A Directional Equispaced interpolation-based Fast Multipole Method for oscillatory kernels

Igor Chollet (1, 2) , Xavier Claeys (3, 2) , Pierre Fortin (4, 5) , Laura Grigori (3, 2)

1 ISCD - Institut des Sciences du Calcul et des Données
2 ALPINES - Algorithms and parallel tools for integrated numerical simulations
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
5 PEQUAN - Performance et Qualité des Algorithmes Numériques

Fast Multipole Methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from Boundary Integral Equations (BIEs) in acoustic or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm - directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach, and show how symmetries can be exploited in the Fourier domain. We also describe the algorithmic design of defmm, well-suited for the BIE non-uniform particle distributions, and present performance optimizations on one CPU core. Finally, we exhibit important performance gains on all test cases for defmm over a state-of-the-art FMM library for oscillatory kernels.

Mots clés: Fast multipole method FMM; Symmetries; Fast Fourier Transform FFT; High performance computing on one CPU core; Highly oscillatory kernels; Non-uniform particle distributions

Domaines: Analyse numérique [math.NA]

Igor Chollet, Xavier Claeys, Pierre Fortin, Laura Grigori. A Directional Equispaced interpolation-based Fast Multipole Method for oscillatory kernels. SIAM Journal on Scientific Computing, 2023, 45 (1), pp.C20-C48. ⟨10.1137/22M1472930⟩. ⟨hal-03563005v2⟩