zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Efficient simulation of nonlinear parabolic SPDEs with additive noise. (English) Zbl 1223.60050
The article discusses a method for the numerical approximation of nonlinear SPDEs with additive noise which allow for a mild solution. In the presented setting, the nonlinearity satisfies standard global Lipschitz assumptions. The numerical method is an extension of the scheme developed by {\it A. Jentzen} and {\it P. E. Kloeden} [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, No. 2102, 649--667 (2009; Zbl 1186.65011)] to the present class of SPDEs. It is an exponential Euler scheme using suitable functionals of the noise process which yield an improvement in the order of convergence. It is noted that the applicability of the method is restricted to SPDEs for which the eigenfunctions of the linear operator and the covariance operator of the driving noise process coincide and are explicitly known. The authors prove the strong convergence of this problem, i.e., the mean-square convergence at the endpoint of a time interval $[0,T]$ in the appropriate Hilbert space of the solution, and obtain an expression for the order of convergence. The advantage of the presented method is that it is asymptotically more efficient than, e.g., the linear implicit Euler method, i.e., a given precision $\varepsilon>0$ is obtained with a smaller number of computational operations and independent standard normal random variables. In particular, the new method possesses an effort of $O(\varepsilon^{-2})$ compared to $O(\varepsilon^{-3})$ of the linear-implicit Euler method. Numerical examples are presented to illustrate the theoretical results: the improvement in convergence order of the new method as well as its higher efficiency. For comparison, a linear implicit Euler-spectral Galerkin method is used. The numerical examples are simple reaction diffusion equations on the one-dimensional unit interval with additive noise and twice continuously differentiable nonlinearity with bounded derivatives. Finally, a numerical example with nonglobally Lipschitz continuous nonlinearity is implemented and the pathwise error of the method is calculated. Again, a higher convergence order is experimentally found compared to the linear-implicit Euler method.

60H35Computational methods for stochastic equations
60H15Stochastic partial differential equations
35R60PDEs with randomness, stochastic PDE
65C30Stochastic differential and integral equations
Full Text: DOI arXiv
[1] Blömker, D. and Jentzen, A. (2009). Galerkin approximations for the stochastic burgers equation. Preprint, Institute for Mathematics, Univ. Augsburg. Available at . · Zbl 1267.60071 · 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. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[3] Gyöngy, I. (1998). A note on Euler’s approximations. Potential Anal. 8 205-216. · Zbl 0946.60059 · 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. · Zbl 0944.60074 · doi:10.1023/A:1008699504438
[5] Gyöngy, I. and Millet, A. (2005). On discretization schemes for stochastic evolution equations. Potential Anal. 23 99-134. · Zbl 1067.60049 · 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. · Zbl 1168.60025 · doi:10.1007/s11118-008-9105-5
[7] Hausenblas, E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18 141-186. · Zbl 1015.60053 · doi:10.1023/A:1020552804087
[8] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840 . Springer, Berlin. · Zbl 0456.35001
[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. · Zbl 1228.65014 · doi:10.1098/rspa.2010.0348
[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. · Zbl 1186.65011 · 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. · Zbl 1220.35202 · doi:10.1214/09-AOP500
[12] Pettersson, R. and Signahl, M. (2005). Numerical approximation for a white noise driven SPDE with locally bounded drift. Potential Anal. 22 375-393. · Zbl 1065.35221 · 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. · Zbl 1123.60001
[14] Sell, G. R. and You, Y. (2002). Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143 . Springer, New York. · Zbl 1254.37002
[15] Walsh, J. B. (2005). Finite element methods for parabolic stochastic PDE’s. Potential Anal. 23 1-43. · Zbl 1065.60082 · doi:10.1007/s11118-004-2950-y