Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Naurois, Ladislas Jacobe de; Jentzen, Arnulf; Welti, Timo

doi:10.1007/s00245-020-09744-6

Cited by 5 publications

(11 citation statements)

References 44 publications

(54 reference statements)

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“…The next result, Lemma 3.3 below, can be found in, e.g., Jacobe de Naurois et al [JdNJW,Lemma 2.5]. A proof of Lemma 3.3 can be found in, e.g., Lindgren [Lin12, Section 5.3].…”

Section: Basic Results For the Linear Wave Equationmentioning

confidence: 91%

“…The statement and the proof of the next result, Lemma 3.2 below, can be found in, e.g., Jacobe de Naurois et al [JdNJW,Lemma 2.4].…”

Section: Basic Results For the Linear Wave Equationmentioning

confidence: 98%

“…The proof of the next result, Corollary 3.9 below, is a minor modification of the third step in the proof of Jacobe de Naurois et al [JdNJW,Lemma 3.7].…”

Section: Upper Bounds For the Strong Approximation Errorsmentioning

confidence: 90%

“…(cf., for example, Theorem 2.1 above, Lemma 3.3 below, and Jacobe de Naurois et al [JdNJW,Remark 3.1] for sufficient conditions which ensure the existence of such a process).…”

Section: Settingmentioning

confidence: 99%

“…This would allow one to prove optimal weak rates for more general semilinear drift and diffusion coefficients; see [HJK] for analogous results for parabolic SPDEs. For completeness we note that optimal weak convergence rates for spatial spectral Galerkin approximations of stochastic wave equations have been established in [JdNJW18,JdNJW]. The approach taken in [JdNJW18,JdNJW] essentially relies on the specific structure of the spatial spectral Galerkin approximations and can thus neither be extended to temporal approximations nor to other more complicated spatial approximations such as the finite element method.…”

Section: Introductionmentioning

confidence: 99%

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Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

Cox,

Jentzen,

Lindner

2019

Preprint

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In numerical analysis for stochastic partial differential equations one distinguishes between weak and strong convergence rates. Often the weak convergence rate is twice the strong convergence rate. However, there is no standard way to prove this: to obtain optimal weak convergence rates for stochastic partial differential equations requires specially tailored techniques, especially if the noise is multiplicative. In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. The weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula.

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“…The next result, Lemma 3.3 below, can be found in, e.g., Jacobe de Naurois et al [JdNJW,Lemma 2.5]. A proof of Lemma 3.3 can be found in, e.g., Lindgren [Lin12, Section 5.3].…”

Section: Basic Results For the Linear Wave Equationmentioning

confidence: 91%

“…The statement and the proof of the next result, Lemma 3.2 below, can be found in, e.g., Jacobe de Naurois et al [JdNJW,Lemma 2.4].…”

Section: Basic Results For the Linear Wave Equationmentioning

confidence: 98%

“…The proof of the next result, Corollary 3.9 below, is a minor modification of the third step in the proof of Jacobe de Naurois et al [JdNJW,Lemma 3.7].…”

Section: Upper Bounds For the Strong Approximation Errorsmentioning

confidence: 90%

“…(cf., for example, Theorem 2.1 above, Lemma 3.3 below, and Jacobe de Naurois et al [JdNJW,Remark 3.1] for sufficient conditions which ensure the existence of such a process).…”

Section: Settingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

Cox,

Jentzen,

Lindner

2019

Preprint

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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 and simulation of the stochastic wave equation on the sphere

Cohen

Lang

2022

Calcolo

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Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrödinger equation on the unit sphere.

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Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Cited by 5 publications

References 44 publications

Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

Temporal approximation of stochastic evolution equations with irregular nonlinearities

Numerical approximation and simulation of the stochastic wave equation on the sphere

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