Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity

Hong, Jialin; Zhai, Shuxing; Zhang, Jingjing

doi:10.1137/090771880

Cited by 36 publications

(43 citation statements)

References 18 publications

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“…From (17) we obtain that Since M1(0) = ∂ h M1(0) = 0, then |M1(h)| ≤ M2(h)h 2 with M2(h) := max k∈[0,h] |∂ 2 h M1(k)|, and, finally, I1 − I N 1 2 ≤ |b − cd|h 1/2 M2(h)G1(T ) where G1(T ) is given by (19), according with (18). From (21) we obtain…”

Section: Appendixmentioning

confidence: 93%

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A symmetry-adapted numerical scheme for SDEs

Vecchi

Romanò

Ugolini

2019

Journal of Geometric Mechanics

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We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry-adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the symmetry properties of the equation is introduced. Our algorithmic procedure is applied to the family of general linear SDEs for which two theoretical estimates of the numerical forward error are established.

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Section: Appendixmentioning

confidence: 93%

“…[1,31,36]), some conserved quantities (see e.g. [5,19,28]) or the variational structure (see e.g. [2,3,18,38]) of the considered SDEs.…”

Section: Introductionmentioning

confidence: 99%

A symmetry-adapted numerical scheme for SDEs

Vecchi

Romanò

Ugolini

2019

Journal of Geometric Mechanics

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“…We observe that MAVF method for system (5.4) preserves exactly these three methods (4.7) using quadrature formula (4.2), (4.3), (4.4) and (4.5) are called MAVF-Q2 method, MAVF-Q3 method, MAVF-Q4 method and MAVF-Q6 method respectively. Consider the following SDE with commutative noises (see [3]) 6) where c 1 , c 2 are constants, and W 1 (t), W 2 (t) are two independent Brownian motions. This system has I(p, q) = 1 2 p 2 − cos(q) as its invariant.…”

Section: (55)mentioning

confidence: 99%

Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

2020

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A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order 1 of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation.

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“…Based on the particular SG formula (7) of (1), we are able to replace the gradient in (7) with the discrete gradient, which leads to the discrete gradient method (see [11] for details). It is beneficial to approximate the solution to SDE (1) and preserve the general conserved quantity I(x) simultaneously.…”

Section: Single Conserved Quantitymentioning

confidence: 99%

Projection methods for stochastic differential equations with conserved quantities

et al. 2016

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In this paper, we consider the numerical methods preserving single or multiple conserved quantities, and these methods are able to reach high order of strong convergence simultaneously based on some kinds of projection methods. The mean-square convergence orders of these methods under certain conditions are given, which can reach order 1.5 or even 2 according to the supporting methods embedded in the projection step. Finally, three numerical experiments are taken into account to show the superiority of the projection methods.

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Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity

Cited by 36 publications

References 18 publications

A symmetry-adapted numerical scheme for SDEs

A symmetry-adapted numerical scheme for SDEs

Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

Projection methods for stochastic differential equations with conserved quantities

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