计算矩阵指数对向量作用的蒙特卡罗方法
A Monte Carlo method for computing the action of a matrix exponential on a vector
摘要：
提出了一种计算某类矩阵的矩阵指数在向量上作用的蒙特卡罗方法。该方法是基于生成随机路径，通过矩阵的指数，由一个给定的连续时间马尔可夫链。通过在适当的乘法泛函上求平均值来概率地计算向量解。这种表示扩展了现有的基于线性代数蒙特卡洛的方法，并在实践中用于开发一种有效的算法，能够计算两个，一个单一的条目或完整的矢量解决方案。最后，通过相关的基准测试对算法的性能进行了评估.与Krylov为基础的方法得到的结果的比较显示了显着的性能的算法解决大规模的问题。
英文摘要：
A Monte Carlo method for computing the action of a matrix exponential for a certain class of matrices on a vector is proposed. The method is based on generating random paths, which evolve through the indices of the matrix, governed by a given continuous-time Markov chain. The vector solution is computed probabilistically by averaging over a suitable multiplicative functional. This representation extends the existing linear algebra Monte Carlo-based methods, and was used in practice to develop an efficient algorithm capable of computing both, a single entry or the full vector solution. Finally, several relevant benchmarks were executed to assess the performance of the algorithm. A comparison with the results obtained with a Krylov-based method shows the remarkable performance of the algorithm for solving large-scale problems.
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