Efficient Numerical Simulation for High-Dimensional Stochastic Systems: a Parallel Waveform Relaxation Approach Based on Stochastic Runge-Kutta Methods

Abstract

In statistical mechanics and its applications, stochastic differential equations (SDEs) are crucial for modeling complex systems. This study proposes a parallel waveform relaxation (WR) framework for high-dimensional SDEs, based on the stochastic Runge-Kutta (SRK) method. The scheme, designed for parallel computing, enhances computational efficiency, particularly in applications such as the simulation of stochastic particle dynamics. Integrating the SRK method with a limit-based iterative scheme ensures convergence and high-order convergence for some SDEs. A rigorous analysis confirms the method’s reliability and effectiveness. Numerical experiments on linear and non-linear high-dimensional SDEs show the proposed scheme outperforms the original SRK method in both convergence rate and computational time. These results are beneficial for applications in statistical mechanics and reduce the computational load in large-scale system analysis.

Keywords

Stochastic differential equations (SDEs) / Stochastic Runge-Kutta (SRK) method / Waveform relaxation (WR) method / Statistical mechanics and its applications / 60H10 / 65L06 / 65P10

References

[1] Al-Khaleel, M.D., Gander, M.J., Kumbhar, P.M.: Optimized Schwarz waveform relaxation methods for the telegrapher equation. SIAM J. Sci. Comput. 46(6), 3528–3551 (2024). https://doi.org/10.1137/24M1642962

[2] Alfredo B, Marino Z. The use of Runge-Kutta formulae in waveform relaxation methods. Appl. Numer. Math., 1993, 11: 95-114

[3] Bolten M, Hahne J. An SDE waveform-relaxation method with application in distributed neural network simulations. Proc. Appl. Math. Mech., 2019, 19(1 e201900373

[4] Burrage K, Burrage PM. Order conditions of stochastic Runge-Kutta methods by B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B$$\end{document}-series. SIAM J. Numer. Anal., 2000, 385): 1626-1646

[5] Chouly F, Gander MJ, Martin V. Optimized Schwarz waveform relaxation methods for wave-heat coupling in one dimensional bounded domains. Numer. Algorithms, 2025, 100: 1739-1763

[6] Fan ZC. Discrete time waveform relaxation method for stochastic delay differential equations. Appl. Math. Comput., 2010, 217: 3903-3909

[7] Fan ZC. Waveform relaxation method for stochastic differential equations with constant delay. Appl. Numer. Math., 2011, 61: 229-240

[8] Fan ZC. SOR waveform relaxation methods for stochastic differential equations. Appl. Math. Comput., 2013, 219: 4992-5003

[9] Fan ZC. Using waveform relaxation methods to approximate neutral stochastic functional differential equation systems. Appl. Math. Comput., 2015, 263: 151-164

[10] Fan ZC. Convergence of discrete time waveform relaxation methods. Numer. Algorithms, 2019, 80: 469-483

[11] Garrappa R. An analysis of convergence for two-stage waveform relaxation methods. J. Comput. Appl. Math., 2004, 169(2): 377-392

[12] Giona M, Venditti C, Adrover A. On the long-term simulation of stochastic differential equations for predicting effective dispersion coefficients. Physica A: Statistical Mechanics and Its Applications, 2020, 543 123392

[13] Hahne, J., Dahmen, D., Schuecker, J., Frommer, A., Bolten, M., Helias, M., Diesmann, M.: Integration of continuous-time dynamics in a spiking neural network simulator. Front. Neuroinform. 11(34) (2017). https://doi.org/10.3389/fninf.2017.00034

[14] Hahne J, Helias M, Kunkel S, Igarashi J, Bolten M, Frommer A, Diesmann M. A unified framework for spiking and gap-junction interactions in distributed neuronal network simulations. Front. Neuroinform., 2015, 9(22): 22

[15] Hout, K.J.I.: On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations. Appl. Numer. Math. 18(1/2/3), 175–190 (1995). https://doi.org/10.1016/0168-9274(95)00052-V

[16] Janssen J, Vandewalle S. On SOR waveform relaxation methods. SIAM J. Numer. Anal., 1997, 3462456-2481

[17] Keller SHA, Harald N. Monte Carlo and Quasi-Monte Carlo Methods 2006, 2007, New York, Springer

[18] Kloeden PE, Platen E. Numerical Solution of Stochastic Differential Equations, 1992, New York, Springer

[19] Lelarasmee E, Ruehli A, Sangiovanni-Vincentelli A. The waveform relaxaiton method for time domain analysis of large scale integrated circuits. IEEE Trans. CAD IC Syst., 1982, 1: 131-145

[20] Liu MZ, Cao WR, Fan ZC. Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math., 2004, 170: 255-268

[21] Malacarne, M.F., Marcio, A.V., Franco, S.R.: A parallelizable method for two-dimensional wave propagation using subdomains in time with multigrid and waveform relaxation. Acta Sci.-Technol. 47(1) (2025). https://doi.org/10.4025/actascitechnol.v47i1.70187

[22] Mandal BC, Tomer D. Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods for PDEs with time delay. IMA J. Appl. Math., 2025, 90(2): 115-139

[23] Mao X. Stochastic Differential Equations and Their Applications, 1997, Chichester, Harwood

[24] Nevanlinna O. Remarks on Picard-Lindelöf iteration Part I. BIT Numer. Math., 1989, 29(2): 328-346

[25] Nichols NK. On the convergence of two-stage iterative processes for solving linear equations. SIAM J. Numer. Anal., 1973, 10(3): 460-469

[26] Rossler A. Runge-Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal., 2010, 48(3): 922-952

[27] Sanda J, Burrage K. A Jacobi waveform relaxation method for ODEs. SIAM J. Sci. Comput., 1998, 20(2): 534-552

[28] Schurz H, Schneider KR. Waveform relaxation methods for stochastic differential equations. Int. J. Numer. Anal. Model., 2006, 3(2): 232-254

[29] Wang J, Bai ZZ. Convergence analysis of two-stage waveform relaxation method for the initial value problems. Appl. Math. Comput., 2006, 172(2): 797-808

[30] Wang XL, Feng J, Liu Q, Li YG, Xu Y. Neural network-based parameter estimation of stochastic differential equations driven by Lévy noise. Physica A: Statistical Mechanics and Its Applications, 2022, 606 128146

[31] Wang XJ, Wu JY, Dong BZ. Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition. BIT Numer. Math., 2020, 60: 759-790

Funding

Natural Science Foundation of Shandong Province(ZR2022QA051)

National Outstanding Youth Science Fund Project of National Natural Science Foundation of China(12401519)

RIGHTS & PERMISSIONS

Shanghai University