American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jcd.2021023
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Numerical preservation issues in stochastic dynamical systems by $ \vartheta $-methods

Raffaele D'Ambrosio , and  Stefano Di Giovacchino

Department of Information Engineering and Computer Science and Mathematics, University of L'Aquila, Via Vetoio, Loc. Coppito - 67100 L'Aquila, Italy

* Corresponding author: Raffaele D'Ambrosio

Received  February 2021 Revised  August 2021 Early access December 2021

Fund Project: This work is supported by GNCS-INDAM project and by PRIN2017-MIUR project 2017JYCLSF entitled "Structure preserving approximation of evolutionary problems". The authors are member of the INDAM Research group GNCS

This paper analyzes conservation issues in the discretization of certain stochastic dynamical systems by means of stochastic $ \vartheta $-mehods. The analysis also takes into account the effects of the estimation of the expected values by means of Monte Carlo simulations. The theoretical analysis is supported by a numerical evidence on a given stochastic oscillator, inspired by the Duffing oscillator.

Keywords: Stochastic oscillators, stochastic $ \vartheta $-methods.
Mathematics Subject Classification: 65C30.
Citation: Raffaele D'Ambrosio, Stefano Di Giovacchino. Numerical preservation issues in stochastic dynamical systems by $ \vartheta $-methods. Journal of Computational Dynamics, doi: 10.3934/jcd.2021023
References:
[1]

G. Berkolaiko, E. Buckwar, C. Kelly and A. Rodkina, Almost sure asymptotic stability analysis of the $\theta$-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations, LMS J. Comp. Math., 15 (2012), 71-83.  doi: 10.1112/S1461157012000010.  Google Scholar

[2]

E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods, Math. Comput. Simul., 81 (2011), 1110-1127.  doi: 10.1016/j.matcom.2010.09.015.  Google Scholar

[3]

E. Buckwar and T. Sickenberger, A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems, Appl. Numer. Math., 62 (2012), 842-859.  doi: 10.1016/j.apnum.2012.03.002.  Google Scholar

[4]

K. Burrage, I. Lenane and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.  doi: 10.1137/050646032.  Google Scholar

[5]

P. M. Burrage and K. Burrage, Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algorithms, 65 (2014), 519-532.  doi: 10.1007/s11075-013-9796-6.  Google Scholar

[6]

C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math., 46 (2020), art. no. 27. doi: 10.1007/s10444-020-09771-5.  Google Scholar

[7]

V. Citro and R. D'Ambrosio, Long-term analysis of stochastic $\theta$-methods for damped stochastic oscillators, Appl. Numer. Math., 150 (2020), 18-26.  doi: 10.1016/j.apnum.2019.08.011.  Google Scholar

[8]

D. Conte, R. D'Ambrosio and B. Paternoster, On the stability of $\vartheta$-methods for stochastic Volterra integral equations, Disc. Cont. Dyn. Sys. - Series B, 23 (2018), 2695-2708.  doi: 10.3934/dcdsb.2018087.  Google Scholar

[9]

R. D'Ambrosio, G. Giordano, B. Paternoster and A. Ventola, Perturbative analysis of stochastic Hamiltonian problems under time discretizations, Appl. Math. Lett., 120 (2021), article number 107223. doi: 10.1016/j.aml.2021.107223.  Google Scholar

[10]

R. D'Ambrosio and S. D. Giovacchino, Mean-square contractivivity of stochastic $\theta$-methods, Commun. Nonlinear Sci. Numer. Simul, 96 (2021), 105671.  doi: 10.1016/j.cnsns.2020.105671.  Google Scholar

[11]

R. D'Ambrosio, M. Moccaldi and B. Paternoster, Numerical preservation of long-term dynamics by stochastic two-step methods, Discr. Cont. Dyn. Sys. B, 23 (2018), 2763-2773.  doi: 10.3934/dcdsb.2018105.  Google Scholar

[12]

R. D'Ambrosio and C. Scalone, On the numerical structure preservation of nonlinear damped stochastic oscillators, Numer. Algorithm, 86 (2021), 933-952.  doi: 10.1007/s11075-020-00918-5.  Google Scholar

[13]

H. de la Cruz, J. C. Jimenez and J. P. Zubelli, Locally linearized methods for the sim- ulation of stochastic oscillators driven by random forces, BIT, 57 (2017), 123-151.  doi: 10.1007/s10543-016-0620-2.  Google Scholar

[14]

G. Failla and A. Pirrotta, On the stochastic response of a fractionally-damped Duffing oscillator, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 5131-5142.  doi: 10.1016/j.cnsns.2012.03.033.  Google Scholar

[15]

C. W. Gardiner, Handbook of Stochastic Methods, for Physics, 3$^{rd}$ edition, Chemistry and the Natural Sciences, 13, Springer-Verlag, 2004.  Google Scholar

[16]

