On the discretization in time of parabolic stochastic partial differential equations

Printems, Jacques

doi:10.1051/m2an:2001148

Cited by 127 publications

(185 citation statements)

References 22 publications

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“…See Gyöngy (1999) and Hausenblas (2002Hausenblas ( , 2003 for results involving further (semi-)norms in suitable (function) spaces. Gyöngy, Nualart (1997), Gyöngy (1998), Printems (2001), and Yoo (2000) focus on orders of convergence for semi-discretizations. The discretizations of W that are analyzed in the literature so far may be called uniform, since a fixed step-size is used for all one-dimensional components of W that are evaluated.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

Müller-Gronbach¹,

Ritter

2006

Found Comput Math

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Abstract. We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[ d . The error of an algorithm is defined in L 2 -sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W . For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of non-uniform time discretizations is crucial.

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Section: Introductionmentioning

confidence: 99%

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

Müller-Gronbach¹,

Ritter

2006

Found Comput Math

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“…In spite of the recent contributions in the convergence analysis of the finite element method for second order stochastic parabolic problems with additive space-time white noise (see, e.g., [1,3,14,19,25,27,29]), the order of convergence of the finite element method for the fourth order problem (1.1) is not conclusive. Also, we refer to [6] for the convergence analysis of a finite difference method, to [15,16,23] for the convergence analysis of time-discretization methods for a wide family of evolution problems that includes (1.1), while the finite element method is not among the space-discretization techniques considered in [15,16].…”

Section: Main Results Of the Papermentioning

confidence: 99%

Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

Kossioris¹,

Zouraris²

2010

ESAIM: M2AN

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Abstract.We consider an initial and Dirichlet boundary value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. Discretizing the space-time white noise a modelling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a Galerkin finite element method based on C 0 or C 1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modelling error and for the approximation error to the solution of the regularized problem.Mathematics Subject Classification. 65M60, 65M15, 65C20.

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“…For the splitting up method of Zakai equation and SPDE, see [4], [11], [23], [18]. See also [34] for a time discretization analysis of θ-schemes of parabolic type SPDEs driven by a(n infinite dimensional) Wiener process.…”

Section: A Short Discussion Of Related Literaturementioning

confidence: 99%

Discretization and Simulation of the Zakai Equation

Gobet¹,

Pagès²,

Pham³

et al. 2006

SIAM J. Numer. Anal.

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This paper is concerned with numerical approximations for stochastic partial differential Zakai equation of nonlinear filtering problem. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order √ δ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. This formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from correlated Kalman-Bucy filter.

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On the discretization in time of parabolic stochastic partial differential equations

Cited by 127 publications

References 22 publications

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

Discretization and Simulation of the Zakai Equation

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