Weak Approximation for Semilinear Stochastic Evolution Equations

Hausenblas, Erika

doi:10.1007/978-3-0348-8020-6_5

Cited by 93 publications

(151 citation statements)

References 21 publications

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“…However, the standard Euler-Maruyama scheme Eq. (3.4) has worse stability properties (see [7,8]), and a restriction is required that h < 2/N 2 . As a consequence, this scheme is unlikely to be used for practical computations.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The rate of convergence in N cannot be expected to be faster than the rate of decay of b n . Strong convergence for the implicit Euler-Maruyama scheme has been considered for Galerkin approximations of SPDEs by [8] (who also examines other discretisations in space and time) and in [11].…”

Section: Proposition 33 Eq(36) Converges Strongly In H To a Solutimentioning

confidence: 99%

“…The standard Euler Maruyama scheme was first considered by [7] (see Section 3), and for this scheme a stability restriction is required on the time step (similar to the deterministic case). Strong convergence for the standard implicit Euler-Maruyama scheme is considered in both [11] and [8] which also considers a number of different spatial and time discretizations.…”

Section: Introductionmentioning

confidence: 99%

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A numerical scheme for stochastic PDEs with Gevrey regularity

Lord¹,

Rougemont²

2004

IMA Journal of Numerical Analysis

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We consider strong approximations to parabolic stochastic PDEs. We assume the noise lies in a Gevrey space of analytic functions. This type of stochastic forcing includes the case of forcing in a finite number of Fourier modes. We show that with Gevrey noise our numerical scheme has solutions in a discrete equivalent of this space and prove a strong error estimate. Finally we present some numerical results for a stochastic PDE with a GinzburgLandau nonlinearity and compare to the more standard implicit Euler-Maruyama scheme.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Proposition 33 Eq(36) Converges Strongly In H To a Solutimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A numerical scheme for stochastic PDEs with Gevrey regularity

Lord¹,

Rougemont²

2004

IMA Journal of Numerical Analysis

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“…¿From (9) and using the fact that S t is the semigroup generated by Laplacian operator, ∆, we conclude…”

Section: Time Discretizationmentioning

confidence: 92%

“…Hausenblas investigated the discretization error of semilinear stochastic evolution equations in L p -spaces, Banach spaces and quasi linear evolution equations driven by nuclear or space time white noise in [8,9]. Gyöngy and Shardlow in [18,7] apply finite differences in order to approximate the mild solution of parabolic SPDEs driven by space-time white noise.…”

Section: Introductionmentioning

confidence: 99%

Full discretization of the stochastic Burgers equation with correlated noise

Blömker

Kamrani

Hosseini

2013

IMA Journal of Numerical Analysis

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The main purpose of this paper is to investigate the spectral Galerkin method for spatial discretization. We combine it with the method introduced by Kloeden, Jentzen & Winkel in [12] for temporal discretization of stochastic partial differential equations and study pathwise convergence. We consider the case of colored noise, instead of the usual space-time white noise that was used before for the spatial discretization. The rate of convergence in uniform topology is estimated for the stochastic Burgers equation. Numerical examples illustrate the estimated convergence rate.

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On the discretisation in time of the stochastic Allen–Cahn equation

Kovács

Larsson

Lindgren

2018

Mathematische Nachrichten

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Abstract. We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretisation in time of the equation by an Euler type split-step method with step size k > 0. We show that the method converges strongly with a rate O(k 1 2 ). By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

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Weak Approximation for Semilinear Stochastic Evolution Equations

Cited by 93 publications

References 21 publications

A numerical scheme for stochastic PDEs with Gevrey regularity

A numerical scheme for stochastic PDEs with Gevrey regularity

Full discretization of the stochastic Burgers equation with correlated noise

On the discretisation in time of the stochastic Allen–Cahn equation

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