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Kernel aggregated fast multipole method

Efficient summation of Laplace and Stokes kernel functions

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Abstract

Many different simulation methods for Stokes flow problems involve a common computationally intense task—the summation of a kernel function over O(N2) pairs of points. One popular technique is the kernel independent fast multipole method (KIFMM), which constructs a spatial adaptive octree for all points and places a small number of equivalent multipole and local equivalent points around each octree box, and completes the kernel sum with O(N) cost, using these equivalent points. Simpler kernels can be used between these equivalent points to improve the efficiency of KIFMM. Here we present further extensions and applications to this idea, to enable efficient summations and flexible boundary conditions for various kernels. We call our method the kernel aggregated fast multipole method (KAFMM), because it uses different kernel functions at different stages of octree traversal. We have implemented our method as an open-source software library STKFMM based on the high-performance library PVFMM, with support for Laplace kernels, the Stokeslet, regularized Stokeslet, Rotne-Prager-Yamakawa (RPY) tensor, and the Stokes double-layer and traction operators. Open and periodic boundary conditions are supported for all kernels, and the no-slip wall boundary condition is supported for the Stokeslet and RPY tensor. The package is designed to be ready-to-use as well as being readily extensible to additional kernels.

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Acknowledgements

We thank Dhairya Malhotra and Alex Barnett for inspiring discussions, and thank Bryce Palmer for comments on the manuscript.

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Authors and Affiliations

  1. Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, NY, 10010, USA

    Wen Yan & Robert Blackwell

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  1. Wen Yan
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Correspondence to Wen Yan.

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Communicated by: Michael O’Neil

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Appendices

Appendix A: All kernels supported by the open-source STKFMM package

Table 12 All kernels currently supported by the STKFMM package. All are supported for open boundary conditions and for singly, doubly, and triply periodic boundary conditions.
Full size table

Appendix B. Overhead of periodic boundary condition

In contrast to our previous implementation [24], we have merged our postfix of applying the M → L operator to implement periodic boundary condition into the downward pass (L → L and L → T ) of running KIFMM itself, with the help of Dr. Malhotra. Therefore, the extra overhead of running KIFMM with periodicity compared to open boundary condition is significantly reduced. Figure 9 shows the comparison of tree construction time and FMM evaluation time for four different kernels under different boundary conditions. The overhead of implementing periodic boundary conditions is usually 5% of the evaluation time with free boundary condition.

Fig. 9
figure 9

Timing results for selected kernels with different periodic boundary conditions. None, PX, PXY, and PXYZ refer to open boundary, singly periodic, doubly periodic, and triply periodic, respectively

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Appendix C. Note about our SIMD implementation

Although the benchmarks presented in Section 3 were on CPUs (Intel Skylake and Cascade Lake) that support AVX-512 instructions, our kernels are hand-optimized with AVX2 instructions. Since the AVX-512 instruction set is still evolving and new subsets of instructions are still being added, we plan to support AVX-512 in the future when the instruction set is more stable.

Appendix D. Parallel scaling

Detailed parallel scaling benchmarks and analysis are beyond the scope of this paper because we implemented our package based on the PVFMM library without modifying any of its advanced parallelization facilities. Since the parallel scaling of PVFMM has been thoroughly tested [9], we present only some brief test results here. We tested strong and weak scaling for the low-precision case m = 8 and the high-precision case m = 12. The scaling benchmarks were measured on a cluster where each node was equipped with two Intel Cascade Lake 8268 CPUs @ 2.9 GHz. Each CPU had 24 cores, hyper-threading was disabled, and the machines were set to “performance” mode. We launched two MPI ranks on each CPU. Each MPI rank launched 12 OpenMP threads and each thread was bound to one CPU core. The number of source and target points was increased compared to the convergence tests in the previous sections, and we increased the box size to L = 100. We again used a non-uniform distribution of points picked from the log-normal distribution in Eq. (36).

We tested strong scaling using the RPY kernel with a doubly periodic boundary condition and 2.7 × 107 target points and source points. Figure 10a shows the results, which are comparable to 70% parallel efficiency (dashed black line). We tested weak scaling using the StokesPVel kernel GP with 4 million target points per node, i.e., 1 million points for each MPI rank. As shown in Fig. 10b, the trun results are again comparable to 70% parallel efficiency. These results are completely determined by the PVFMM library, and are close to its benchmarks [9].

Fig. 10
figure 10

Strong and weak scaling benchmarks for our KAFMM package. Each node has 48 cores. The left panel shows strong scaling results for a fixed-size RPY tensor problem with 2.7 × 107 source and target points, with doubly periodic (DP) boundary conditions. The right panel shows weak scaling results for a variable-size StokesPVel problem with 4 × 106 points per node, with non-periodic (NP) boundary conditions. In both cases the dashed line is a reference line of 70% parallel efficiency

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Yan, W., Blackwell, R. Kernel aggregated fast multipole method. Adv Comput Math 47, 69 (2021). https://doi.org/10.1007/s10444-021-09896-1

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