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.Comment: Submitted to the Journal of Computational Physic