A Stochastic Method for Solving Time-Fractional Differential Equations

TL;DR

This paper introduces a stochastic Monte Carlo method utilizing Mittag-Leffler distributed Markov chains to efficiently solve time-fractional PDEs modeling anomalous subdiffusion, demonstrating high scalability and performance.

Contribution

It presents a novel stochastic algorithm for solving time-fractional PDEs using Mittag-Leffler matrix functions and Markov chains, enabling efficient computation at large scales.

Findings

High efficiency in computing solutions at selected domain points

Remarkable scalability up to 16,384 CPU cores

Effective for large-scale anomalous diffusion problems

Abstract

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.