Adaptive Wavelet Methods for SPDEs

Cioica, Petru A.; Dahlke, Stephan; Döhring, Nicolas; Kinzel, Stefan; Lindner, Felix; Raasch, Thorsten; Ritter, Klaus; Schilling, René L.

doi:10.1007/978-3-319-08159-5_5

Cited by 3 publications

(2 citation statements)

References 34 publications

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“…We remark that the use of spatially adaptive schemes is useful especially for stochastic equations, where singularities appear naturally near the boundary due to the irregular behaviour of the noise, cf. [8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

et al. 2016

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This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.MSC 2010: Primary: 60H15, 60H35; secondary: 65M22.

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Section: Introductionmentioning

confidence: 99%

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

et al. 2016

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“…By constructing a stochastic integral in quasi-Banach spaces we deliver the first building block towards a semigroup approach to SPDEs in quasi-Banach spaces, thus, in particular, to a regularity analysis in Besov spaces from the scale (1.1). For more background information on the relationship between regularity, optimal convergence rates and adaptivity in the context of SPDEs, we refer to the introduction of [22] and the survey article [21], see also [20].…”

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confidence: 99%

Stochastic integration in quasi-Banach spaces

Cioica-Licht¹,

Cox²,

Veraar³

2018

Preprint

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In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent p ∈ (0, 1]. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations. Contents14 4. Stochastic integration I: deterministic integrands 22 5. Stochastic integration II: random integrands 29 6. The stochastic heat equation 38 Appendix A. Vector-valued Besov spaces 41 Appendix B. The Hardy-Littlewood maximal function and a technical lemma 43 References 49

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A posteriori error estimation and space-time adaptivity for a linear stochastic PDE with additive noise

Majee

Prohl

2021

IMA Journal of Numerical Analysis

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We present a strong residual-based a posteriori error estimate for a finite element-based space-time discretization of the linear stochastic convected heat equation with additive noise. This error estimate is used for an adaptive algorithm that automatically selects deterministic mesh parameters in space and time. For every $n \geq 0$, we find a new time-step $\tau _n$, a new spatial mesh ${\mathcal M}_{n}$ terminating within finitely many iterations and a finite element value approximation $Y^n_h$ on this spatial mesh, which then approximates strongly the solution of the stochastic partial differential equation (SPDE) within a prescribed tolerance.

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Adaptive Wavelet Methods for SPDEs

Cited by 3 publications

References 34 publications

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

Stochastic integration in quasi-Banach spaces

A posteriori error estimation and space-time adaptivity for a linear stochastic PDE with additive noise

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