Wave atoms and time upscaling of wave equations

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 packe...

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Published in:	Numerische Mathematik Vol. 113; no. 1; pp. 1 - 71
Main Authors:	Demanet, Laurent, Ying, Lexing
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer-Verlag 01.07.2009 Springer
Subjects:	Exact sciences and technology Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical analysis. Scientific computation Numerical and Computational Physics Numerical linear algebra Ordinary differential equations Partial differential equations Sciences and techniques of general use Simulation Special functions Theoretical Primary 65M99 Secoundary 42C99 Wave equation Grid pattern Initial condition Numerical linear algebra Acoustics Algorithm Green function Wavefront Numerical analysis Pseudospectral method Numerical solution Hyperbolic equation Oscillation A priori estimation Condition number Two-dimensional calculations Finite difference method
ISSN:	0029-599X, 0945-3245
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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 , 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 3 log N ) and O ( N 4 log N ) steps depending on the separability of the acoustic medium. The complexity of upscaled timestepping, however, beats the O ( N 3 log 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.
AbstractList	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 , 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 3 log N ) and O ( N 4 log N ) steps depending on the separability of the acoustic medium. The complexity of upscaled timestepping, however, beats the O ( N 3 log 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.
Author	Demanet, Laurent Ying, Lexing
Author_xml	– sequence: 1 givenname: Laurent surname: Demanet fullname: Demanet, Laurent email: demanet@gmail.com organization: Department of Mathematics, Stanford University – sequence: 2 givenname: Lexing surname: Ying fullname: Ying, Lexing organization: Department of Mathematics, University of Texas at Austin
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Cites_doi	10.1215/S0012-7094-57-02471-7 10.1137/0915048 10.1016/j.wavemoti.2004.05.008 10.1137/S00361445024180 10.2307/2944346 10.1006/acha.1998.0248 10.1002/cpa.3160440202 10.1007/BF02771772 10.1016/S0165-2125(99)00026-8 10.1090/S0025-5718-00-01252-7 10.1137/040604959 10.1002/cpa.20078 10.1080/03605307808820083 10.1137/0729059 10.1023/A:1010469004645 10.1007/s006070050015 10.1002/cpa.10116 10.1016/S1631-073X(03)00095-5 10.1017/CBO9780511791253 10.1137/05064182X 10.1109/MCSE.2006.49 10.1016/j.acha.2007.03.003 10.1051/m2an/1992260707931 10.1016/j.jcp.2006.05.008 10.1016/S1570-579X(03)80030-7 10.1190/1.1845326 10.1007/978-1-4612-1015-3 10.5802/aif.1640 10.1007/BF02921717
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Keywords	Primary 65M99 Secoundary 42C99 Wave equation Grid pattern Initial condition Numerical linear algebra Acoustics Algorithm Green function Wavefront Numerical analysis Pseudospectral method Numerical solution Hyperbolic equation Oscillation A priori estimation Condition number Two-dimensional calculations Finite difference method
