2007, University of Reading, UK, 2007.
  • 27. Mohammad Motamed and Olof Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion 47 (2010), no. 7, 421–439. MR 2684020, 10.1016/j.wavemoti.2010.02.001
  • 28. M. M. Popov, A new method of computation of wave fields using Gaussian beams, Wave Motion 4 (1982), no. 1, 85–97. MR 638867, 10.1016/0165-2125(82)90016-6
  • 29. Jianliang Qian and Lexing Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation, J. Comput. Phys. 229 (2010), no. 20, 7848–7873. MR 2674307, 10.1016/
  • 30. James Ralston, Gaussian beams and the propagation of singularities, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 206–248. MR 716507
  • 31. V. Rousse and T. Swart.
    Global $ {L}^2$-boundedness theorems for semiclassical Fourier integral operators with complex phase.
    arxiv:0710.4200v3, 2007.
  • 32. Torben Swart and Vidian Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys. 286 (2009), no. 2, 725–750. MR 2472042, 10.1007/s00220-008-0681-4
  • 33. Olof Runborg, Mathematical models and numerical methods for high frequency waves, Commun. Comput. Phys. 2 (2007), no. 5, 827–880. MR 2355631
  • 34. Nicolay M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci. 6 (2008), no. 2, 449–475. MR 2433704
  • 35. Nicolay M. Tanushev, Björn Engquist, and Richard Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys. 228 (2009), no. 23, 8856–8871. MR 2558781, 10.1016/
  • 36. Nicolay M. Tanushev, Jianliang Qian, and James V. Ralston, Mountain waves and Gaussian beams, Multiscale Model. Simul. 6 (2007), no. 2, 688–709 (electronic). MR 2338499, 10.1137/060673667

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35J10, 35L05, 35A35, 41A60, 35L30

Retrieve articles in all journals with MSC (2010): 35J10, 35L05, 35A35, 41A60, 35L30

Additional Information

Hailiang Liu
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Olof Runborg
Affiliation: Department of Numerical Analysis, CSC, KTH, 100 44 Stockholm, Sweden – and – Swedish e-Science Research Center (SeRC), KTH, 100 44 Stockholm, Sweden

Nicolay M. Tanushev
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712
Address at time of publication: Z-Terra Inc., 17171 Park Row, Suite 247, Houston, Texas 77084

Keywords: High-frequency wave propagation, error estimates, Gaussian beams
Received by editor(s): August 6, 2010
Received by editor(s) in revised form: June 2, 2011, and August 31, 2011
Published electronically: November 6, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.