Abstract
We present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying, two-dimensional time-independent periodic media. The method consists in representing the time-dependent Green’s function in wave atoms, a tight frame of multiscale, directional wave packets obeying a precise parabolic balance between oscillations and support size, namely wavelength ~(diameter).2 Wave atoms offer a uniquely structured representation of the Green’s function in the sense that
-
the resulting matrix is universally sparse over the class of C ∞ coefficients, even for “large” times;
-
the matrix has a natural low-rank block-structure after separation of the spatial indices.
The parabolic scaling is essential for these properties to hold. As a result, it becomes realistic to accurately build the full matrix exponential in the wave atom frame, using repeated squaring up to some time typically of the form \({\Delta t \sim \sqrt{\Delta x}}\) , which is bigger than the standard CFL timestep. Once the “expensive” precomputation of the Green’s function has been carried out, it can be used to perform unusually large, upscaled, “cheap” time steps. The algorithm is relatively simple in that it does not require an underlying geometric optics solver. We prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks. On a N-by-N grid, the “expensive” precomputation takes somewhere between O(N 3log N) and O(N 4log N) steps depending on the separability of the acoustic medium. The complexity of upscaled timestepping, however, beats the O(N 3log N) bottleneck of pseudospectral methods on an N-by-N grid, for a wide range of physically relevant situations. In particular, we show that a naive version of the wave atom algorithm provably runs in O(N 2+δ) operations for arbitrarily small δ—but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant. As a result, we get estimates between O(N 2.5 log N) and O(N 3 log N) for upscaled timestepping. We also show several numerical examples. In practice, the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions, as in reflection seismology. In academic examples of accurate propagation of bandlimited wavefronts, if the precomputation step is factored out, then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3–5 at N = 512, and a factor 10–20 at N = 1024, for the same accuracy. Very similar gains are obtained in comparison versus a finite difference method.
Similar content being viewed by others
References
Babenko Yu.V.: Pointwise inequalities of Landau-Kolmogorov-type for functions defined on a finite segment. Ukr. Math. J. 53(2), 270–275 (2001)
Bacry E., Mallat S., Papanicolaou G.: A wavelet based space-time adaptive numerical method for partial differential equations. Math. Model. Num. Anal. 26(7), 793 (1992)
Beylkin G., Coifman R.R., Rokhlin V.: Fast wavelet transforms and numerical algorithms. Comm. Pure Appl. Math. 44, 141–183 (1991)
Beylkin G., Mohlenkamp M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26(6), 2133–2159 (2005)
Beylkin G., Sandberg K.: Wave propagation using bases for bandlimited functions. Wave Motion 41(3), 263–291 (2005)
Candès E.J.: Harmonic analysis of neural networks. Appl. Comput. Harmon. Anal. 6, 197–218 (1999)
Candès E.J., Demanet L.: Curvelets and Fourier integral operators. C. R. Acad. Sci. Paris Ser. I 336, 395–398 (2003)
Candès E.J., Demanet L.: The curvelet representation of wave propagators is optimally sparse. Comm. Pure Appl. Math. 58(11), 1472–1528 (2005)
Candès, E.J., Demanet, L., Donoho, D.L., Ying, L.: Fast discrete curvelet transforms. SIAM Mult. Model. Sim. (2005, submitted)
Candès, E.J., Donoho, D.L.: Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In: Rabut, C., Cohen, A., Schumaker, L.L. (eds.) Curves and Surfaces, pp.105–120. Vanderbilt University Press, Nashville (2000)
Candès E.J., Donoho D.L.: New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Comm. Pure Appl. Math. 57, 219–266 (2004)
Cohen A.: Numerical Analysis of Wavelet Methods. North-Holland/Elsevier, Amsterdam (2003)
Cohen A., Dahmen W., DeVore R.: Adaptive wavelet methods for elliptic operator equations— Convergence rates. Math. Comp. 70(233), 27–75 (2000)
Córdoba A., Fefferman C.: Wave packets and Fourier integral operators. Comm. PDE 3(11), 979–1005 (1978)
Demanet, L.: Curvelets, wave atoms, and wave equations. Ph.D. Thesis, California Institute of Technology, May (2006)
Demanet, L., Ying, L.: Wave atoms and sparsity of oscillatory patterns (2006, submitted)
Douma, H., de Hoop, M.V.: Wave-character preserving prestack map migration using curvelets. Presentation at the Society of Exploration Geophysicists, Denver, CO (2004)
Duistermaat J.: Fourier Integral Operators. Birkhauser, Boston (1996)
Engquist B., Osher S., Zhong S.: Fast wavelet based algorithms for linear evolution equations. SIAM J. Sci. Comput. 15(4), 755–775 (1994)
Fefferman C.: A note on spherical summation multipliers. Israel J. Math. 15, 44–52 (1973)
Flesia, A.G., Hel-Or, H., Averbuch, A., Candès, E.J., Coifman, R.R., Donoho, D.L.: Digital implementation of ridgelet packets. In: Stoeckler, J., Welland, G.V. (eds.) Beyond Wavelets. Academic Press, London (2003)
Hackbusch W.: A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices. Computing 62, 89–108 (1999)
Hennenfent, G., Herrmann, F.J.: Seismic denoising with unstructured curvelets. Comput. Sci. Eng. (2006, to appear)
Herrmann, F.J., Moghaddam, P.P., Stolk, C.C.: Sparsity- and continuity-promoting seismic image recovery with curvelet frames. (2006, submitted)
Hoop M.V., Rousseau J.H., Wu R.: Generlization of the phase-screen approximation for the scattering of acoustic waves. Wave Motion 31(1), 43–70 (2000)
Hörmander L.: The Analysis of Linear Partial Differential Operators, vol.4. Springer, Heidelberg (1985)
Jaffard S.: Wavelet methods for fast resolution of elliptic problems. SIAM J. Numer. Anal. 29(4), 965–986 (1992)
Lax P.: Asymptotic solutions of oscillatory initial value problems. Duke Math J. 24, 627–646 (1957)
LeVeque R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, London (2002)
Meyer Y.: Ondelettes et Opérateurs. Hermann, Paris (1990)
Meyer Y., Coifman R.R.: Wavelets, Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997)
Moler C., Van Loan C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)
Seeger A., Sogge C., Stein E.: Regularity properties of Fourier integral operators. Ann. Math. 134, 231–251 (1991)
Smith H.: A Hardy space for Fourier integral operators. J. Geom. Anal. 8, 629–653 (1998)
Smith H.: A parametrix construction for wave equations with C 1,1 coefficients. Ann. Inst. Fourier (Grenoble) 48, 797–835 (1998)
Stein E.: Harmonic Analysis. Princeton University Press, Princeton (1993)
Ying, L., Candès, E.J.: The phase-flow method. J. Comput. Phys. (2006, to appear)
Ziemer W.: Weakly Differentiable Functions. Springer, Heidelberg (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Demanet, L., Ying, L. Wave atoms and time upscaling of wave equations. Numer. Math. 113, 1–71 (2009). https://doi.org/10.1007/s00211-009-0226-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-009-0226-6