A comparative study of wavelet matrix transformations for the solution of integral equations

The application of wavelets for the solution of electromagnetic field integral equation yields sparse matrix equations which can be solved efficiently by using sparse matrix techniques. In this paper, the wavelet matrix transforms using the semi-orthogonal wavelets (SOW) and the Daubechies orthogonal wavelets (DOW) are applied to the solution of integral equations and their performance is compared by investigating the convergence rate when the conjugate gradient (CG) method is employed. Since the SOW transform yields a matrix with a larger condition number than that corresponding to the DOW transform, it is expected that the convergence rate is much better in the latter case. Numerical simulations are conducted for the transverse magnetic (TM) scattering by conducting cylinders and the convergence rate of the CG iterative process is determined for the matrix equations transformed using the SOW and the DOW. The computed results confirm the theoretical expectations.