A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths which evolve through the indices of the matrix according to a suitable probability law. The computational complexity is proved in this paper to be significantly better than the classical Monte Carlo method, which allows the computation of much more accurate solutions. Furthermore, the positive features of the algorithm in terms of parallelism were exploited in practice to develop a highly scalable implementation capable of solving some test problems very efficiently using high performance supercomputers equipped with a large number of cores. For the specific case of shared memory architectures the performance of the algorithm was compared with the results obtained using an available Krylov-based algorithm, outperforming the latter in all benchmarks analyzed so far.

提出了一种计算矢量上矩阵指数作用的新颖算法。该算法基于多级蒙特卡洛方法，并且向量概率的计算是根据适当的概率定律通过矩阵的索引演化而生成的适当的随机路径。本文证明了计算复杂度明显优于经典的蒙特卡洛方法，后者可以计算更精确的解。此外，在实践中利用了算法在并行性方面的积极特性，以开发一种高度可扩展的实现，该实现可使用配备有大量核的高性能超级计算机非常有效地解决一些测试问题。