New upper bounds to determine the sampling rate in 2-D MLMFA

The error controllability of fast algorithms, such as the fast multipole method (FMM) and the multilevel fast multipole algorithm (MLFMA), is an important issue. On the one hand, it is crucial that the error sources can be clearly identified. On the other hand, heuristics need to be implemented that allow an easy automatic error setting of the fast algorithms. An error analysis has already been conducted for 2D MLFMAs in lossless homogeneous background media. Here, however, the focus is on MLMFAs for very lossy background media, and more specifically, on the (quasi-)bandlimitedness of the 2D radiation patterns. We rigorously derive upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy. On the one hand, these formulas can immediately be used in 2D homogeneous (lossy) media MLFMAs to estimate the required number of samples by means of heuristic approaches, using these upper bounds as a starting point. On the other hand, these upper bounds can be used in MLMFAs that are based on the decomposition of 3D Green functions into sets of cylindrical modes, each mode corresponding to a 2D Green function. Especially in the PML-MLMFA, some of these cylindrical modes´ wavenumbers have large negative imaginary parts, and hence, correspond to very lossy 2D homogeneous space problems.