Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 16, 2017

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Raphael Kruse and Yue Wu

Abstract

This paper contains an error analysis of two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the Lp(Ω;d)-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: FOR 2402

Funding statement: This research was carried out in the framework of Matheon supported by Einstein Foundation Berlin. The authors also gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft through the research unit FOR 2402 (Rough paths, stochastic partial differential equations and related topics) at TU Berlin.

Acknowledgements

The authors like to thank Wolf-Jürgen Beyn, Monika Eisenmann, Mihály Kovács, and Stig Larsson for inspiring discussions and helpful comments.

References

[1] H. Bauer, Measure and Integration Theory, De Gruyter Stud. Math. 26, De Gruyter, Berlin, 2001. 10.1515/9783110866209Search in Google Scholar

[2] D. L. Burkholder, Martingale transforms, Ann. Math. Stat. 37 (1966), 1494–1504. 10.1214/aoms/1177699141Search in Google Scholar

[3] J. C. Butcher, On Runge–Kutta processes of high order, J. Aust. Math. Soc. 4 (1964), 179–194. 10.1017/S1446788700023387Search in Google Scholar

[4] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley & Sons, Chichester, 2008. 10.1002/9780470753767Search in Google Scholar

[5] D. L. Cohn, Measure Theory, 2nd ed., Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, New York, 2013. 10.1007/978-1-4614-6956-8Search in Google Scholar

[6] I. Coulibaly and C. Lécot, A quasi-randomized Runge–Kutta method, Math. Comp. 68 (1999), no. 226, 651–659. 10.1090/S0025-5718-99-01056-XSearch in Google Scholar

[7] T. Daun, On the randomized solution of initial value problems, J. Complexity 27 (2011), no. 3–4, 300–311. 10.1016/j.jco.2010.07.002Search in Google Scholar

[8] T. Daun and S. Heinrich, Complexity of parametric initial value problems in Banach spaces, J. Complexity 30 (2014), no. 4, 392–429. 10.1016/j.jco.2014.01.002Search in Google Scholar

[9] E. Emmrich, Discrete versions of Gronwall’s lemma and their application to the numerical analysis of parabolic problems, Report Number 637-1999, Institute of Mathematics, Technische Universität Berlin, 1999. Search in Google Scholar

[10] M. Evans and T. Swartz, Approximating Integrals via Monte Carlo and Deterministic Methods, Oxford Statist. Sci. Ser., Oxford University Press, Oxford, 2000. Search in Google Scholar

[11] P. K. Friz and M. Hairer, A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext, Springer, Cham, 2014. 10.1007/978-3-319-08332-2Search in Google Scholar

[12] R. D. Grigorieff, Numerik gewöhnlicher Differentialgleichungen. Band 1: Einschrittverfahren, B. G. Teubner, Stuttgart, 1972. Search in Google Scholar

[13] I. Gyöngy, A note on Euler’s approximations, Potential Anal. 8 (1998), no. 3, 205–216. 10.1023/A:1016557804966Search in Google Scholar

[14] S. Haber, A modified Monte-Carlo quadrature, Math. Comp. 20 (1966), 361–368. 10.1090/S0025-5718-1966-0210285-0Search in Google Scholar

[15] S. Haber, A modified Monte-Carlo quadrature. II, Math. Comp. 21 (1967), 388–397. 10.1090/S0025-5718-1967-0234606-9Search in Google Scholar

[16] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems, 2nd ed., Springer Ser. Comput. Math. 8, Springer, Berlin, 1993. Search in Google Scholar

[17] J. K. Hale, Ordinary Differential Equations, 2nd ed., Robert E. Krieger, Huntington, 1980. Search in Google Scholar

[18] S. Heinrich, Complexity of initial value problems in Banach spaces, Zh. Mat. Fiz. Anal. Geom. 9 (2013), no. 1, 73–101, 116. Search in Google Scholar

[19] S. Heinrich and B. Milla, The randomized complexity of initial value problems, J. Complexity 24 (2008), no. 2, 77–88. 10.1016/j.jco.2007.09.002Search in Google Scholar

[20] A. Jentzen and A. Neuenkirch, A random Euler scheme for Carathéodory differential equations, J. Comput. Appl. Math. 224 (2009), no. 1, 346–359. 10.1016/j.cam.2008.05.060Search in Google Scholar

[21] B. Kacewicz, Optimal solution of ordinary differential equations, J. Complexity 3 (1987), no. 4, 451–465. 10.1016/0885-064X(87)90011-2Search in Google Scholar

[22] B. Kacewicz, Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity 22 (2006), no. 5, 676–690. 10.1016/j.jco.2006.03.001Search in Google Scholar

[23] O. Kallenberg, Foundations of Modern Probability, 2nd ed., Probab. Appl. (N. Y.), Springer, New York, 2002. 10.1007/978-1-4757-4015-8Search in Google Scholar

[24] A. Klenke, Probability Theory. A Comprehensive Course, 2nd ed., Universitext, Springer, London, 2014. 10.1007/978-1-4471-5361-0Search in Google Scholar

[25] P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math. 10 (2007), 235–253. 10.1112/S1461157000001388Search in Google Scholar

[26] X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing, Chichester, 2008. 10.1533/9780857099402Search in Google Scholar

[27] T. Müller-Gronbach, E. Novak and K. Ritter, Monte Carlo-Algorithmen, Springer-Lehrbuch, Springer, Heidelberg, 2012. 10.1007/978-3-540-89141-3Search in Google Scholar

[28] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, 6th ed., Universitext, Springer, Berlin, 2003. 10.1007/978-3-642-14394-6Search in Google Scholar

[29] P. Przybyłowicz and P. Morkisz, Strong approximation of solutions of stochastic differential equations with time-irregular coefficients via randomized Euler algorithm, Appl. Numer. Math. 78 (2014), 80–94. 10.1016/j.apnum.2013.12.003Search in Google Scholar

[30] G. Stengle, Numerical methods for systems with measurable coefficients, Appl. Math. Lett. 3 (1990), no. 4, 25–29. 10.1016/0893-9659(90)90040-ISearch in Google Scholar

[31] G. Stengle, Error analysis of a randomized numerical method, Numer. Math. 70 (1995), no. 1, 119–128. 10.1007/s002110050113Search in Google Scholar

Received: 2017-1-16
Accepted: 2017-1-18
Published Online: 2017-2-16
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow