Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients

Jentzen, Arnulf; Kurniawan, Ryan

doi:10.1007/s10208-020-09448-x

Cited by 14 publications

(7 citation statements)

References 48 publications

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“…Further results on lower bounds for strong and weak errors for stochastic parabolic equations can be found, e.g., in Davie & Gaines [15], Müller-Gronbach et al [38], Müller-Gronbach & Ritter [37], Conus et al [9], and Jentzen & Kurniawan [25]. Third, we illustrate Theorem 1.1 by a simple example (cf.…”

Section: Introductionmentioning

confidence: 84%

“…However, the situation is different in the case of the infinitedimensional stochastic partial differential equations (cf., e.g., Walsh [43], Da Prato & Zabczyk [13], Liu & Röckner [35]). In the case of stochastic partial differential equations with regular nonlinearities, strong convergence rates are essentially well understood, whereas a proper understanding of weak convergence rates has still not been reached (cf., e.g., [1,2,[4][5][6]9,11,[16][17][18][19][20][21][22]25,[27][28][29]31,32,34,41,[45][46][47] for several weak convergence results in the literature). In this work we derive weak convergence rates for stochastic wave equations.…”

Section: Introductionmentioning

confidence: 99%

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

Naurois¹,

Jentzen

Welti

2021

Appl Math Optim

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Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Hölder-inequality for Schatten norms.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

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

Naurois¹,

Jentzen

Welti

2021

Appl Math Optim

Self Cite

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“…In this proof, we use the second interpretation (32) of the scheme. Note that the inequality (108) obtained above ensures the existence of an invariant distribution by the Krylov-Bogoliubov criterion, and also gives the bound (46). It thus suffices to check the uniqueness of the invariant distribution and the inequality (47).…”

Section: Properties Of the Modified Euler Schemementioning

confidence: 93%

“…For other weak approximation results using similar techniques, we refer for instance to [4] (finite element approximation), to [15] (standard Euler scheme in the multiplicative noise case) or to [53]. Other strategies are used to prove weak error results for instance in [3], in [28,46] or in [19]. The accelerated exponential Euler scheme has been introduced and studied in [45,44].…”

Section: Contributionsmentioning

confidence: 99%

Analysis of a modified Euler scheme for parabolic semilinear stochastic PDEs

Bréhier¹

2022

Preprint

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We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. The new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein-Uhlenbeck process obtained when the nonlinearity is absent, for any time-step size. The weak order of convergence of the proposed method is shown to be equal to 1{2 in a general setting, like for the standard Euler scheme. A stronger weak approximation result is obtained when considering the approximation of a Gibbs invariant distribution, when the nonlinearity is a gradient: one obtains an approximation in total variation distance of order 1{2, which does not hold for the standard method. This is the first result of this type in the literature. A key point in the analysis is the interpretation of the proposed modified Euler scheme as the accelerated exponential Euler scheme applied to a modified stochastic evolution equation. Finally, it is shown that the proposed method can be applied to design an asymptotic preserving scheme for a class of slow-fast multiscale systems, and to construct a Markov Chain Monte Carlo method which is well-defined in infinite dimension. We also revisit the analysis of the standard and the accelerated exponential Euler scheme, and we prove new results with approximation in the total variation distance, which serve to illustrate the behavior of the proposed modified Euler scheme.' % B t Xpt, ξq " B ξ `apξqB ξ Xpt, ξq ˘`f pXpt, ξqq `9 W pt, ξq, @t ą 0, ξ P p0, 1q, Xpt, 0q " Xpt, 1q " 0, @t ą 0, Xp0, ξq " x 0 pξq, @ξ P p0, 1q,

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“…We recall that strong convergence usually refers to convergence in mean-square sense: see for instance [14,22,23,26,27,28,29,35,36,37,46,54,57,60]. Weak convergence refers to convergence in distribution: recent contributions are [7,12,15,39].…”

Section: Charles-edouard Bréhier and Ludovic Goudenègementioning

confidence: 99%

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

Bréhier¹,

Goudenège²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L 2 (Ω)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients

Cited by 14 publications

References 48 publications

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

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

Analysis of a modified Euler scheme for parabolic semilinear stochastic PDEs

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

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