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Scattering in flatland: Efficient representations via wave atoms. (English) Zbl 1197.65192

Summary: This paper presents a numerical compression strategy for the boundary integral equation of acoustic scattering in two dimensions. These equations have oscillatory kernels that we represent in a basis of wave atoms, and compress by thresholding the small coefficients to zero.
This phenomenon was perhaps first observed in 1993 by B. Bradie, R. Coifman and A. Grossman, in the context of local Fourier bases [Appl. Comput. Harmon. Anal. 1, No. 1, 94–99 (1993; Zbl 0791.76060)]. Their results have since then been extended in various ways. The purpose of this paper is to bridge a theoretical gap and prove that a well-chosen fixed expansion, the non-standard wave atom form, provides a compression of the acoustic single- and double-layer potentials with wave number \(k\) as \(O(k)\)-by-\(O(k)\) matrices with \(C_{\varepsilon \delta } k^{1+\delta }\) non-negligible entries, with \(\delta >0\) arbitrarily small, and \(\varepsilon \) the desired accuracy. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that allow this result to be written as a theorem are a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling (wavelength of the wave packet) \(\sim\) (essential diameter)\(^{2}\). Numerical experiments support the estimate and show that this wave atom representation may be of interest for applications where the same scattering problem needs to be solved for many boundary conditions, for example, the computation of radar cross sections.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
65T99 Numerical methods in Fourier analysis

Citations:

Zbl 0791.76060

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References:

[1] E. Abbott, Flatland: A Romance in Many Dimensions, 3rd edn. (Dover, New York, 1992) (1884).
[2] J.P. Antoine, R. Murenzi, Two-dimensional directional wavelets and the scale-angle representation, Signal Process. 52, 259–281 (1996). · Zbl 0875.94074 · doi:10.1016/0165-1684(96)00065-5
[3] A. Averbuch, E. Braverman, R. Coifman, M. Israeli, A. Sidi, Efficient computation of oscillatory integrals via adaptive multiscale local Fourier bases, Appl. Comput. Harmon. Anal. 9(1), 19–53 (2000). · Zbl 0958.65148 · doi:10.1006/acha.2000.0305
[4] G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms. I, Commun. Pure Appl. Math. 44(2), 141–183 (1991). · Zbl 0722.65022 · doi:10.1002/cpa.3160440202
[5] B. Bradie, R. Coifman, A. Grossmann, Fast numerical computations of oscillatory integrals related to acoustic scattering, Appl. Comput. Harmon. Anal. 1, 94–99 (1993). · Zbl 0791.76060 · doi:10.1006/acha.1993.1007
[6] E.J. Candès, D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise-C 2 singularities, Commun. Pure Appl. Math. 57, 219–266 (2004). · Zbl 1038.94502 · doi:10.1002/cpa.10116
[7] H. Cheng, W.Y. Crutchfield, Z. Gimbutas, L.F. Greengard, J.F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, J. Zhao, A wideband fast multipole method for the Helmholtz equation in three dimensions, J. Comput. Phys. 216, 300–325 (2006). · Zbl 1093.65117 · doi:10.1016/j.jcp.2005.12.001
[8] A. Córdoba, C. Fefferman, Wave packets and Fourier integral operators, Commun. PDE 3(11), 979–1005 (1978). · Zbl 0389.35046 · doi:10.1080/03605307808820083
[9] L. Demanet, L. Ying, Curvelets and wave atoms for mirror-extended images, in Proc. SPIE Wavelets XII Conf., 2007. · Zbl 1132.68068
[10] L. Demanet, L. Ying, Wave atoms and sparsity of oscillatory patterns, Appl. Comput. Harmon. Anal. 23(3), 368–387 (2007). · Zbl 1132.68068 · doi:10.1016/j.acha.2007.03.003
[11] H. Deng, H. Ling, Fast solution of electromagnetic integral equations using adaptive wavelet packet transform, IEEE Trans. Antennas Propag. 47(4), 674–682 (1999). · Zbl 0949.78022 · doi:10.1109/8.768807
[12] H. Deng, H. Ling, On a class of predefined wavelet packet bases for efficient representation of electromagnetic integral equations, IEEE Trans. Antennas Propag. 47(12), 1772–1779 (1999). · Zbl 0949.78023
[13] B. Engquist, L. Ying, Fast directional multilevel algorithms for oscillatory kernels, SIAM J. Sci. Comput. 29(4), 1710–1737 (2007). · Zbl 1180.65006 · doi:10.1137/07068583X
[14] B. Engquist, L. Ying, Fast directional computation for the high frequency Helmholtz kernel in two dimensions, Commun. Math. Sci. 7(2), 327–345 (2009). · Zbl 1182.65178
[15] W.L. Golik, Wavelet packets for fast solution of electromagnetic integral equations, IEEE Trans. Antennas Propag. 46(5), 618–624 (1998). · doi:10.1109/8.668902
[16] D. Huybrechs, S. Vandewalle, A two-dimensional wavelet-packet transform for matrix compression of integral equations with highly oscillatory kernel, J. Comput. Appl. Math. 197(1), 218–232 (2006). · Zbl 1101.65113 · doi:10.1016/j.cam.2005.11.001
[17] S. Kapur, V. Rokhlin, High-order corrected trapezoidal quadrature rules for singular functions, SIAM J. Numer. Anal. 34, 1331–1356 (1997). · Zbl 0891.65019 · doi:10.1137/S0036142995287847
[18] R. Kress, Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering, Q. J. Mech. Appl. Math. 38(2), 323–341 (1985). · Zbl 0559.73095 · doi:10.1093/qjmam/38.2.323
[19] S. Mallat, A Wavelet Tour of Signal Processing, 2nd edn. (Academic Press, Orlando-San Diego, 1999). · Zbl 0998.94510
[20] F.G. Meyer, R.R. Coifman, Brushlets: a tool for directional image analysis and image compression, Appl. Comput. Harmon. Anal. 4, 147–187 (1997). · Zbl 0879.68117 · doi:10.1006/acha.1997.0208
[21] V. Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions, J. Comput. Phys. 86(2), 414–439 (1990). · Zbl 0686.65079 · doi:10.1016/0021-9991(90)90107-C
[22] V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal. 1, 82–93 (1993). · Zbl 0795.35021 · doi:10.1006/acha.1993.1006
[23] L. Villemoes, Wavelet packets with uniform time-frequency localization, C. R. Math. 335(10), 793–796 (2002). · Zbl 1015.42026
[24] G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1966). · Zbl 0174.36202
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