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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 \aligned & du(t) = [Au(t) + F (u(t))] dt + dW(t); t\ge 0;\\ & u(0) = u_0; \endaligned on the Hilbert space $H = L_2([a; b]^d)$, where $A$ is the generator of an analytic semigroup $H$, $u_0\in D(A)$, $W$ 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 $p$th-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:
 65C30 Stochastic differential and integral equations 60H15 Stochastic partial differential equations 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 60H35 Computational methods for stochastic equations 35R60 PDEs with randomness, stochastic PDE
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##### 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) · Zbl 0946.60059 [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. · Zbl 1267.60071 [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) · Zbl 0516.47023 [14] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions, (1992) · Zbl 0761.60052