Efficient Evaluation of the Rokhlin Translator in Multilevel Fast Multipole Algorithm

In multilevel fast multipole algorithm (MLFMA) the direct evaluation of the Rokhlin translator is computationally expensive, and usually the cost is lowered by using local Lagrange interpolation in the evaluation, which requires oversampling of the translator. In this paper we improve the interpolation procedure by introducing a new, accurate, and fast oversampling technique based on the fast Fourier transform (FFT). In addition to speeding up the oversampling this also allows the use of lower number of points in the interpolation stencils improving the efficiency of the evaluation of the Rokhlin translator. We have optimized the interpolation parameters, i.e., the number of the stencil points and the oversampling factor, by using as the error criterion the accuracy in the translated (incoming) field rather than the usually used interpolation error. This choice leads to better optimized parameter pairs which further lowers the interpolation cost. We have computed and tabulated the optimized pairs for a wide range of target accuracies and the MLFMA division levels. These tables can be used for a good error control and maximal speed-up in practical computations.