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References	de Hoop, le Rousseau, Wu (CR25) 2000; 31 Duistermaat (CR18) 1996 Smith (CR35) 1998; 48 Candès (CR6) 1999; 6 Jaffard (CR27) 1992; 29 Meyer (CR30) 1990 CR17 Hackbusch (CR22) 1999; 62 CR16 Candès, Demanet (CR7) 2003; 336 CR15 CR37 Beylkin, Coifman, Rokhlin (CR3) 1991; 44 Moler, Van Loan (CR32) 2003; 45 Stein (CR36) 1993 CR10 Lax (CR28) 1957; 24 Candès, Donoho (CR11) 2004; 57 Smith (CR34) 1998; 8 Beylkin, Mohlenkamp (CR4) 2005; 26 Seeger, Sogge, Stein (CR33) 1991; 134 Babenko (CR1) 2001; 53 Córdoba, Fefferman (CR14) 1978; 3 Fefferman (CR20) 1973; 15 Hörmander (CR26) 1985 LeVeque (CR29) 2002 Meyer, Coifman (CR31) 1997 Bacry, Mallat, Papanicolaou (CR2) 1992; 26 Ziemer (CR38) 1989 Cohen, Dahmen, DeVore (CR13) 2000; 70 Candès, Demanet (CR8) 2005; 58 CR9 Cohen (CR12) 2003 CR24 CR23 CR21 Beylkin, Sandberg (CR5) 2005; 41 Engquist, Osher, Zhong (CR19) 1994; 15 A. Cohen (226_CR12) 2003 E. Stein (226_CR36) 1993 226_CR10 M.V. Hoop de (226_CR25) 2000; 31 A. Cohen (226_CR13) 2000; 70 226_CR17 E.J. Candès (226_CR7) 2003; 336 226_CR9 226_CR15 226_CR37 226_CR16 G. Beylkin (226_CR3) 1991; 44 B. Engquist (226_CR19) 1994; 15 S. Jaffard (226_CR27) 1992; 29 W. Hackbusch (226_CR22) 1999; 62 Y. Meyer (226_CR30) 1990 H. Smith (226_CR35) 1998; 48 E.J. Candès (226_CR8) 2005; 58 E.J. Candès (226_CR11) 2004; 57 H. Smith (226_CR34) 1998; 8 J. Duistermaat (226_CR18) 1996 P. Lax (226_CR28) 1957; 24 C. Fefferman (226_CR20) 1973; 15 C. Moler (226_CR32) 2003; 45 G. Beylkin (226_CR5) 2005; 41 W. Ziemer (226_CR38) 1989 E.J. Candès (226_CR6) 1999; 6 226_CR21 E. Bacry (226_CR2) 1992; 26 226_CR24 226_CR23 Yu.V. Babenko (226_CR1) 2001; 53 Y. Meyer (226_CR31) 1997 L. Hörmander (226_CR26) 1985 A. Seeger (226_CR33) 1991; 134 R.J. LeVeque (226_CR29) 2002 A. Córdoba (226_CR14) 1978; 3 G. Beylkin (226_CR4) 2005; 26
References_xml	– volume: 24 start-page: 627 year: 1957 end-page: 646 ident: CR28 article-title: Asymptotic solutions of oscillatory initial value problems publication-title: Duke Math J. doi: 10.1215/S0012-7094-57-02471-7 – volume: 15 start-page: 755 issue: 4 year: 1994 end-page: 775 ident: CR19 article-title: Fast wavelet based algorithms for linear evolution equations publication-title: SIAM J. Sci. Comput. doi: 10.1137/0915048 – volume: 8 start-page: 629 year: 1998 end-page: 653 ident: CR34 article-title: A Hardy space for Fourier integral operators publication-title: J. Geom. Anal. – ident: CR16 – ident: CR37 – volume: 41 start-page: 263 issue: 3 year: 2005 end-page: 291 ident: CR5 article-title: Wave propagation using bases for bandlimited functions publication-title: Wave Motion doi: 10.1016/j.wavemoti.2004.05.008 – year: 1997 ident: CR31 publication-title: Wavelets, Calderón-Zygmund and Multilinear Operators – volume: 48 start-page: 797 year: 1998 end-page: 835 ident: CR35 article-title: A parametrix construction for wave equations with coefficients publication-title: Ann. Inst. Fourier (Grenoble) – ident: CR10 – volume: 45 start-page: 3 issue: 1 year: 2003 end-page: 49 ident: CR32 article-title: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later publication-title: SIAM Rev. doi: 10.1137/S00361445024180 – volume: 134 start-page: 231 year: 1991 end-page: 251 ident: CR33 article-title: Regularity properties of Fourier integral operators publication-title: Ann. Math. doi: 10.2307/2944346 – year: 2002 ident: CR29 publication-title: Finite Volume Methods for Hyperbolic Problems – volume: 6 start-page: 197 year: 1999 end-page: 218 ident: CR6 article-title: Harmonic analysis of neural networks publication-title: Appl. Comput. Harmon. Anal. doi: 10.1006/acha.1998.0248 – volume: 44 start-page: 141 year: 1991 end-page: 183 ident: CR3 article-title: Fast wavelet transforms and numerical algorithms publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.3160440202 – volume: 26 start-page: 793 issue: 7 year: 1992 ident: CR2 article-title: A wavelet based space-time adaptive numerical method for partial differential