M. B. Giles, Multi-level Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.  Google Scholar

[17]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta mehods, SIAM J. Numer. Anal., 38 (2000), 753-769.  doi: 10.1137/S003614299834736X.  Google Scholar

[18]

D. J. Higham and P. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.  Google Scholar

[19]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[20]

K. Nouri, H. Ranjbar and J. C. Cortés López, Modifying the split-step $\theta$-method with harmonic-mean term for stochastic differential equations, Int. J. Numer. Anal. Model., 17 (2020), 662-678.   Google Scholar

[21]

D. Roy, A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 539-566.  doi: 10.1098/rspa.2000.0681.  Google Scholar

[22]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409.  Google Scholar

[23]

J. Yalim, B. D. Welfert and J. M. Lopez, Evaluation of closure strategies for a periodically-forced Duffing oscillator with slowly modulated frequency subject to Gaussian white noise, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 144-158.  doi: 10.1016/j.cnsns.2016.08.003.  Google Scholar

[24]

J. Zhang and L. Wang, A new symplectic method for a linear stochastic oscillator via stochastic variational integrators, AIP Conference Proceedings, 1479 (2012), 1772-1775.  doi: 10.1063/1.4756519.  Google Scholar

show all references

References:
[1]

G. Berkolaiko, E. Buckwar, C. Kelly and A. Rodkina, Almost sure asymptotic stability analysis of the $\theta$-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations, LMS J. Comp. Math., 15 (2012), 71-83.  doi: 10.1112/S1461157012000010.  Google Scholar

[2]

E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods, Math. Comput. Simul., 81 (2011), 1110-1127.  doi: 10.1016/j.matcom.2010.09.015.  Google Scholar

[3]

E. Buckwar and T. Sickenberger, A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems, Appl. Numer. Math., 62 (2012), 842-859.  doi: 10.1016/j.apnum.2012.03.002.  Google Scholar

[4]

K. Burrage, I. Lenane and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.  doi: 10.1137/050646032.  Google Scholar

[5]

P. M. Burrage and K. Burrage, Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algorithms, 65 (2014), 519-532.  doi: 10.1007/s11075-013-9796-6.  Google Scholar

[6]

C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math., 46 (2020), art. no. 27. doi: 10.1007/s10444-020-09771-5.  Google Scholar

[7]

V. Citro and R. D'Ambrosio, Long-term analysis of stochastic $\theta$-methods for damped stochastic oscillators, Appl. Numer. Math., 150 (2020), 18-26.  doi: 10.1016/j.apnum.2019.08.011.  Google Scholar

[8]

D. Conte, R. D'Ambrosio and B. Paternoster, On the stability of $\vartheta$-methods for stochastic Volterra integral equations, Disc. Cont. Dyn. Sys. - Series B, 23 (2018), 2695-2708.  doi: 10.3934/dcdsb.2018087.  Google Scholar

[9]

R. D'Ambrosio, G. Giordano, B. Paternoster and A. Ventola, Perturbative analysis of stochastic Hamiltonian problems under time discretizations, Appl. Math. Lett., 120 (2021), article number 107223. doi: 10.1016/j.aml.2021.107223.  Google Scholar

[10]

R. D'Ambrosio and S. D. Giovacchino, Mean-square contractivivity of stochastic $\theta$-methods, Commun. Nonlinear Sci. Numer. Simul, 96 (2021), 105671.  doi: 10.1016/j.cnsns.2020.105671.  Google Scholar

[11]

R. D'Ambrosio, M. Moccaldi and B. Paternoster, Numerical preservation of long-term dynamics by stochastic two-step methods, Discr. Cont. Dyn. Sys. B, 23 (2018), 2763-2773.  doi: 10.3934/dcdsb.2018105.  Google Scholar

[12]

R. D'Ambrosio and C. Scalone, On the numerical structure preservation of nonlinear damped stochastic oscillators, Numer. Algorithm, 86 (2021), 933-952.  doi: 10.1007/s11075-020-00918-5.  Google Scholar

[13]

H. de la Cruz, J. C. Jimenez and J. P. Zubelli, Locally linearized methods for the sim- ulation of stochastic oscillators driven by random forces, BIT, 57 (2017), 123-151.  doi: 10.1007/s10543-016-0620-2.  Google Scholar

[14]

G. Failla and A. Pirrotta, On the stochastic response of a fractionally-damped Duffing oscillator, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 5131-5142.  doi: 10.1016/j.cnsns.2012.03.033.  Google Scholar

[15]

C. W. Gardiner, Handbook of Stochastic Methods, for Physics, 3$^{rd}$ edition, Chemistry and the Natural Sciences, 13, Springer-Verlag, 2004.  Google Scholar

[16]

M. B. Giles, Multi-level Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.  Google Scholar

[17]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta mehods, SIAM J. Numer. Anal., 38 (2000), 753-769.  doi: 10.1137/S003614299834736X.  Google Scholar

[18]

D. J. Higham and P. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.  Google Scholar

[19]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[20]

