We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to CPU cores.

中文翻译：

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求解时间分数阶微分方程的随机方法

我们提出了一种随机方法，用于有效计算模拟次扩散类型的反常扩散问题的时间分数偏微分方程（fPDE）的解。在空间中离散化 fPDE 后，接下来的分数线性方程组通过相应 Mittag-Leffler 矩阵函数的蒙特卡罗评估来求解。这是通过随机过程的适当乘法函数的期望值的近似来实现的，该随机过程由马尔可夫链组成，其在每个状态的停留时间都是 Mittag-Leffler 分布。由此产生的算法能够在域中方便选择的点上高效地计算解。此外，我们还介绍了如何推广该算法以计算完整的解决方案。对于几个大规模数值问题，我们的方法在共享内存和分布式内存系统中都表现出了卓越的性能，实现了近乎完美的 CPU 内核可扩展性。