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Direct MR Preconditioner for MoM based fast methods

Matrices arising from field problems generally have very wide eigenvalue spectra, and therefore the use of a suitable preconditioner becomes necessary to reduce the total solution time, especially when the mesh is dense, as typical in antenna and circuit problems. In this paper, the preconditioning issue is addressed by having recourse to the multiresolution (MR) scheme that manages to generate "wavelet-like" hierarchical multiscale vector functions for any meshed geometry and well controls the method-of-moments (MoM) condition number, by the simple application of a diagonal preconditioner. The MR functions are expressed in terms of sub-domain functions, therefore their generation leads to the construction of a changing basis matrix T that is very sparse, and therefore it has a low memory occupation and its multiplication for any matrix in the RWG basis is low cost. For this reason, the computational cost required for the construction and the application of the MR preconditioner is very low and smaller than the one needed for the application of other preconditioners