A Fourier-Series-Based Kernel-Independent Fast Multipole Method

We present in this talk a new kernel-independent fast multipole method, named as FKI-FMM, for pair-wise particle interactions with translation-invariant kernel functions. The FKI-FMM creates, using numerical techniques, sufficiently accurate representations of a given kernel function over multiscale interaction regions in the form of a truncated Fourier series. It also provides, using analytical approaches, the operators for the multipole-to-multipole, multipole-to-local, and local-to-local translations in the FMM framework. The multipole-to-local translation operator, in particular, is readily diagonal. We present complexity analysis and numerical results to demonstrate the accuracy and efficiency features of the FKI-FMM algorithm, which is applicable to a broad class of translation-invariant kernel functions in science and engineering applications, including the study of anisotropic elastic materials and multi-physics biological systems.