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 16,384 CPU cores.

翻译：我们提出一种随机方法，用于高效计算描述亚扩散型反常扩散问题的时间分数阶偏微分方程的数值解。在空间离散化后，通过蒙特卡洛方法评估相应的Mittag-Leffler矩阵函数，求解由此产生的分数阶线性方程组。该方法通过逼近某随机过程适当乘性泛函的期望值来实现，该随机过程由马尔可夫链构成，其在每个状态的停留时间服从Mittag-Leffler分布。所提出的算法能够在域内任意选取的点上高效计算解，并进一步推广以求解完整解。针对多个大规模数值问题，该方法在共享内存与分布式内存系统中均展现出卓越性能，在多达16,384个CPU核心上实现了近乎完美的可扩展性。