arXiv subjects

Clément Guillet

Publications and source records attributed to Clément Guillet.

4 recordsLinked to original sources

arXiv2026

Exact hierarchical algorithms for accelerating particle--mesh coupling in sparse-grid particle-in-cell methods

In this paper, we propose two hierarchical algorithms for charge deposition and electric-field interpolation that apply to both the sparse-grid combination technique (SGCT-PIC) and hierarchical sparse-grid (HSG-PIC) particle-in-cell methods. The two algorithms are inspired by the fast multipole method (FMM) and exploit clusters of particles associated with a directed acyclic graph (DAG) of particle-populated boxes to reduce the number of particle--mesh interactions. The particle--mesh interactions are governed by piecewise-polynomial kernels, so that the associated multipole expansions are exact, requiring neither truncation nor approximation, and are valid in both near- and far-field regions, thereby eliminating the need for multipole-to-local translations. The arithmetic complexity of the charge deposition and field interpolation steps is reduced from $Ø(p^d n^{d-1}N)$ to $Ø(p^d(N+M))$, where $M=2^{dn}$ denotes the number of full-grid mesh nodes and is typically no larger than the particle population in the considered regime, $M\lesssim N$. Numerical experiments in two-dimensional configurations demonstrate charge-deposition speedups of $8.2\times$--$66.9\times$ for SGCT-PIC and $3.1\times$--$18.8\times$ for HSG-PIC, and field-interpolation speedups of $4.1\times$--$62.6\times$ and $4.2\times$--$13.7\times$, respectively, depending on the particle-per-cell ratio, while preserving the exact particle--mesh interactions. The speedups increase with the particle-per-cell ratio, reflecting the reduced dependence of the hierarchical algorithms on the number of particles and their increasing advantage for large particle populations.

math.NA↗

arXiv2026

A hierarchical sparse-grid particle method for the Vlasov--Poisson system

We introduce a hierarchical sparse-grid (HSG) particle method for the numerical solution of the Vlasov--Poisson system. Sparse-grid PIC methods have so far been formulated within finite-difference frameworks, most notably through the sparse-grid combination technique (SGCT), which ties them to tensor-product Cartesian grids and globally defined component grids. This paper brings sparse-grid particle methods into the Galerkin setting: the field equation is solved in variational form on a hierarchical sparse-grid space spanned by B-splines of arbitrary degree, and the traditional charge deposition step is replaced by a direct Galerkin projection of the raw Monte Carlo density estimator onto this space. Beyond preserving the mesh-complexity and noise-reduction benefits of sparse grids, this reformulation substantially extends the versatility of the approach, opening the way to spatial adaptivity and to non-rectangular geometries, which are notoriously difficult to accommodate within the SGCT framework. We carry out a probabilistic error analysis decomposing the numerical error into a grid-based bias and a statistical noise component. Under mixed-derivative regularity assumptions on the particle distribution, the bias of the charge density in the $\mathrm{L}^2$-norm is shown to scale as $\mathcal{O}(h^{p+1}|\log h|^{d-1})$, where $p$ is the B-spline degree and $d$ the spatial dimension, and the statistical error in the $\mathrm{L}^1$-norm as $\mathcal{O}(|\log h|^{(d-1)/2}(Nh)^{-1/2})$, matching the accuracy of high-order SGCT-PIC methods. Corresponding bounds are derived for the electric field. The theoretical estimates are validated on classical kinetic plasma benchmarks, including configurations with limited regularity and strong anisotropies that are known to be challenging for sparse-PIC approximations.

math.NA↗

arXiv2026

Error Estimates for Sparse Tensor Products of B-spline Approximation Spaces

This work introduces and analyzes B-spline approximation spaces defined on general geometric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, employing either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive approximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution -- particularly concerning isotropic (non-mixed) derivatives -- are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor-product domains.

math.NA↗

arXiv2026

Efficient Fine-Scale Simulation of Nonlinear Hyperelastic Lattice Structures

With the growing maturity of additive manufacturing, the fabrication of architected or lattice-based metamaterials has become a reality for industrial applications. These materials combine lightweight design with tailored mechanical properties, most of which exhibit pronounced nonlinear, especially large-deformation, behaviors. The main numerical challenge therefore lies in performing nonlinear simulations of such lattice structures, which may contain thousands of geometrically intricate unit cells, while lacking sufficient scale separation for multiscale homogenization schemes to be applicable straightforwardly. In this work, we propose a dedicated solver for the full volumetric fine-scale simulation of nonlinear hyperelastic lattice structures that drastically reduces both memory and computational costs. The key idea is to exploit the intrinsic self-similarity of the cells through a reduced-order modeling strategy applied within a domain-decomposition framework. At each Newton iteration, a limited set of principal cells is identified through a dedicated, weakly intrusive, EIM-like approach, allowing all local tangent operators to be expressed as linear combinations of a few principal ones. This enables fast and memory-efficient operator assembly, and then feeds an efficient inexact FETI-DP based preconditioner at the solution stage, resulting in a quasi matrix-free algorithm for the nonlinear analysis. Numerical experiments in two and three dimensions demonstrate significant computational gains, with runtime reductions from several hours to a few tens of minutes and memory savings by factors of about three, while maintaining full fine-scale accuracy. Notably, the proposed strategy enables the computation of problems involving thousands of cells (i.e., millions of degrees of freedom) within a few minutes on an off-the-shelf laptop.

math.NA↗