On the use of Coifman intervallic wavelets in the method of moments for fast construction of wavelet sparsified matrices

Orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained. However, in many practical problems, the solution domain is confined in a bounded interval, while the wavelets are originally defined on the entire real line. To overcome this problem, periodic wavelets have been described in the literature. Nonetheless, the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions. In this paper, we present the intervallic Coifman wavelets (coiflets) for the method of moments (MoM). The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets. The intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis (MRA) of other usual unbounded wavelets. The coiflets possesses a special property that their scaling functions have many vanishing moments. As a result, the zero entries of the matrices are identified directly, without using a truncation scheme with an artificially established threshold. Further, the majority of matrix elements are evaluated directly without performing numerical integration procedures such as Gaussian quadrature. For an n×x matrix, the number of actual numerical integrations is reduced from n² to the order of 3n(2L - 1), when the coiflets of order L is employed. The construction of intervallic wavelets will be presented. Numerical examples of scattering problems are discussed and the relative error of this method is studied analytically.