Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

Kovács, Mihály; Larsson, Stig; Lindgren, Fredrik

doi:10.1007/s10543-012-0405-1

Cited by 34 publications

(73 citation statements)

References 24 publications

(52 reference statements)

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“…In the past decade, plenty of work has been done on the strong convergence of numerical methods for SPDEs ( [2,[5][6][7][8][9][10][11][12][13][14][15][16] and see the review article [17] for more references). On the contrary, just a few literature [18][19][20][21][22][23] focus on the weak convergence, which is sometimes more interesting in many applications. This work will investigate weak convergence order of semi-discretization in time by the method (1.8) applied to (1.1) and weak convergence of full discretization will be our future work.…”

Section: Introductionmentioning

confidence: 99%

Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise

Wang

Gan

2013

Journal of Mathematical Analysis and Applications

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a b s t r a c tIn this paper, we analyze the weak error of a semi-discretization in time by the linear implicit Euler method for semilinear stochastic partial differential equations (SPDEs) with additive noise. The main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence. In particular, the linear implicit Euler method for SPDEs driven by trace class noise achieves an almost optimal order 1 − ϵ for arbitrarily small ϵ > 0.

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Section: Introductionmentioning

confidence: 99%

Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise

Wang

Gan

2013

Journal of Mathematical Analysis and Applications

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“…for all t ∈ [0, T ] holds uniformly in h, ∆t ∈ (0, 1], see, e.g., [17]. Moreover, the following error estimate with respect to its first component holds.…”

Section: Approximation and Convergencementioning

confidence: 97%

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

2020

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We consider the numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise. For the spatial approximation we consider a standard finite element method and for the temporal approximation, a rational approximation of the exponential function. We first show strong convergence of this approximation in both positive and negative order norms. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.1991 Mathematics Subject Classification. 60H15, 65M12, 60H35, 65C30, 65M60, 60H07. Key words and phrases. Stochastic partial differential equations, stochastic wave equations, stochastic hyperbolic equations, weak convergence, finite element methods, Galerkin methods, rational approximations of semigroups, Crank-Nicolson method, Malliavin calculus.

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“…Results concerning weak convergence rates have essentially been obtained in the last decade, using different approaches. In the case of the stochastic equation with additive noise (F = 0 in (1)), see [15], [16], [22], [23]. For semilinear equations, see [2], [6], [14], [30], [32], [33], for an approach related to the Kolmogorov equation.…”

Section: Introductionmentioning

confidence: 99%

Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs

Bréhier

2020

Journal of Complexity

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This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form dX(t) = AX(t)dt + F (X(t))dt + dW (t), driven by space-time white noise. In previous results, test functions are assumed (at least) of class C 2 with bounded derivatives, and the weak order is twice the strong order.We prove, in the case F = 0, that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even when the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by 1, does not converge to 0 as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by 2, i.e. it is not better than the strong order. This is in contrast with the finite dimensional case, where the Euler-Maruyama discretization of elliptic SDEs dY (t) = f (Y (t))dt + dB t has weak order of convergence 1 even for bounded continuous functions.1991 Mathematics Subject Classification. 60H15,60H35,65C30.

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Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

Cited by 34 publications

References 24 publications

Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise

Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs

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