A version-1 arXiv preprint dated 20 August 2026 reports substantial speedups for two sparse-grid particle–in-cell methods. Across the reported tests, full simulation-cycle speedups ranged from 7× to 45× for SGCT-PIC and from 3× to 11× for HSG-PIC. The standard and FMM implementations produced identical results up to roundoff errors in the simulations described.
The findings come from a computational benchmark restricted to two-dimensional, sequential executions on a single CPU core using a diocotron-instability configuration. They therefore describe performance in that reported setup.
A different way to organize the calculation
The paper investigates whether two hierarchical, FMM-inspired algorithms can accelerate charge deposition and electric-field interpolation in SGCT-PIC and HSG-PIC. These are the two particle–mesh operations examined in the comparison with standard implementations.
The proposed algorithms organize particles in a directed acyclic graph of particle-populated boxes. They use exact, globally valid polynomial expansions, avoiding truncation and the multipole-to-local translation used in approximate hierarchical schemes.
The paper’s arithmetic analysis reports a change from O(p_d n^(d−1) N) to O(p_d (N + M)), where M = 2dn denotes the number of full-grid mesh nodes. In plain terms, the reported formulation replaces the original sparse-grid-level dependence with a form tied to the particle count and the full grid.
For the timing comparisons, component times were averaged over 10 iterations. The tests used mesh resolutions n = 8 and n = 10, particle-to-cell ratios Pc = 50, Pc = 500 and Pc = 2,500, and piecewise-linear shape functions.
The implementation combined Python and C++, used scipy.sparse Cholesky factorization for linear systems, and ran on an Apple M4 core with 32 GB of RAM. The executions were sequential rather than parallel.
The biggest gains appeared in SGCT-PIC
For SGCT-PIC, the paper reports charge-deposition speedups of 8.2× at n = 8 and Pc = 50, 38.3× at n = 8 and Pc = 500, and 66.9× at n = 8 and Pc = 2,500. At n = 10 and Pc = 500, the reported speedup was 22.7×.
Field interpolation also became faster in the reported SGCT-PIC cases. The speedups were 4.1×, 6.3× and 62.6× at n = 8 with Pc = 50, 500 and 2,500, respectively, and 32.3× at n = 10 and Pc = 500.
At the level of a complete PIC cycle, SGCT-PIC speedups ranged from 7× to 45× across the reported cases. The paper also reports that the speedup increased as the particle-to-cell ratio rose.
HSG-PIC showed smaller, still sizeable gains
The gains were more modest for HSG-PIC charge deposition but followed the same reported pattern. At n = 8, the speedup rose from 3.1× at Pc = 50 to 12.4× at Pc = 500 and 18.8× at Pc = 2,500. At n = 10 and Pc = 500, it was 7.1×.
For HSG-PIC field interpolation, the reported speedup ranged from 4.2× to 13.7× as the particle-to-cell ratio varied.
The overall HSG-PIC cycle speedup was reported as 3× to 11×. The authors link the rise in speedup at higher particle-to-cell ratios to weaker dependence of the hierarchical algorithms on particle number, although the supplied analysis reports no formal trend test.
What the benchmark does not settle
The numerical check is a comparison between implementations: across all simulations, the FMM and standard versions produced identical results up to roundoff errors. That finding concerns agreement in the reported calculations; it does not establish improved plasma-physics accuracy or long-term simulation fidelity.
The evidence is limited to one two-dimensional diocotron configuration and sequential execution on one CPU core. The timing experiments also used piecewise-linear shape functions, so the supplied results cover that stated choice rather than a broader set of shape-function degrees.
The reported speedups are point comparisons: timing variability, confidence intervals and inferential statistical tests were not reported. Exact particle counts for the individual timing cases were also not explicitly listed in the supplied analysis.
Three-dimensional performance and parallel scalability remain untested, as do performance changes across other hardware, plasma configurations, shape-function choices and particle populations. Those tests would be needed to determine how widely the reported speedups generalize.
The preprint states that its code is available at https://gitlab.inria.fr/cguillet/sg-pic. It reports support from the French National Research Agency through the MATURATION project, reference ANR-22-CE46-0012.
Paper data and sources
Original title: Exact hierarchical algorithms for accelerating particle--mesh coupling in sparse-grid particle-in-cell methods
Authors: Clément Guillet
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text