Stochastic Partial Differential Equations

Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng

doi:10.1007/978-1-4684-9215-6

Cited by 305 publications

(255 citation statements)

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“…At this point, it is stressed that the method proposed in this paper is independent of the type of stochastic approximation used. [14,3,37]. These mathematical considerations are beyond the scope of this article.…”

Section: Stochastic Discretizationmentioning

confidence: 97%

Generalized spectral decomposition for stochastic nonlinear problems

Nouy

Maı̂tre

2009

Journal of Computational Physics

121

127

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We present an extension of the Generalized Spectral Decomposition method for the resolution of non-linear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multiwavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional non-linear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space.

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Section: Stochastic Discretizationmentioning

confidence: 97%

Generalized spectral decomposition for stochastic nonlinear problems

Nouy

Maı̂tre

2009

Journal of Computational Physics

121

127

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“…Note that this is the same definition as the Brownian motion in [12], the difference being the probability measure assigned to (Ω, B).…”

Section: The Process B Mmentioning

confidence: 99%

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Alpay¹,

Kipnis²

2013

OpMath

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Abstract. Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

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“…One effective and popular regularization procedure for these models is to replace equation by its Wick product version − ∇ (a (x, ω) ∇u(x)) = f (x), x ∈ O ⊂ R d , u |∂O = g (x) ; (1.4) see [8] for a discussion of the Wick product as a modeling tool. Admittedly, "Wick product" is anything but product, it is rather a stochastic convolution.…”

Section: )mentioning

confidence: 99%

“…where f ∈ V is non-random and denotes the Wick product [8]. In particular, A special case of (3.1) is…”

Section: Stochastic Elliptic Equations: Existence and Uniqueness Of Smentioning

confidence: 99%

“…In this paper, we limit our considerations to coefficients that are (possibly nonlinear) functions of Gaussian random fields. A popular example of an equation with random coefficients is Darcy equation for pressure with lognormal permeability field (see [8] and the references therein). It can be specified as the following Dirichlet problem:…”

Section: Introductionmentioning

confidence: 99%

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Elliptic equations of higher stochastic order

Lototsky¹,

Rozovskiĭ²,

Wan³

2010

ESAIM: M2AN

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Abstract. This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.Mathematics Subject Classification. 35R60, 65L60, 60H15, 60H35.

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Stochastic Partial Differential Equations

Cited by 305 publications

References 0 publications

Generalized spectral decomposition for stochastic nonlinear problems

Generalized spectral decomposition for stochastic nonlinear problems

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Elliptic equations of higher stochastic order

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