A two-dimensional wavelet-packet transform for matrix compression of integral equations with highly oscillatory kernel

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.