Sparsity and conditioning of impedance matrices obtained with semi-orthogonal and bi-orthogonal wavelet bases

Wavelet and wavelet packet transforms are often used to sparsify dense matrices arising in the discretization of CEM integral equations. This paper compares orthogonal, semi-orthogonal, and bi-orthogonal wavelet and wavelet packet transforms with respect to the condition numbers, matrix sparsity, and number of iterations for the transformed systems. The best overall results are obtained with the orthogonal wavelet packet transforms that produce highly sparse matrices requiring fewest iterations. Among wavelet transforms the semi-orthogonal wavelet transforms lead to the sparsest matrices, but require too many iterations due to high condition numbers. The bi-orthogonal wavelets produce very poor sparsity and require many iterations and should not be used in these applications.