equations publication-title: Math. Model. Num. Anal. – ident: CR23 – volume: 15 start-page: 44 year: 1973 end-page: 52 ident: CR20 article-title: A note on spherical summation multipliers publication-title: Israel J. Math. doi: 10.1007/BF02771772 – year: 1990 ident: CR30 publication-title: Ondelettes et Opérateurs – year: 2003 ident: CR12 publication-title: Numerical Analysis of Wavelet Methods – ident: CR21 – year: 1993 ident: CR36 publication-title: Harmonic Analysis – volume: 31 start-page: 43 issue: 1 year: 2000 end-page: 70 ident: CR25 article-title: Generlization of the phase-screen approximation for the scattering of acoustic waves publication-title: Wave Motion doi: 10.1016/S0165-2125(99)00026-8 – volume: 336 start-page: 395 year: 2003 end-page: 398 ident: CR7 article-title: Curvelets and Fourier integral operators publication-title: C. R. Acad. Sci. Paris Ser. I – ident: CR15 – volume: 70 start-page: 27 issue: 233 year: 2000 end-page: 75 ident: CR13 article-title: Adaptive wavelet methods for elliptic operator equations— Convergence rates publication-title: Math. Comp. doi: 10.1090/S0025-5718-00-01252-7 – ident: CR17 – year: 1985 ident: CR26 publication-title: The Analysis of Linear Partial Differential Operators, vol.4 – volume: 26 start-page: 2133 issue: 6 year: 2005 end-page: 2159 ident: CR4 article-title: Algorithms for numerical analysis in high dimensions publication-title: SIAM J. Sci. Comput. doi: 10.1137/040604959 – volume: 58 start-page: 1472 issue: 11 year: 2005 end-page: 1528 ident: CR8 article-title: The curvelet representation of wave propagators is optimally sparse publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20078 – ident: CR9 – volume: 3 start-page: 979 issue: 11 year: 1978 end-page: 1005 ident: CR14 article-title: Wave packets and Fourier integral operators publication-title: Comm. PDE doi: 10.1080/03605307808820083 – volume: 29 start-page: 965 issue: 4 year: 1992 end-page: 986 ident: CR27 article-title: Wavelet methods for fast resolution of elliptic problems publication-title: SIAM J. Numer. Anal. doi: 10.1137/0729059 – volume: 53 start-page: 270 issue: 2 year: 2001 end-page: 275 ident: CR1 article-title: Pointwise inequalities of Landau-Kolmogorov-type for functions defined on a finite segment publication-title: Ukr. Math. J. doi: 10.1023/A:1010469004645 – year: 1996 ident: CR18 publication-title: Fourier Integral Operators – volume: 62 start-page: 89 year: 1999 end-page: 108 ident: CR22 article-title: A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices publication-title: Computing doi: 10.1007/s006070050015 – volume: 57 start-page: 219 year: 2004 end-page: 266 ident: CR11 article-title: New tight frames of curvelets and optimal representations of objects with piecewise singularities publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.10116 – ident: CR24 – year: 1989 ident: CR38 publication-title: Weakly Differentiable Functions – volume: 3 start-page: 979 issue: 11 year: 1978 ident: 226_CR14 publication-title: Comm. PDE doi: 10.1080/03605307808820083 – volume: 15 start-page: 755 issue: 4 year: 1994 ident: 226_CR19 publication-title: SIAM J. Sci. Comput. doi: 10.1137/0915048 – volume: 336 start-page: 395 year: 2003 ident: 226_CR7 publication-title: C. R. Acad. Sci. Paris Ser. I doi: 10.1016/S1631-073X(03)00095-5 – volume-title: Finite Volume Methods for Hyperbolic Problems year: 2002 ident: 226_CR29 doi: 10.1017/CBO9780511791253 – volume: 45 start-page: 3 issue: 1 year: 2003 ident: 226_CR32 publication-title: SIAM Rev. doi: 10.1137/S00361445024180 – volume: 57 start-page: 219 year: 2004 ident: 226_CR11 