Leveraging Trace Theory, we investigate the efficient parallelization of direct solvers for large linear equation systems. Our focus lies on a multi-frontal algorithm, and we present a methodology for achieving near-optimal scheduling on modern massively parallel machines. By employing trace theory with Diekert Graphs and Foata Normal Form, we rigorously validate the effectiveness of our proposed solution. To establish a strong link between the mesh and elimination tree of the multi-frontal solver, we conduct extensive testing on matrices derived from the Finite Element Method (FEM). Furthermore, we assess the performance of computations on both GPU and CPU platforms, employing practical implementation strategies.

翻译：基于迹理论，我们研究了大型线性方程组直接求解器的高效并行化方法。重点聚焦于多波前算法，并提出了一种在现代大规模并行机器上实现近似最优调度的技术方案。通过采用带有Diekert图和Foata标准型的迹理论，我们严格验证了所提方案的有效性。为建立多波前求解器中网格与消去树之间的强关联性，我们对基于有限元方法生成的矩阵进行了广泛测试。此外，我们采用实际实现策略，评估了在GPU和CPU平台上的计算性能表现。