## Abstract

This article studies an infinite-dimensional analog of Milstein’s scheme for finite-dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite-dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite-dimensional commutativity condition. In particular, a suitable infinite-dimensional analog of Milstein’s algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation, stochastic reaction diffusion equations and a stochastic Burgers equation, showing significant computational savings.

This is a preview of subscription content, access via your institution.

## References

Alabert, A., and Gyöngy, I. On numerical approximation of stochastic Burgers’ equation. In

*From stochastic calculus to mathematical finance*. Springer, Berlin, 2006, pp. 1–15.Allen, E. J., Novosel, S. J., and Zhang, Z. Finite element and difference approximation of some linear stochastic partial differential equations.

*Stochastics Stochastics Rep. 64*, 1-2 (1998), 117–142.Babuška, I., Tempone, R., and Zouraris, G. E. Galerkin finite element approximations of stochastic elliptic partial differential equations.

*SIAM J. Numer. Anal. 42*, 2 (2004), 800–825 (electronic).Barth, A.

*Stochastic Partial Differential Equations: Approximations and Applications*. University of Oslo, Oslo, Norway, 2009. Dissertation.Barth, A., and Lang, A. \( {L}^p \) and almost sure convergence of a Milstein scheme for stochastic partial differential equations.

*Stochastic Process. Appl.**123*, 5 (2013), 1563–1587.Barth, A., and Lang, A. Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises.

*Appl. Math. Optim*.*66*, 3 (2012), 387–413.Barth, A., Lang, A., and Schwab, C. Multilevel Monte Carlo method for parabolic stochastic partial differential equations.

*BIT**53*, 1 (2013), 3–27.Bensoussan, A., Glowinski, R., and Răşcanu, A. Approximation of the Zakai equation by the splitting up method.

*SIAM J. Control Optim. 28*, 6 (1990), 1420–1431.Bensoussan, A., Glowinski, R., and Răşcanu, A. Approximation of some stochastic differential equations by the splitting up method.

*Appl. Math. Optim. 25*, 1 (1992), 81–106.Blömker, D., and Jentzen, A. Galerkin approximations for the stochastic Burgers equation.

*SIAM J. Numer. Anal.**51*, 1 (2013), 694–715.Da Prato, G., Debussche, A., and Temam, R. Stochastic Burgers’ equation.

*NoDEA Nonlinear Differential Equations Appl. 1*, 4 (1994), 389–402.Da Prato, G., and Gatarek, D. Stochastic Burgers equation with correlated noise.

*Stochastics Stochastics Rep. 52*, 1-2 (1995), 29–41.Da Prato, G., Jentzen, A., and Röckner, M. A mild Itô formula for SPDEs. arXiv:1009.3526 (2012).

Da Prato, G., and Zabczyk, J. Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.

Da Prato, G., and Zabczyk, J. Ergodicity for infinite-dimensional systems, vol. 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1996.

Debussche, A. Weak approximation of stochastic partial differential equations: the nonlinear case.

*Math. Comp. 80*, 273 (2011), 89–117.Debussche, A., and Printems, J. Weak order for the discretization of the stochastic heat equation.

*Math. Comp. 78*, 266 (2009), 845–863.Du, Q., and Zhang, T. Numerical approximation of some linear stochastic partial differential equations driven by special additive noises.

*SIAM J. Numer. Anal. 40*, 4 (2002), 1421–1445 (electronic).Dunford, N., and Schwartz, J. T.

*Linear operators. Part II*. Wiley Classics Library. John Wiley & Sons Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication.Florchinger, P., and Le Gland, F. Time-discretization of the Zakai equation for diffusion processes observed in correlated noise.

*Stochastics Stochastics Rep. 35*, 4 (1991), 233–256.Geissert, M., Kovács, M., and Larsson, S. Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise.

*BIT 49*, 2 (2009), 343–356.Giles, M. B. Multilevel Monte Carlo path simulation.

*Oper. Res. 56*, 3 (2008), 607–617.Grecksch, W., and Kloeden, P. E. Time-discretised Galerkin approximations of parabolic stochastic PDEs.

*Bull. Austral. Math. Soc. 54*, 1 (1996), 79–85.Gyöngy, I. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I.

*Potential Anal. 9*, 1 (1998), 1–25.Gyöngy, I. A note on Euler’s approximations.

*Potential Anal. 8*, 3 (1998), 205–216.Gyöngy, I. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II.

*Potential Anal. 11*, 1 (1999), 1–37.Gyöngy, I., and Krylov, N. On the rate of convergence of splitting-up approximations for SPDEs. In

*Stochastic inequalities and applications*, vol. 56 of*Progr. Probab.*Birkhäuser, Basel, 2003, pp. 301–321.Gyöngy, I., and Krylov, N. On the splitting-up method and stochastic partial differential equations.

