Layer methods for stochastic Navier–Stokes equations using simplest characteristics

Milstein, Grigori N.; Tretyakov, Michael V.

doi:10.1016/j.cam.2016.01.051

Cited by 6 publications

(3 citation statements)

References 43 publications

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“…It is, however, possible to extend the method to some classes of non-linear PDEs. On the one hand, we refer to [30] for a layer method based on stochastic representations for a class of stochastic Navier Stokes equations. On the other hand, there are representations of more general non-linear PDEs by backward stochastic differential equations, more precisely systems of second order forward-backward SDEs.…”

Section: Stochastic Representationsmentioning

confidence: 99%

SDE Based Regression for Linear Random PDEs

Anker¹,

Bayer²,

Eigel³

et al. 2017

SIAM J. Sci. Comput.

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A simulation based method for the numerical solution of PDEs with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behavior.2010 Mathematics Subject Classification. 35R60, 47B80, 60H35, 65C20, 65N12, 65N22, 65J10,65C05 . Key words and phrases. partial differential equations with random coefficients, random PDE, uncertainty quantification, Feynman-Kac, stochastic differential equations, stochastic simulation, stochastic regression, Monte-Carlo, Euler-Maruyama.We would like to thank the anonymous referees for their helpful comments. We are also grateful to Anders Szepessy for clarifying discussions.

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Section: Stochastic Representationsmentioning

confidence: 99%

SDE Based Regression for Linear Random PDEs

Anker¹,

Bayer²,

Eigel³

et al. 2017

SIAM J. Sci. Comput.

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show abstract

“…in [9,16,18]. The literature on numerics for deterministic NSE is extensive [11,27,30] (see also references therein) while the literature on numerics for stochastic NSE is still rather sparse, let us mention [5,3,2,7,26].…”

Section: Introductionmentioning

confidence: 99%

Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation

Milstein,

Tretyakov

2018

Preprint

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We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.

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“…There have also been some works on numerics for SPDEs in the sense of time-dependent PDEs driven by temporal (Brownian) noise. In particular, we refer to [22] for applications to stochastic Navier-Stokes equations. Due to the non-linearity, the Feynman-Kac representation cannot be directly applied in that problem, but a layer method based on linearized problems is constructed.…”

Section: Introductionmentioning

confidence: 99%

A Fully Adaptive Interpolated Stochastic Sampling Method for Linear Random Pdes

Anker

Bayer

Eigel

et al. 2017

Int. J. UncertaintyQuantification

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A numerical method for the fully adaptive sampling and interpolation of linear PDEs with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The spatial domain is decomposed by a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.• the pointwise solution in the spatial domain is determined by an appropriate SDE and solved by an adaptive Euler scheme as in [1], 2010 Mathematics Subject Classification. 35R60, 47B80, 60H35, 65C20, 65N12, 65N22, 65J10,65C05 .

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Layer methods for stochastic Navier–Stokes equations using simplest characteristics

Cited by 6 publications

References 43 publications

SDE Based Regression for Linear Random PDEs

SDE Based Regression for Linear Random PDEs

Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation

A Fully Adaptive Interpolated Stochastic Sampling Method for Linear Random Pdes

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