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

Wang, Xiaojie; Gan, Siqing

doi:10.1016/j.jmaa.2012.08.038

Cited by 55 publications

(57 citation statements)

References 25 publications

(49 reference statements)

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“…Results related to (40) can, e.g., be found in Debussche [ (40) for all δ 1 , δ k ∈ (− 1 /2, 0], k ∈ {1, 2} without the constraint that δ 1 + δ 2 > − 1 /2 but under the additional assumption (2). Moreover, very roughly speaking, Lemma 3.3 in [50] establishes (40) for all δ 1 , δ k ∈ (−1, 0], k ∈ {1, 2} with the constraint that δ 1 + δ 2 > −1 in the case of additive noise. Note that condition (2) is obviously satisfied in the case of additive noise.…”

Section: Regularity Properties For Solutions Of Infinite Dimensional mentioning

confidence: 98%

“…Strong convergence rates for (temporal, spatial, and noise) numerical approximations for SEEs of the form (1) are well understood. Weak convergence rates for numerical approximations of SEEs of the form (1) have been investigated for about two decades; cf., e.g., [45,24,18,20,22,25,19,32,21,35,36,33,50,34,6,48,4,5,7,49]. Except for Debussche & De Bouard [18], Debussche [19], and Andersson & Larsson [5], all of the above mentioned references assume, beside further assumptions, that the considered SEE is driven by additive noise.…”

Section: Introductionmentioning

confidence: 99%

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Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Conus¹,

Jentzen²,

Kurniawan³

2019

Ann. Appl. Probab.

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Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak convergence problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Itô type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak convergence problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs.

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Section: Regularity Properties For Solutions Of Infinite Dimensional mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Conus¹,

Jentzen²,

Kurniawan³

2019

Ann. Appl. Probab.

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“…These tools have also been used in [28] to treat the time-discretization in a slightly more-general setting, and in [1] where discretization in space with a finite element method is studied. Basically, the two ingredients are the following:…”

Section: Introductionmentioning

confidence: 99%

Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

Bréhier¹,

Kopec²

2016

IMA J Numer Anal

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Abstract. We study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure µ. We show that the average of regular enough test functions with respect to the (possibly non unique) invariant laws of the approximations are close to the corresponding quantity for µ.More precisely, we analyze the rate of the convergence with respect to the different discretization parameters. Here we focus on the discretization in time thanks to a scheme of Euler type, and on a Finite Element discretization in space.The results rely on the use of a Poisson equation, generalizing the approach of [21]; we obtain that the rates of convergence for the invariant laws are given by the weak order of the discretization on finite time intervals: order 1/2 with respect to the time-step and order 1 with respect to the mesh-size.

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“…for all t n = nh ≤ T , where C(u 0 , φ, T ) is independent of n, h. The weak order of accuracy (7) has been analyzed in [14] for (6) applied to the heat equation in the case F = 0 and in [12] in the semilinear case; see also [39] for the case of colored noise in dimension d > 1 and [23] where different techniques are used. Concerning the approximation of the invariant distribution of the heat equation, it is proved in [4] that for d = 1 and Q = I, one has for all time t n , similarly to (5), the exponential convergence property…”

Section: Introductionmentioning

confidence: 99%

High Order Integrator for Sampling the Invariant Distribution of a Class of Parabolic Stochastic PDEs with Additive Space-Time Noise

Bréhier¹,

Vilmart²

2016

SIAM J. Sci. Comput.

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We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear SPDEs driven by an additive space-time noise. Combined with a postprocessor, the new method is a modification with negligible overhead of the standard linearized implicit Euler-Maruyama method. We first provide an analysis of the integrator when applied for SDEs (finite dimension), where we prove that the method has order 2 for the approximation of the invariant distribution, instead of 1. We then perform a stability analysis of the integrator in the semilinear SPDE context, and we prove in a linear case that a higher order of convergence is achieved. Numerical experiments, including the semilinear heat equation driven by space-time white noise, confirm the theoretical findings and illustrate the efficiency of the approach.

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

Cited by 55 publications

References 25 publications

Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

High Order Integrator for Sampling the Invariant Distribution of a Class of Parabolic Stochastic PDEs with Additive Space-Time Noise

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