*Ann. Probab. 31*, 2 (2003), 564–591.Gyöngy, I., and Krylov, N. An accelerated splitting-up method for parabolic equations.

*SIAM J. Math. Anal. 37*, 4 (2005), 1070–1097 (electronic).Gyöngy, I., and Martínez, T. On numerical solution of stochastic partial differential equations of elliptic type.

*Stochastics 78*, 4 (2006), 213–231.Hairer, M., and Voss, J. Approximations to the stochastic Burgers equation.

*J. Nonlinear Sci*.*21*, 6 (2011), 897–920.Hausenblas, E. Numerical analysis of semilinear stochastic evolution equations in Banach spaces.

*J. Comput. Appl. Math. 147*, 2 (2002), 485–516.Hausenblas, E. Approximation for semilinear stochastic evolution equations.

*Potential Anal. 18*, 2 (2003), 141–186.Hausenblas, E. Finite element approximation of stochastic partial differential equations driven by Poisson random measures of jump type.

*SIAM J. Numer. Anal. 46*, 1 (2007/08), 437–471.Heinrich, S. Monte Carlo complexity of global solution of integral equations.

*J. Complexity 14*, 2 (1998), 151–175.Hutzenthaler, M., Jentzen, A., and Kloeden, P. E. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients.

*Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467*(2011), 1563–1576.Ito, K., and Rozovskii, B. Approximation of the Kushner equation for nonlinear filtering.

*SIAM J. Control Optim. 38*, 3 (2000), 893–915 (electronic).Jentzen, A. Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients.

*Potential Anal. 31*, 4 (2009), 375–404.Jentzen, A. Taylor expansions of solutions of stochastic partial differential equations.

*Discrete Contin. Dyn. Syst. Ser. B 14*, 2 (2010), 515–557.Jentzen, A., and Kloeden, P. E. Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise.

*Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465*, 2102 (2009), 649–667.Jentzen, A., and Kloeden, P. E. Taylor expansions of solutions of stochastic partial differential equations with additive noise.

*Ann. Probab. 38*, 2 (2010), 532–569.Jentzen, A., Kloeden, P. E., and Neuenkirch, A. Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients.

*Numer. Math. 112*, 1 (2009), 41–64.Jentzen, A., and Röckner, M. Regularity analysis of stochastic partial differential equations with nonlinear multiplicative trace class noise.

*J. Differential Equations 252*, 1 (2012), 114–136.Katsoulakis, M. A., Kossioris, G. T., and Lakkis, O. Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem.

*Interfaces Free Bound. 9*, 1 (2007), 1–30.Kloeden, P. E., and Platen, E.

*Numerical solution of stochastic differential equations*, vol. 23 of*Applications of Mathematics (New York)*. Springer-Verlag, Berlin, 1992.Kloeden, P. E., and Shott, S. Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs.

*J. Appl. Math. Stochastic Anal. 14*, 1 (2001), 47–53. Special issue: Advances in applied stochastics.Kossioris, G., and Zouraris, G. E. Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise.

*M2AN Math. Model. Numer. Anal*.*44*, 2 (2010), 289–322.Kovács, M., Larsson, S., and Lindgren, F. Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise.

*Numer. Algorithms 53*, 2-3 (2010), 309–320.Kovács, M., Larsson, S., and Saedpanah, F. Finite element approximation of the linear stochastic wave equation with additive noise.

*SIAM J. Numer. Anal. 48*, 2 (2010), 408–427.Lang, A., Chow, P.-L., and Potthoff, J. Almost sure convergence for a semidiscrete Milstein scheme for SPDEs of Zakai type.

*Stochastics 82*, 1-3 (2010), 315–326.Lang, A., Chow, P.-L., and Potthoff, J. Erratum: Almost sure convergence for a semidiscrete Milstein scheme for SPDEs of Zakai type.

*Stochastics*(2011). http://www.tandfonline.com/doi/abs/10.1080/17442508.2011.618884.Lord, G. J., and Rougemont, J. A numerical scheme for stochastic PDEs with Gevrey regularity.

*IMA J. Numer. Anal. 24*, 4 (2004), 587–604.Lord, G. J., and Shardlow, T. Postprocessing for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 45, 2 (2007), 870–889 (electronic).

Lord, G. J., and Tambue, A. A modified semi-implicit Euler–Maruyama Scheme for finite element discretization of SPDEs. arXiv:1004.1998 (2010), 23 pages.

Lunardi, A. Interpolation theory, second ed. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 2009.

Millet, A., and Morien, P.-L. On implicit and explicit discretization schemes for parabolic SPDEs in any dimension.

