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 BCG . 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 nonstandard wave atom form, provides a compression of the acoustic single and double layer potentials with wave number @math as @math -by- @math matrices with @math nonnegligible entries, with a constant that depends on the relative @math accuracy @math in an acceptable way. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that enable to write this result as a theorem is a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling wavelength @math (essential diameter) @math . 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.
There has been a lot of work on sparsifying the integral operator of , or some variants of it, in appropriate bases. In @cite_3 , showed that the operator becomes sparse in a local cosine basis. They proved that the number of coefficients with absolute value greater than any fixed @math is bounded by @math when the constant depends on @math . Notice that our result in Theorem is stronger as the @math norm is used instead in . In @cite_7 , extended the work in @cite_3 by performing best basis search in the class of adaptive hierarchical local cosine bases.
@cite_7 @cite_3
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Abstract The integral ∫0Leiνφ(s,t)f(s)dswith a highly oscillatory kernel (large ν, ν is up to 2000) is considered. This integral is accurately evaluated with an improved trapezoidal rule and effectively transcribed using local Fourier basis and adaptive multiscale local Fourier basis. The representation of the oscillatory kernel in these bases is sparse. The coefficients after the application of local Fourier transform are smoothed. Sometimes this enables us to obtain further compression with wavelets. Abstract We prove that certain oscillatory boundary integral operators occurring in acoustic scattering computations become sparse when represented in the appropriate local cosine transform orthonormal basis.