zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds. (English) Zbl 1208.65017

The authors consider the pathwise numerical approximation of the stochastic evolution equation

du(t)=[Au(t)+F(u(t))]dt+dW(t);t0;u(0)=u 0 ;

on the Hilbert space H=L 2 ([a;b] d ), where A is the generator of an analytic semigroup H, u 0 D(A), W is an H-valued Q-Wiener process, the mapping F is nonlinear with bounded first and second derivatives. They also assume that A and Q have common eigenfunctions. However, the covariance operator Q needs not to be a trace-class operator.

To treat this equation numerically, the authors discretise in space by a Galerkin method and in time by a stochastic exponential integrator. They estimate the rates of convergence for both pth-mean and pathwise errors of approximations. It appears that for spatially regular noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces. The results are illustrated by the two-dimensional Allen–Cahn equation.

MSC:
65C30Stochastic differential and integral equations
60H15Stochastic partial differential equations
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
60H35Computational methods for stochastic equations
35R60PDEs with randomness, stochastic PDE
References:
[1]Lord, G. J.; Rougemont, J.: A numerical scheme for stochastic pdes with gevrey regularity, IMA J. Numer. anal. 24, 587-604 (2004) · Zbl 1073.65008 · doi:10.1093/imanum/24.4.587
[2]Lord, G. J.; Shardlow, T.: Postprocessing for stochastic parabolic partial differential equations, SIAM J. Numer. anal. 45, 870-899 (2007) · Zbl 1140.60036 · doi:10.1137/050640138
[3]Gyöngy, I.: A note on Euler’s approximations, Potential anal. 8, 205-216 (1998)
[4]Kloeden, P. E.; Neuenkirch, A.: The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. math. 10, 235-253 (2007) · Zbl 1223.60051 · doi:10.1112/S1461157000001388
[5]Milstein, G. V.; Tretyakov, M. V.: Solving parabolic stochastic partial differential equations via averaging over characteristics, Math. comp. 78, 2075-2106 (2009) · Zbl 1198.65033 · doi:10.1090/S0025-5718-09-02250-9
[6]Gyöngy, I.; Nualart, D.: Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space–time white noise, Stochastic process. Appl. 58, 57-72 (1995) · Zbl 0832.60068 · doi:10.1016/0304-4149(95)00010-5
[7]Gyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space–time white noise I, Potential anal. 9, 1-25 (1998) · Zbl 0915.60069 · doi:10.1023/A:1008615012377
[8]Gyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space–time white noise II, Potential anal. 11, 1-37 (1999) · Zbl 0944.60074 · doi:10.1023/A:1008699504438
[9]D. Blömker, A. Jentzen, Galerkin approximations for the stochastic Burgers equation, Working Paper, 2010.
[10]Jentzen, A.: Pathwise numerical approximations of spdes with additive noise under non-global Lipschitz coefficients, Potential anal. 31, 375-404 (2009) · Zbl 1176.60051 · doi:10.1007/s11118-009-9139-3
[11]A. Barth, A finite element method for martingale-driven stochastic partial differential equations, COSA (in press).
[12]A. Barth, A. Lang, Almost sure convergence of a Galerkin–Milstein approximation for stochastic partial differential equations, Working Paper, 2010.
[13]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983)
[14]Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions, (1992) · Zbl 0761.60052