Many scientific applications require the evaluation of the action of the matrix function over a vector and the most common methods for this task are those based on the Krylov subspace. Since the orthogonalization cost and memory requirement can quickly become overwhelming as the basis grows, the Krylov method is often restarted after a few iterations. This paper proposes a new acceleration technique for restarted Krylov methods based on randomization. The numerical experiments show that the randomized method greatly outperforms the classical approach with the same level of accuracy. In fact, randomization can actually improve the convergence rate of restarted methods in some cases. The paper also compares the performance and stability of the randomized methods proposed so far for solving very large finite element problems, complementing the numerical analyses from previous studies.

翻译：众多科学应用需要计算矩阵函数作用于向量的结果，基于Krylov子空间的方法是该任务最常用的计算手段。随着基向量的增长，正交化计算成本与内存需求可能急剧增加，因此Krylov方法通常在若干次迭代后重启。本文提出一种基于随机化的重启Krylov方法加速技术。数值实验表明，在相同精度水平下，随机化方法显著优于经典方法。事实上，在某些情况下随机化甚至能提升重启方法的收敛速率。本文还比较了现有随机化方法在求解超大规模有限元问题时的性能与稳定性，对先前研究的数值分析形成了补充。