*Stochastic Process. Appl. 115*, 7 (2005), 1073–1106.Milstein, G. N. Approximate integration of stochastic differential equations.

*Teor. Verojatnost. i Primenen. 19*(1974), 583–588.Mishura, Y. S., and Shevchenko, G. M. Approximation schemes for stochastic differential equations in a Hilbert space.

*Teor. Veroyatn. Primen. 51*, 3 (2006), 476–495.Müller-Gronbach, T., and Ritter, K. An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise.

*BIT 47*, 2 (2007), 393–418.Müller-Gronbach, T., and Ritter, K. Lower bounds and nonuniform time discretization for approximation of stochastic heat equations.

*Found. Comput. Math. 7*, 2 (2007), 135–181.Müller-Gronbach, T., Ritter, K., and Wagner, T. Optimal pointwise approximation of a linear stochastic heat equation with additive space-time white noise. In

*Monte Carlo and quasi-Monte Carlo methods 2006*. Springer, Berlin, 2007, pp. 577–589.Müller-Gronbach, T., Ritter, K., and Wagner, T. Optimal pointwise approximation of infinite-dimensional Ornstein-Uhlenbeck processes.

*Stoch. Dyn. 8*, 3 (2008), 519–541.Pettersson, R., and Signahl, M. Numerical approximation for a white noise driven SPDE with locally bounded drift.

*Potential Anal. 22*, 4 (2005), 375–393.Prévôt, C., and Röckner, M.

*A concise course on stochastic partial differential equations*, vol. 1905 of*Lecture Notes in Mathematics*. Springer, Berlin, 2007.Printems, J. On the discretization in time of parabolic stochastic partial differential equations.

*M2AN Math. Model. Numer. Anal. 35*, 6 (2001), 1055–1078.Renardy, M., and Rogers, R. C.

*An introduction to partial differential equations*, vol. 13 of*Texts in Applied Mathematics*. Springer-Verlag, New York, 1993.Roth, C. Difference methods for stochastic partial differential equations.

*ZAMM Z. Angew. Math. Mech. 82*, 11-12 (2002), 821–830. 4th GAMM-Workshop “Stochastic Models and Control Theory” (Lutherstadt Wittenberg, 2001).Roth, C. A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations.

*Stoch. Anal. Appl. 24*, 1 (2006), 221–240.Roth, C. Weak approximations of solutions of a first order hyperbolic stochastic partial differential equation.

*Monte Carlo Methods Appl. 13*, 2 (2007), 117–133.Sell, G. R., and You, Y.

*Dynamics of evolutionary equations*, vol. 143 of*Applied Mathematical Sciences*. Springer-Verlag, New York, 2002.Shardlow, T. Numerical methods for stochastic parabolic PDEs.

*Numer. Funct. Anal. Optim. 20*, 1-2 (1999), 121–145.Walsh, J. B. Finite element methods for parabolic stochastic PDE’s.

*Potential Anal. 23*, 1 (2005), 1–43.Walsh, J. B. On numerical solutions of the stochastic wave equation.

*Illinois J. Math. 50*, 1-4 (2006), 991–1018 (electronic).Wang, X., and Gan, S. A Runge–Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise.

*Numer. Algorithms 62*, 2 (2013), 193–223.Yan, Y. Galerkin finite element methods for stochastic parabolic partial differential equations.

*SIAM J. Numer. Anal. 43*, 4 (2005), 1363–1384 (electronic).

## Acknowledgments

This work has been supported by the BiBoS Research Center, by the research project “Numerical solutions of stochastic differential equations with non-globally Lipschitz continuous coefficients” funded by the German Research Foundation, by the Collaborative Research Centre 701 “Spectral Structures and Topological Methods in Mathematics” funded by the German Research Foundation and by the International Graduate School “Stochastics and Real World Models” funded by the German Research Foundation. The support of Issac Newton Institute for Mathematical Sciences in Cambridge is also gratefully acknowledged where part of this was done during the special semester on “Stochastic Partial Differential Equations”. In addition, we are very grateful to Sebastian Becker for his help with the numerical simulations and to Carlo Marinelli for his helpful advice concerning Schatten norms.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by Peter E. Kloeden.

## Rights and permissions

## About this article

### Cite this article

Jentzen, A., Röckner, M. A Milstein Scheme for SPDEs.
*Found Comput Math* **15**, 313–362 (2015). https://doi.org/10.1007/s10208-015-9247-y

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10208-015-9247-y

### Keywords

- Milstein scheme
- Stochastic partial differential equation
- SPDE
- Stochastic differential equation
- SDE
- Numerical approximation
- Higher-order approximation

### Mathematics Subject Classification

- 60H35
- 65C30