The Annals of Applied Probability

Efficient simulation of nonlinear parabolic SPDEs with additive noise

Arnulf Jentzen, Peter Kloeden, and Georg Winkel

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Abstract

Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649–667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The key idea was to use suitable linear functionals of the noise process in the numerical scheme which allow a higher approximation order to be obtained. Following this approach, a new simplified version of the scheme in the above named reference is proposed and analyzed in this article. The main advantage of the convergence result given here is the higher convergence order for nonlinear parabolic SPDEs with additive noise, although the used numerical scheme is very simple to simulate and implement.

Article information

Source
Ann. Appl. Probab. Volume 21, Number 3 (2011), 908-950.

Dates
First available in Project Euclid: 2 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1307020387

Digital Object Identifier
doi:10.1214/10-AAP711

Mathematical Reviews number (MathSciNet)
MR2830608

Zentralblatt MATH identifier
1223.60050

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 65C30: Stochastic differential and integral equations

Keywords
Exponential Euler scheme linear implicit Euler scheme computational cost stochastic reaction diffusion equations

Citation

Jentzen, Arnulf; Kloeden, Peter; Winkel, Georg. Efficient simulation of nonlinear parabolic SPDEs with additive noise. Ann. Appl. Probab. 21 (2011), no. 3, 908--950. doi:10.1214/10-AAP711. https://projecteuclid.org/euclid.aoap/1307020387.


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References

  • [1] Blömker, D. and Jentzen, A. (2009). Galerkin approximations for the stochastic burgers equation. Preprint, Institute for Mathematics, Univ. Augsburg. Available at http://opus.bibliothek.uni-augsburg.de/volltexte/2009/1444/.
  • [2] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1207136
  • [3] Gyöngy, I. (1998). A note on Euler’s approximations. Potential Anal. 8 205–216.
    Mathematical Reviews (MathSciNet): MR1625576
    Digital Object Identifier: doi:10.1023/A:1008605221617
  • [4] Gyöngy, I. (1999). Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space–time white noise. II. Potential Anal. 11 1–37.
    Mathematical Reviews (MathSciNet): MR1699161
    Digital Object Identifier: doi:10.1023/A:1008699504438
  • [5] Gyöngy, I. and Millet, A. (2005). On discretization schemes for stochastic evolution equations. Potential Anal. 23 99–134.
    Mathematical Reviews (MathSciNet): MR2139212
    Digital Object Identifier: doi:10.1007/s11118-004-5393-6
  • [6] Gyöngy, I. and Millet, A. (2009). Rate of convergence of space–time approximations for stochastic evolution equations. Potential Anal. 30 29–64.
    Mathematical Reviews (MathSciNet): MR2465711
    Digital Object Identifier: doi:10.1007/s11118-008-9105-5
  • [7] Hausenblas, E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18 141–186.
    Mathematical Reviews (MathSciNet): MR1953619
    Digital Object Identifier: doi:10.1023/A:1020552804087
  • [8] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR610244
  • [9] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2011). Strong and weak divergence in finite time of Euler’s method for SDEs with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 1563–1576.
  • [10] Jentzen, A. and Kloeden, P. E. (2009). 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. 465 649–667.
    Mathematical Reviews (MathSciNet): MR2471778
    Zentralblatt MATH: 1186.65011
    Digital Object Identifier: doi:10.1098/rspa.2008.0325
  • [11] Jentzen, A. and Kloeden, P. E. (2010). Taylor expansions of solutions of stochastic partial differential equations with additive noise. Ann. Probab. 38 532–569.
    Mathematical Reviews (MathSciNet): MR2642885
    Zentralblatt MATH: 05695670
    Digital Object Identifier: doi:10.1214/09-AOP500
    Project Euclid: euclid.aop/1268143526
  • [12] Pettersson, R. and Signahl, M. (2005). Numerical approximation for a white noise driven SPDE with locally bounded drift. Potential Anal. 22 375–393.
    Mathematical Reviews (MathSciNet): MR2135265
    Digital Object Identifier: doi:10.1007/s11118-004-1329-4
  • [13] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2329435
  • [14] Sell, G. R. and You, Y. (2002). Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1873467
    Zentralblatt MATH: 01710797
  • [15] Walsh, J. B. (2005). Finite element methods for parabolic stochastic PDE’s. Potential Anal. 23 1–43.
    Mathematical Reviews (MathSciNet): MR2136207
    Digital Object Identifier: doi:10.1007/s11118-004-2950-y

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