Joar Bagge

Yunhui Cai, Joar Bagge, Per-Gunnar Martinsson

The paper describes a numerical method for solving acoustic multibody scattering problems in two and three dimensions. The idea is to compute a highly accurate approximation to the scattering operator for each body through a local computation, and then use these scattering matrices to form a global linear system. The resulting coefficient matrix is relatively well-conditioned, even for problems involving a very large number of scatterers. The linear system is amenable to iterative solvers, and can readily be accelerated via fast algorithms for the matrix-vector multiplication such as the fast multipole method. The key point of the work is that the local scattering matrices can be constructed using potentially ill-conditioned techniques such as the method of fundamental solutions (MFS), while still maintaining scalability and numerical stability of the global solver. The resulting algorithm is simple, as the MFS is far simpler to implement than alternative techniques based on discretizing boundary integral equations using Nyström or Galerkin.

Gabriel Kosmacher, Ziyu Du, Joar Bagge et al.

We present GPU algorithms for Ewald summation methods for accelerating N-body Stokes flow problems in periodic domains. Like most N-body codes, Ewald sums use a near-field/far-field decomposition. The near field involves particle-to-particle (P2P) interactions. The far field primarily involves particle-to-grid (P2G) and grid-to-particle (G2P) interactions, as well as Fast Fourier Transforms. For each interaction, we investigate several algorithmic variants. Our implementation uses PyKokkos, a Python interface for the Kokkos C++ parallel programming framework, which supports portability to AMD/NVIDIA GPU and ARM/x86 CPU architectures. Double and single-precision numerical results, alongside analytical performance models, confirm the efficiency of our algorithms on AMD and NVIDIA GPU and on ARM and AMD CPU architectures. The P2P interaction achieves around 73% compute efficiency on NVIDIA H200, 84% on NVIDIA A100, 60% on AMD MI300, 52% on Grace CPU, and 68% on AMD Epyc CPU. A straightforward implementation of the P2G kernel can become a computational bottleneck. We introduce a novel P2G algorithm that achieves up to 16$\times$ speedup compared to a baseline GPU implementation. The overall Ewald sum code processes approximately 8 million particles per second on a H200 GPU, and about a half-million particles per second on a Grace CPU, for nine digits of accuracy. We also perform a multi-GPU weak scaling test on up to 256 million particles (64 GPUs) that shows bounded communication cost for all stages except the all-to-all particle sorting, which can be reduced to neighbor communication in the relevant time-stepping regime.