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.10116 – ident: 226_CR24 – volume: 6 start-page: 197 year: 1999 ident: 226_CR6 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1006/acha.1998.0248 – ident: 226_CR9 doi: 10.1137/05064182X – volume: 53 start-page: 270 issue: 2 year: 2001 ident: 226_CR1 publication-title: Ukr. Math. J. doi: 10.1023/A:1010469004645 – ident: 226_CR23 doi: 10.1109/MCSE.2006.49 – volume: 41 start-page: 263 issue: 3 year: 2005 ident: 226_CR5 publication-title: Wave Motion doi: 10.1016/j.wavemoti.2004.05.008 – ident: 226_CR16 doi: 10.1016/j.acha.2007.03.003 – ident: 226_CR10 – volume-title: Harmonic Analysis year: 1993 ident: 226_CR36 – volume: 134 start-page: 231 year: 1991 ident: 226_CR33 publication-title: Ann. Math. doi: 10.2307/2944346 – volume: 26 start-page: 793 issue: 7 year: 1992 ident: 226_CR2 publication-title: Math. Model. Num. Anal. doi: 10.1051/m2an/1992260707931 – volume: 26 start-page: 2133 issue: 6 year: 2005 ident: 226_CR4 publication-title: SIAM J. Sci. Comput. doi: 10.1137/040604959 – volume: 70 start-page: 27 issue: 233 year: 2000 ident: 226_CR13 publication-title: Math. Comp. doi: 10.1090/S0025-5718-00-01252-7 – volume: 31 start-page: 43 issue: 1 year: 2000 ident: 226_CR25 publication-title: Wave Motion doi: 10.1016/S0165-2125(99)00026-8 – volume: 44 start-page: 141 year: 1991 ident: 226_CR3 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.3160440202 – ident: 226_CR37 doi: 10.1016/j.jcp.2006.05.008 – volume-title: Fourier Integral Operators year: 1996 ident: 226_CR18 – volume-title: The Analysis of Linear Partial Differential Operators, vol.4 year: 1985 ident: 226_CR26 – ident: 226_CR21 doi: 10.1016/S1570-579X(03)80030-7 – volume: 24 start-page: 627 year: 1957 ident: 226_CR28 publication-title: Duke Math J. doi: 10.1215/S0012-7094-57-02471-7 – volume: 15 start-page: 44 year: 1973 ident: 226_CR20 publication-title: Israel J. Math. doi: 10.1007/BF02771772 – ident: 226_CR17 doi: 10.1190/1.1845326 – volume-title: Ondelettes et Opérateurs year: 1990 ident: 226_CR30 – volume-title: Wavelets, Calderón-Zygmund and Multilinear Operators year: 1997 ident: 226_CR31 – volume-title: Weakly Differentiable Functions year: 1989 ident: 226_CR38 doi: 10.1007/978-1-4612-1015-3 – ident: 226_CR15 – volume: 48 start-page: 797 year: 1998 ident: 226_CR35 publication-title: Ann. Inst. Fourier (Grenoble) doi: 10.5802/aif.1640 – volume: 58 start-page: 1472 issue: 11 year: 2005 ident: 226_CR8 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20078 – volume-title: Numerical Analysis of Wavelet Methods year: 2003 ident: 226_CR12 – volume: 62 start-page: 89 year: 1999 ident: 226_CR22 publication-title: Computing doi: 10.1007/s006070050015 – volume: 29 start-page: 965 issue: 4 year: 1992 ident: 226_CR27 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0729059 – volume: 8 start-page: 629 year: 1998 ident: 226_CR34 publication-title: J. Geom. Anal. doi: 10.1007/BF02921717
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Snippet	We present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying, two-dimensional time-independent periodic...
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SubjectTerms	Exact sciences and technology Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical analysis. Scientific computation Numerical and Computational Physics Numerical linear algebra Ordinary differential equations Partial differential equations Sciences and techniques of general use Simulation Special functions Theoretical
Title	Wave atoms and time upscaling of wave equations
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