The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.

通过采用混合精度迭代细化，可以提高线性系统基于 LU 分解的标准求解过程的速度或精度。最近的工作集中在密集系统上。我们研究了混合精度迭代细化的潜力，以增强基于近似稀疏分解的稀疏系统的方法。为此，我们首先在 LU 分解的一般模型下为基于 LU 和 GMRES 的迭代细化开发了一种新的误差分析，该模型考虑了现代稀疏求解器通常使用的近似方法，例如低秩近似或松弛的枢轴策略. 然后，我们对不同算法的执行时间和内存消耗进行了详细的性能分析，基于一组选定的迭代细化变体和近似稀疏分解。我们的性能研究使用多前沿求解器 MUMPS，它可以利用块低秩分解和静态旋转。我们评估了算法在来自各种现实生活和工业应用的大型稀疏问题上的性能，表明混合精度迭代细化与近似稀疏分解相结合可以显着减少时间和内存消耗。