Stochastic Parareal Algorithm for Stochastic Differential Equations

TL;DR

The paper develops and analyzes the Stochastic Parareal (P_s) method for stochastic differential equations, showing that stochastic perturbations can yield linear convergence over unbounded time horizons compared to the classical Parareal. It provides a mean-square stability framework based on the stochastic θ-method, derives explicit linear convergence bounds under four sampling rules, and validates the approach with extensive numerical experiments on linear and nonlinear SDEs. The results indicate that increasing the number of samples $m$ generally improves convergence (reducing the required iterations $k$), though diminishing returns and higher computational cost arise beyond a threshold. Overall, P_s offers a promising, parallel-in-time alternative for solving SDEs with notable efficiency gains, while also suggesting future work in nonlinear settings and robust time-integration schemes.

Abstract

This paper analyzes the SParareal algorithm for stochastic differential equations (SDEs). Compared to the classical Parareal algorithm, the SParareal algorithm accelerates convergence by introducing stochastic perturbations, achieving linear convergence over unbounded time intervals. We first revisit the classical Parareal algorithm and stochastic Parareal algorithm. Then we investigate mean-square stability of the SParareal algorithm based on the stochastic $θ$-method for SDEs, deriving linear error bounds under four sampling rules. Numerical experiments demonstrate the superiority of the SParareal algorithm in solving both linear and nonlinear SDEs, reducing the number of iterations required compared to the classical Parareal algorithm.

Figures (11)

• Figure 1: Illustration of the sampling and propagation process of the SParareal algorithm after $k = 1$. The fine solution is represented by the blue line, the coarse solution at $k = 0$ by the green line, the fine solution at $k = 0$ by the cyan line, the coarse solution at $k = 1$ by the red line, and the predicted corrected solution at $k = 1$ by red dots. For $m = 5$, four samples $\alpha_{n,m}^1$ (purple dots) are drawn from distributions with means $U^1_2$ and $U^1_3$, and some finite standard deviations respectively. The best-selected sample $\widehat{\alpha}^1_n$ is then propagated using $\mathcal{G}_{\Delta T}$ (orange-yellow line).
• Figure 2: \ref{['fig:subfig1']} Numerical solutions obtained by $\mathcal{F}_{\Delta t}$ and $P_s$. Only one time-independent simulation is shown for clarity. \ref{['fig:subfig2']} Numerical errors for $P$ (black line) and $P_s$ (blue line). The horizontal red dashed line indicates the error threshold $\rho = 10^{-12}$.
• Figure 3: Linear error bounds from Corollary \ref{['Corollary']} (red line) compared with numerical errors. \ref{['fig:subfig3']} Errors for sampling rules 1 and 3 (blue and green lines). \ref{['fig:subfig4']} Errors for sampling rules 2 and 4 (purple and cyan lines).
• Figure 4: Numerical errors for $P_s$ with varying sample sizes $m$. Sampling rules 1/3 are represented in blue and black, while sampling rules 2/4 are shown in purple and cyan. From left to right, $m$ takes values 2, 7, 20, 125, 500, and 1000.
• Figure 5: Numerical errors and convergence behavior for $P_s$ and $P$ with $T \in [0,9]$, $\Delta T = \frac{9}{40}$, and $\Delta t = \frac{9}{80}$. \ref{['fig:subfig6']} Comparsion of $P$ and $P_s$. \ref{['fig:subfig7']} Numerical errors for sampling rules 1 and 3 (blue and green lines). \ref{['fig:subfig8']} Numerical errors for sampling rules 2 and 4 (purple and cyan lines).

Theorems & Definitions (11)

• Lemma 2.1
• Theorem 3.1
• proof
• Remark 3.1
• Corollary 3.1
• proof
• Remark 3.2
• Example 1
• Example 2
• Example 3