Scattering in Flatland: Efficient Representations via Wave Atoms
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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 Bradie, Coifman, and Grossman, in the context of local Fourier bases (Bradie et al. in Appl. Comput. Harmon. Anal. 1:94–99, 1993). 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 ε δ k 1+δ non-negligible entries, with δ>0 arbitrarily small, and ε 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) ∼ (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.
- E. Abbott, Flatland: A Romance in Many Dimensions, 3rd edn. (Dover, New York, 1992) (1884).
- J.P. Antoine, R. Murenzi, Two-dimensional directional wavelets and the scale-angle representation, Signal Process. 52, 259–281 (1996). CrossRef
- 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). CrossRef
- G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms. I, Commun. Pure Appl. Math. 44(2), 141–183 (1991). CrossRef
- B. Bradie, R. Coifman, A. Grossmann, Fast numerical computations of oscillatory integrals related to acoustic scattering, Appl. Comput. Harmon. Anal. 1, 94–99 (1993). CrossRef
- 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). CrossRef
- 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). CrossRef
- A. Córdoba, C. Fefferman, Wave packets and Fourier integral operators, Commun. PDE 3(11), 979–1005 (1978). CrossRef
- L. Demanet, L. Ying, Curvelets and wave atoms for mirror-extended images, in Proc. SPIE Wavelets XII Conf., 2007.
- L. Demanet, L. Ying, Wave atoms and sparsity of oscillatory patterns, Appl. Comput. Harmon. Anal. 23(3), 368–387 (2007). CrossRef
- H. Deng, H. Ling, Fast solution of electromagnetic integral equations using adaptive wavelet packet transform, IEEE Trans. Antennas Propag. 47(4), 674–682 (1999). CrossRef
- 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).
- B. Engquist, L. Ying, Fast directional multilevel algorithms for oscillatory kernels, SIAM J. Sci. Comput. 29(4), 1710–1737 (2007). CrossRef
- B. Engquist, L. Ying, Fast directional computation for the high frequency Helmholtz kernel in two dimensions, Commun. Math. Sci. 7(2), 327–345 (2009).
- W.L. Golik, Wavelet packets for fast solution of electromagnetic integral equations, IEEE Trans. Antennas Propag. 46(5), 618–624 (1998). CrossRef
- 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). CrossRef
- S. Kapur, V. Rokhlin, High-order corrected trapezoidal quadrature rules for singular functions, SIAM J. Numer. Anal. 34, 1331–1356 (1997). CrossRef
- 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). CrossRef
- S. Mallat, A Wavelet Tour of Signal Processing, 2nd edn. (Academic Press, Orlando-San Diego, 1999).
- F.G. Meyer, R.R. Coifman, Brushlets: a tool for directional image analysis and image compression, Appl. Comput. Harmon. Anal. 4, 147–187 (1997). CrossRef
- V. Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions, J. Comput. Phys. 86(2), 414–439 (1990). CrossRef
- V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal. 1, 82–93 (1993). CrossRef
- L. Villemoes, Wavelet packets with uniform time-frequency localization, C. R. Math. 335(10), 793–796 (2002).
- G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1966).
- Scattering in Flatland: Efficient Representations via Wave Atoms
Foundations of Computational Mathematics
Volume 10, Issue 5 , pp 569-613
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Fast algorithm
- Wave propagation
- Boundary integral equation
- Computational harmonic analysis
- Industry Sectors