Weak approximation of the stochastic wave equation

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.2010 Mathematics Subject Classification. 60H15, 60G51, 60H07, 65C30, 65M60.

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“…Further, from (12), (13), (14), the linear growth of F and the negative norm inequality in Proposition 3.6 we obtain…”

Section: 2mentioning

confidence: 82%

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Andersson¹,

Lindner²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…In particular, there are no results on the weak error of the finite element method for the linear stochastic Cahn-Hilliard and wave equations. The papers [4,5,9] consider the stochastic heat equation and so does [7], which proves similar results but under a stronger restriction on the test function G. The results in [3] are concerned with the Schrödinger equation and [8] proves weak convergence of the leap frog scheme for the stochastic wave equation. In all cases it is observed that the rate of weak convergence is twice that of strong convergence.…”

Section: E(t − S)b Dw (S) (12)mentioning

confidence: 58%

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise

2011

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Abstract. A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations. In all cases it is found that the rate of weak convergence is twice the rate of strong convergence, sometimes up to a logarithmic factor, under the same or, essentially the same, regularity requirements.

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“…Note that Assumption 3.6(iv) implies that Λ β−1 2 L 0 2 < ∞ for all β ≤ 1. As a first step towards our weak convergence result, we need the following regularity estimate for the Malliavin derivatives of the first component of the mild solution given by (14) and its approximation.…”

Section: Weak Convergence and Malliavin Calculusmentioning

confidence: 99%

“…Let Assumptions 2.2, 3.3 and 3.6 hold. Let X be the mild solution given by (14) of the stochastic wave equation and letX be the fully discrete approximation given by (29). Then, the weak error of the approximation satisfies…”

Section: Weak Convergence and Malliavin Calculusmentioning

confidence: 99%

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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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Weak approximation of the stochastic wave equation

Cited by 40 publications

References 22 publications

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise

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

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