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Application of wavelet transforms for solving integral equations that arise in rough-surface scattering

We consider the problem of scattering a plane wave from a periodic rough surface. The scattered field is evaluated once the field on the boundary is calculated. The latter is the solution of an integral equation. In fact, different integral equation formulations are available in both coordinate and spectral space. We solve these equations using standard numerical techniques, and compare the results to corresponding solutions of the equations using wavelet transform methods for "sparsification" of the impedance matrix. Using an energy check, the methods are shown to be highly accurate. We limit the discussion to the Dirichlet problem (scalar), or the TE-polarized case for a one-dimensional surface. The boundary unknown is thus the normal derivative of the total (scalar) field or, equivalently, the surface current. We illustrate two conclusions. First, sparsification (using "thresholded" wavelet transforms) can significantly reduce accuracy. Second, the wavelet transform did not speed up the overall solution. For our examples, the solution time was considerably increased when thresholded wavelet transforms were used.