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.
Besides the local cosine transform, adaptive wavelet packets have been used to sparsify the integral operator as well. Deng and Ling @cite_6 applied the best basis algorithm to the integral operator to choose the right one dimensional wavelet packet basis. Golik @cite_4 independently proposed to apply the best basis algorithm on the right hand side of the integral equation . Shortly afterwards, Deng and Ling @cite_14 gave similar results by using a predefined wavelet packet basis that refines the frequency domain near @math . All of these approaches work with the standard form expansion of the integral operator. Recently in @cite_17 , Huybrechs and Vandewalle used the best basis algorithm for two dimensional wavelet packets to construct a nonstandard sparse expansion of the integral operator. In all of these results, the numbers of nonnegligible coefficients in the expansions were reported to scale like @math . However, our result shows that, by using the nonstandard form based on wave atoms, the number of significant coefficients scales like @math .
@cite_14 @cite_4 @cite_6 @cite_17
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The adaptive wavelet packet transform is applied to sparsify the moment matrices for the fast solution of electromagnetic integral equations. In the algorithm, a cost function is employed to adaptively select the optimal wavelet packet expansion testing functions to achieve the maximum sparsity possible in the resulting transformed system. The search for the best wavelet packet basis and the moment matrix transformation are implemented by repeated two-channel filtering of the original moment matrix with a pair of quadrature filters. It is found that the sparsified matrix has above-threshold elements that grow only as O(N sup 1.4 ) for typical scatterers. Consequently the operations to solve the transformed moment equation using the conjugate gradient method scales as O(N sup 1.4 ). The additional computational cost for carrying out the adaptive wavelet packet transform is evaluated and discussed. We introduce atoms" as a variant of 2D wavelet packets obeying the parabolic scaling wavelength (diameter) 2 . We prove that warped oscillatory functions, a toy model for texture, have a signicantly sparser expansion in wave atoms than in other xed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the " technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and ngerprint images demonstrate the potential of the tool introduced. We examine the use of wavelet packets for the fast solution of integral equations with a highly oscillatory kernel. The redundancy of the wavelet packet transform allows the selection of a basis tailored to the problem at hand. It is shown that a well chosen wavelet packet basis is better suited to compress the discretized system than wavelets. The complexity of the matrix-vector product in an iterative solution method is then substantially reduced. A two-dimensional wavelet packet transform is derived and compared with a number of one-dimensional transforms that were presented earlier in literature. By means of some numerical experiments we illustrate the improved efficiency of the two-dimensional approach. We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in "subdomains") that directly lends itself to operator preconditioning via Calderon projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise @math -distributions over each associated interface. Through the use of interior Calderon projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning.