M2L Translation Operators for Kernel Independent Fast Multipole Methods on Modern Architectures

Abstract

Hardware trends favor algorithm designs that maximize data reuse per FLOP. We develop and benchmark high-performance Multipole-to-Local (M2L) translation operators for the kernel-independent Fast Multipole Method (kiFMM), a widely adopted FMM variant that supports a broad class of kernels and has been favored by recent implementations for its simple specification. Naively implemented, M2L is bandwidth-limited and therefore a key bottleneck in the FMM. State-of-the-art FFT-based M2L implementations, though elegant and with a fast setup time, suffer from low operational intensity and require architecture-specific optimizations. We demonstrate that a BLAS-based M2L, combined with randomized low-rank compression, achieves competitive performance with greater portability and a simpler implementation leveraging existing BLAS infrastructure, at the cost of higher setup times-especially for high-accuracy settings in double precision. Our Rust-based implementation enables seamless switching between strategies for fair benchmarking. Results on CPUs show that FFT-based M2L is favorable in low-accuracy settings or dynamic particle simulations, while BLAS-based M2L is favored for high-accuracy settings for static particle distributions, where its higher setup costs are amortized in many practical applications of the FMM.