Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

Bessaih, Hakima; Hausenblas, Erika; Randrianasolo, Tsiry Avisoa; Razafimandimby, Paul André

doi:10.1016/j.cam.2018.04.067

Cited by 7 publications

(4 citation statements)

References 35 publications

(104 reference statements)

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“…Such SDEs can usually not be solved explicitly and it is a quite active area of research to design and analyze approximation algorithms which are able to solve SDEs with superlinearly growing nonlinearities approximatively. In particular, we refer, e.g., to [1, 2, 5, 11, 12, 16, 20, 22, 25, 26, 29-31, 34-36, 41, 45-48, 50, 55, 56, 58, 58, 60,63,64,66,68,70,73,81,82,87-90,92,93,97-101,103,105,108-112,115-117,120,122, 127, 128, 132-138, 140, 142-144, 147-151, 153, 154, 156, 164-166, 168, 170, 171, 175-177] for convergence and simulation results for explicit numerical approximation schemes for SODEs with superlinearly growing nonlinearities, we refer, e.g., to [6-8, 13-15, 17-19, 23, 32, 43, 51, 53, 67, 72, 74, 75, 77, 79, 80, 83-85, 91, 121, 155, 158, 167] for convergence and simulation results for explicit numerical approximation schemes for SPDEs with superlinearly growing nonlinearities, we refer, e.g., to [4, 11, 22, 38, 41, 60-62, 65, 73, 90, 107, 118, 119, 123-125, 131, 145, 152, 157, 161, 162, 169, 172-174] for convergence and simulation results for implicit Euler-type numerical approximation schemes for SODEs with superlinearly growing nonlinearities, and we refer, e.g., to [9,10,21,24,27,28,33,39,40,44,49,51,52,86,[94][95][96]106,114,139,167] for convergence and simulation results for implicit Euler-type numerical approximation schemes for SPDEs with superlinearly growing nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

Beccari¹,

Hutzenthaler²,

Jentzen³

et al. 2019

Preprint

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The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature

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Section: Introductionmentioning

confidence: 99%

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

Beccari¹,

Hutzenthaler²,

Jentzen³

et al. 2019

Preprint

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“…For nonlinear parabolic SPDEs, in [6] D. Blömker and A. Jentzen proved a speed of convergence in probability of Galerkin approximations of the stochastic Burgers equation, which is a simpler nonlinear PDE which has some similarity with the Navier-Stokes equation. In [5], an abstract stochastic nonlinear evolution equation in a separable Hilbert space was investigated, including the GOY and Sabra shell models. These adimensional models are phenomenological approximations of the Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

Bessaih

Millet

2018

IMA Journal of Numerical Analysis

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We prove that some time discretization schemes for the 2D Navier-Stokes equations on the torus subject to a random perturbation converge in L 2 (Ω). This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of fully implicit and semi-implicit time Euler discretizations, and of a splitting scheme. The speed of the L 2 (Ω)-convergence depends on the diffusion coefficient and on the viscosity parameter.2000 Mathematics Subject Classification. Primary 60H15, 60H35; Secondary 76D06, 76M35.

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“…(S(t)) t≥0 being an analytic semigroup), regularisation phenomena allow for different proof techniques, resulting in much stronger convergence results. For details on the parabolic case, we refer to [3,5,6,9,18,25,[33][34][35]40,41,43,44] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Temporal approximation of stochastic evolution equations with irregular nonlinearities

Klioba,

Veraar

2024

J. Evol. Equ.

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In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error$$\begin{aligned} \textrm{E}_{k}^{\infty } {:}{=}\Big (\mathbb {E}\sup _{j\in \{0, \ldots , N_k\}} \Vert U(t_j) - U^j\Vert _X^p\Big )^{1/p} \rightarrow 0\quad (k \rightarrow 0), \end{aligned}$$ E k ∞ : = ( E sup j ∈ { 0 , … , N k } ‖ U ( t j ) - U j ‖ X p ) 1 / p → 0 ( k → 0 ) , where $$p \in [2,\infty )$$ p ∈ [ 2 , ∞ ) , U is the mild solution, $$U^j$$ U j is obtained from a time discretisation scheme, k is the step size, and $$N_k = T/k$$ N k = T / k for final time $$T>0$$ T > 0 . This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error$$\begin{aligned} \textrm{E}_k {:}{=}\bigg (\sup _{j\in \{0,\ldots ,N_k\}}\mathbb {E}\Vert U(t_j) - U^{j}\Vert _X^p\bigg )^{1/p}, \end{aligned}$$ E k : = ( sup j ∈ { 0 , … , N k } E ‖ U ( t j ) - U j ‖ X p ) 1 / p , which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

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Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

Cited by 7 publications

References 35 publications

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

Temporal approximation of stochastic evolution equations with irregular nonlinearities

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