Spherical patch discretization of k-space for improved interpolation and parallelization of the multilevel fast multipole method

The multilevel fast multipole method (ML-FMM) has largely become the method of choice for accelerating integral equations system solutions related to electromagnetic problems. To reduce solution times further, efforts have also been made to parallelize the ML-FMM across clusters of distributed processors working in tandem. A fundamental parallelization methodology for ML-FMM is to partition the problem across the k-space samples of the signature function. However, while this approach reduces interprocessor communication, and is thus more amenable to parallelization, it leads to inaccuracies during interpolation/anterpolation due to truncation of the discretized k-space signature. The paper presents a new spherical patch discretization (SPD) approach for the k-space signature. The approach leads to more uniform sampling, simplified parallelization, and the opportunity to leverage rigorous signal-processing techniques for interpolation between levels of the ML-FMM.