Symplectic Integration of Hamiltonian Systems with Additive Noise

Milstein, Grigori N.; Repin, Yu. M.; Tretyakov, Michael V.

doi:10.1137/s0036142901387440

Cited by 137 publications

(159 citation statements)

References 15 publications

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“…, m, are independent standard Wiener processes. The diffusion coefficients σ r , γ r depend on P, Q (i.e., (1.1) is a system with multiplicative noise), in contrast to [3], where stochastic systems with additive noise are treated.…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Milstein¹,

Repin²,

Tretyakov³

2002

SIAM J. Numer. Anal.

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159

227

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Abstract. Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of Wiener processes in these fully implicit schemes are substituted by some truncated random variables. A number of symplectic integrators is constructed. Special attention is paid to systems with separable Hamiltonians. Some results of numerical experiments are presented. They demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones.

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Section: Introductionmentioning

confidence: 99%

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Milstein¹,

Repin²,

Tretyakov³

2002

SIAM J. Numer. Anal.

Self Cite

159

227

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“…In the stochastic case, such a condition on the time step would be random and too restrictive. A remedy for this problem has been given in [18,19] where it is proposed to truncate the noise when an implicit scheme is used. However, in our numerical experiments, we never encountered any problem and always were able to solve (2).…”

mentioning

confidence: 99%

Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation

Debussche¹,

Printems²

2006

Discrete &Amp; Continuous Dynamical Systems - B

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Abstract. In this article, we prove the convergence of a semi-discrete scheme applied to the stochastic Korteweg-de Vries equation driven by an additive and localized noise. It is the Crank-Nicholson scheme for the deterministic part and is implicit. This scheme was used in previous numerical experiments on the influence of a noise on soliton propagation [8,9]. Its main advantage is that it is conservative in the sense that in the absence of noise, the L 2 norm is conserved. The proof of convergence uses a compactness argument in the framework of L 2 weighted spaces and relies mainly on the path-wise uniqueness in such spaces for the continuous equation. The main difficulty relies in obtaining a priori estimates on the discrete solution. Indeed, contrary to the continuous case, Ito formula is not available for the discrete equation.

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“…Therefore, the amplitude must be calculated precisely, at least, when a periodic coefficient and a white noise term are absent. The system has the symplectic structure even when noise exists if some conditions are satisfied [30]. Taking this property into account, I use the symplectic method developed in ref.…”

Section: The Value Of the Exponent At The Extremum On Parametric Resomentioning

confidence: 99%

Suppression of Growth by Multiplicative White Noise in a Parametric Resonant System

Ishihara

2014

Braz J Phys

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The author studied the growth of the amplitude in a Mathieu-like equation with multiplicative white noise. The approximate value of the exponent at the extremum on parametric resonance regions was obtained theoretically by introducing the width of time interval, and the exponents were calculated numerically by solving the stochastic differential equations by a symplectic numerical method. The Mathieu-like equation contains a parameter α that is determined by the intensity of noise and the strength of the coupling between the variable and the noise. The value of α was restricted not to be negative without loss of generality. It was shown that the exponent decreases with α, reaches a minimum and increases after that. It was also found that the exponent as a function of α has only one minimum at α = 0 on parametric resonance regions of α = 0. This minimum value is obtained theoretically and numerically. The existence of the minimum at α = 0 indicates the suppression of the growth by multiplicative white noise.

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Symplectic Integration of Hamiltonian Systems with Additive Noise

Cited by 137 publications

References 15 publications

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation

Suppression of Growth by Multiplicative White Noise in a Parametric Resonant System

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