Accelerating the Solving of Many Tiny General Linear Systems on GPUs: Application to Constitutive Laws

Abstract

Many applications require solving large numbers of independent linear systems on GPUs. While this need is well addressed for small to large systems, tiny ones, understood here as systems of dimension below 32, remain challenging. This is especially relevant in constitutive law evaluation, where millions of integration points are handled independently, and where each constitutive update generally relies on a Newton iterative method. Each iteration then requires the double-precision solution of a tiny general square linear system using LU factorization with partial pivoting (LUpp). The present study was conducted within a closed-source prototype, which serves as a demonstrator for porting to NVIDIA GPUs constitutive law evaluations currently provided on CPUs by TFEL/MFront, an open-source code generation tool for material knowledge. We compare several double-precision LUpp solvers, including implementations from GPU linear algebra libraries as well as custom-designed CUDA kernels. The comparison is performed first on large batches of standalone linear systems, and then within the full constitutive-law evaluation workflow, where each integration point requires a sequence of distinct linear systems, one per iteration of its own Newton loop. The study shows that the best LUpp solving strategy strongly depends on several factors including system size and application context. We discuss several key aspects, including register pressure, occupancy, the ability to invoke the LUpp solver directly from device code, and whether assigning several threads to each system is the most efficient strategy. Experiments on an NVIDIA H100 GPU show that the specialized LUpp solvers proposed in this work can outperform existing state-of-the-art approaches for this class of workloads, with speedups of up to 6.5x over cuSolverDx, and up to 17.7x over MAGMA.