Mass Matrix Assembly on Tensor Cores for Implicit Particle-In-Cell Methods

2026-04-21

AI Summary

This work addresses the performance bottleneck in implicit particle-in-cell methods, where mass matrix assembly is constrained by reduction operations that poorly utilize modern GPU tensor cores. We present the first formulation that exactly recasts this assembly into a tensor-core-friendly matrix-matrix multiplication (GEMM) form. By integrating high-order B-spline interpolation, particle batching, and support-domain grouping, our approach supports arbitrary interpolation orders and both scalar and tensor-valued block mass matrices. Leveraging NVIDIA’s tensor-core-based matrix multiply-accumulate (MMA) units, the method achieves up to 3× kernel-level speedup over an optimized baseline while maintaining full generality. In end-to-end ECSIM simulations, this translates to a 15% overall acceleration.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Matrix-multiply-accumulate (MMA) units, or tensor cores, are now widespread across modern computing architectures. Yet, their use for particle-grid operators remains limited. In implicit particle methods, mass-matrix assembly is a reduction-dominated kernel in which weighted outer products of interpolation weights are accumulated over particle support. We show that this operation can be reformulated exactly, cell by cell, as a sequence of matrix products matched to hardware MMA tiles. The formulation is general with respect to interpolation order and hardware platform, and applies to both scalar mass matrices and the tensorial block mass matrix arising in implicit in the Energy-Conserving Semi-Implicit Method (ECSIM) for Particle-in-Cell simulations. We introduce particle batching and a support-group decomposition for higher-order shape functions whose stencil extends beyond a single cell, specialize the method to first- and second-order B-spline interpolation, and implement it on NVIDIA tensor cores. The resulting kernels achieve up to 3x over optimized conventional implementations and reduce end-to-end ECSIM runtime by 15%.

Problem

• mass matrix assembly
• tensor cores
• implicit Particle-in-Cell
• matrix-multiply-accumulate
• particle-grid operators

Innovation

• tensor cores
• mass matrix assembly
• Particle-in-Cell
• matrix-multiply-accumulate
• ECSIM