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A Stochastic Method for Solving Time-Fractional Differential Equations

Nicolas L. Guidotti, Juan Acebrón, José Monteiro

TL;DR

The paper tackles solving time-fractional PDEs by computing the Mittag-Leffler matrix function $E_{\alpha}(\mathbf{A}t^{\alpha})$ using a Monte Carlo representation based on a Markov chain with Mittag-Leffler holding times. It derives a rigorous probabilistic description, provides practical algorithms for both single-entry and full-vector solutions, and includes a dedicated random-number generator for Mittag-Leffler variates. Through extensive numerical experiments on 2D and 3D problems, it demonstrates accuracy, favorable memory usage, and near-ideal parallel scalability up to 16,384 cores, outperforming traditional deterministic solvers in large-scale settings. The work highlights the method’s suitability for high-performance computing environments and complex geometries, offering a viable alternative to time-stepping schemes for nonlocal fractional operators.

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.

A Stochastic Method for Solving Time-Fractional Differential Equations

TL;DR

The paper tackles solving time-fractional PDEs by computing the Mittag-Leffler matrix function using a Monte Carlo representation based on a Markov chain with Mittag-Leffler holding times. It derives a rigorous probabilistic description, provides practical algorithms for both single-entry and full-vector solutions, and includes a dedicated random-number generator for Mittag-Leffler variates. Through extensive numerical experiments on 2D and 3D problems, it demonstrates accuracy, favorable memory usage, and near-ideal parallel scalability up to 16,384 cores, outperforming traditional deterministic solvers in large-scale settings. The work highlights the method’s suitability for high-performance computing environments and complex geometries, offering a viable alternative to time-stepping schemes for nonlocal fractional operators.

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.
Paper Structure (14 sections, 1 theorem, 50 equations, 12 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 1 theorem, 50 equations, 12 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Description of the Probabilistic Method
  4. Mathematical description
  5. Implementation
  6. Computing the full solution vector
  7. Generating random numbers from a Mittag-Leffler distribution
  8. Numerical Results
  9. Setup
  10. Time-fractional diffusion equation
  11. Time-fractional convection-diffusion equation
  12. Complex Geometry
  13. Distributed-Memory Performance
  14. Conclusion

Key Result

Theorem 1

Let $\{X_t:t\ge 0\}$ be a stochastic process with finite state space $\Omega~=~\{1,2,\dots,n\}$ given by Such process changes states according to a Markov chain $Z=(Z_m)_{m\in \mathbb{N}}$, which takes values in $\Omega$ and $\mathbf{Q}$ is the corresponding transition matrix. Here, $T_k$ is the time of the $k$-th event, and $\mathds{1}_E$ denotes the indicator function, being $1$ or $0$ dependin

Figures (12)

  • Figure 1: Maximum absolute error as a function of the number of random paths when solving the 2D diffusion equation for $t = 0.1$ and $m = 80$.
  • Figure 2: A histogram of the maximum absolute error for $8000$ runs, each one starting with a different random seed. In each run, we solved the 2D diffusion equation with $t = 0.1$, $m = 80$ and $N_p = 2.5 \times 10^{5}$.
  • Figure 3: Variance $\sigma^2$ between the random paths as a function of the matrix size $N = m^2$ and $\alpha$ when solving the 2D diffusion equation at a single point located at the centre of the mesh for $t = 0.1$. The number of random paths was kept fixed to $10^{10}$.
  • Figure 4: Elapsed time for solving the 2D diffusion equation as a function of the time $t$ and the parameter $\alpha$ for $m = 80$. The number of random paths was kept fixed at $10^5$.
  • Figure 5: Elapsed time for solving the 2D diffusion equation as a function of the matrix size $N = m^2$ and the parameter $\alpha$ for $t~=~0.1$. The number of random paths was kept fixed at $10^6$.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof