Accelerating a restarted Krylov method for matrix functions with randomization

Abstract

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 ill-conditioned problems, complementing the numerical analyses from previous studies.