Cheetah: Optimizing Execution Pipelines for Matrix-Free Finite Element Operators on GPUs

Abstract

Matrix-free finite element methods are widely used in large-scale scientific simulations due to their reduced memory footprint and favorable arithmetic intensity, making them particularly attractive for modern GPU architectures. However, achieving high performance for fully fused, matrix-free GPU kernels remains challenging, as execution is often limited by pipeline inefficiencies rather than floating-point throughput. In this work, we present Cheetah, a set of kernel-level optimizations that systematically redesign the execution pipeline of just-in-time compiled matrix-free finite element operators without modifying the mathematical formulation or user-facing programming model. Cheetah improves register reuse, reduces shared-memory traffic and CTA-level (cooperative thread array) synchronization, overlaps indirect memory accesses with computation using asynchronous copy, and exploits constant memory for invariant operator data. These optimizations are implemented transparently within the libCEED JIT framework and require no changes to user-provided quadrature functions. We evaluate Cheetah on mass-like and diffusion-like operators on an NVIDIA A100 GPU. The results demonstrate substantial performance improvements, particularly at polynomial orders p ≥ 4, with up to 30% higher throughput for mass operators and up to 50% for diffusion operators compared to the baseline libCEED GPU backend. Detailed profiling shows that these gains arise from improved compute utilization, reduced synchronization overhead, increased arithmetic intensity, and more effective latency hiding. Our results highlight the importance of execution pipeline design in fully fused matrix-free GPU kernels and demonstrate that low-level kernel restructuring is essential for unlocking the full performance potential of matrix-free methods on modern GPUs.