K. Nouri, H. Ranjbar and J. C. Cortés López, Modifying the split-step $\theta$-method with harmonic-mean term for stochastic differential equations, Int. J. Numer. Anal. Model., 17 (2020), 662-678.   Google Scholar

[21]

D. Roy, A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 539-566.  doi: 10.1098/rspa.2000.0681.  Google Scholar

[22]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409.  Google Scholar

[23]

J. Yalim, B. D. Welfert and J. M. Lopez, Evaluation of closure strategies for a periodically-forced Duffing oscillator with slowly modulated frequency subject to Gaussian white noise, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 144-158.  doi: 10.1016/j.cnsns.2016.08.003.  Google Scholar

[24]

J. Zhang and L. Wang, A new symplectic method for a linear stochastic oscillator via stochastic variational integrators, AIP Conference Proceedings, 1479 (2012), 1772-1775.  doi: 10.1063/1.4756519.  Google Scholar

Figure 1.  Variance error growth with respect to $ \sigma $ computed by Euler-Maruyama method, that is (8) with $ \vartheta = 0 $, at $ T = 100 $, computed with respect to $ 1000 $ paths
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Variance error growth with respect to the time $ t $ computed by Euler-Maruyama method
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Growth of the errors for the mean and the variance of $ \Theta(T) $ computed by the stochastic trapezoidal rule with $ T = 100 $, $ M = 1000 $, $ h = 10^{-2} $, $ \varepsilon = 10^{-3} $ and $ \omega_0 = 1.3 $
$ \sigma $ $ \Bigl|\mathbb{E}[\Theta_T] - \mathbb{E}[\Theta(T)]\Bigr| $ $ \Bigl|\mathbb{V}[\Theta_T] - \mathbb{V}[\Theta(T)]\Bigr| $
$ 10^{-12} $ $ 3.98 \times 10^{-13} $ $ 5.26 \times 10^{-24} $
$ 10^{-9} $ $ 2.76 \times 10^{-10} $ $ 5.18 \times 10^{-18} $
$ 10^{-6} $ $ 2.76 \times 10^{-7} $ $ 5.18 \times 10^{-12} $
$ 10^{-3} $ $ 2.76 \times 10^{-4} $ $ 5.18 \times 10^{-6} $
$ 10^{0} $ $ 2.76 \times 10^{-1} $ $ 5.18 \times 10^{0} $
Table Options

Download as excel

$ \sigma $ $ \Bigl|\mathbb{E}[\Theta_T] - \mathbb{E}[\Theta(T)]\Bigr| $ $ \Bigl|\mathbb{V}[\Theta_T] - \mathbb{V}[\Theta(T)]\Bigr| $
$ 10^{-12} $ $ 3.98 \times 10^{-13} $ $ 5.26 \times 10^{-24} $
$ 10^{-9} $ $ 2.76 \times 10^{-10} $ $ 5.18 \times 10^{-18} $
$ 10^{-6} $ $ 2.76 \times 10^{-7} $ $ 5.18 \times 10^{-12} $
$ 10^{-3} $ $ 2.76 \times 10^{-4} $ $ 5.18 \times 10^{-6} $
$ 10^{0} $ $ 2.76 \times 10^{-1} $ $ 5.18 \times 10^{0} $
[1]

Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087
[2]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[3]

Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211
[4]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
[5]

Ying-Cheng Lai, Kwangho Park. Noise-sensitive measure for stochastic resonance in biological oscillators. Mathematical Biosciences & Engineering, 2006, 3 (4) : 583-602. doi: 10.3934/mbe.2006.3.583
[6]

Jie Zhang, Yue Wu, Liwei Zhang. A class of smoothing SAA methods for a stochastic linear complementarity problem. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 145-156. doi: 10.3934/naco.2012.2.145
[7]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[8]

Lijin Wang, Pengjun Wang, Yanzhao Cao. Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021095
[9]

Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021044
[10]

Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105
[11]

Christopher Rackauckas, Qing Nie. Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2731-2761. doi: 10.3934/dcdsb.2017133
[12]

Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai, Jie Sun. Variable metric proximal stochastic variance reduced gradient methods for nonconvex nonsmooth optimization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021084
[13]

Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1
[14]

Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021071
[15]

Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233
[16]

Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao. Improving sampling accuracy of stochastic gradient MCMC methods via non-uniform subsampling of gradients. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021157
[17]

Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen, Jens Starke. Stochastic control of traffic patterns. Networks & Heterogeneous Media, 2013, 8 (1) : 261-273. doi: 10.3934/nhm.2013.8.261
[18]

John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843
[19]

Yu Wu, Xiaopeng Zhao, Mingjun Zhang. Dynamics of stochastic mutation to immunodominance. Mathematical Biosciences & Engineering, 2012, 9 (4) : 937-952. doi: 10.3934/mbe.2012.9.937
[20]

David J. Aldous. A stochastic complex network model. Electronic Research Announcements, 2003, 9: 152-161.

 Impact Factor: 

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy