A review on numerical schemes for solving a linear stochastic oscillator

Senosiaín, M. J.; Tocino, Ángel

doi:10.1007/s10543-014-0507-z

Cited by 21 publications

(17 citation statements)

References 9 publications

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“…To inherit the symplecticity of this stochastic oscillator, different kinds of symplectic methods have been constructed (see [7,9,14,22,23,24,28] and references therein).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Chen,

Hong,

Jin

et al. 2019

Preprint

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It is well known that symplectic methods have been rigorously shown to be superior to non-symplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we focus on the superiority of stochastic symplectic methods applied to a linear stochastic oscillator, from the perspective of large deviations principle (LDP). Based on the Gärtner-Ellis theorem, we first study the LDPs of the mean position and the mean velocity for both the exact solution of the stochastic oscillator and its numerical approximations. Then, by giving the conditions which make numerical methods have first order convergence in mean-square sense, we prove that symplectic methods asymptotically preserve these two LDPs, in the sense that the modified rate functions of symplectic methods converge to the rate functions of exact solution. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the "hitting probability" of the mean position and mean velocity of the original system. However, it is shown that non-symplectic methods do not asymptotically preserve these two LDPs by using the tail estimation of Gaussian random variables. To the best of our knowledge, this is the first result about using LDP to show the superiority of stochastic symplectic methods compared with non-symplectic ones.

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“…To inherit the symplecticity of this stochastic oscillator, different kinds of symplectic methods have been constructed (see [7,9,14,22,23,24,28] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…where ∆W n = W t n+1 − W tn , and the real matrix A and the real vector b depend on both the method and the constant step-size h (see [22]). When det(A) = 1, the method (1.5) preserves the discrete symplectic structure, and vice versa.…”

Section: Introductionmentioning

confidence: 99%

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Chen,

Hong,

Jin

et al. 2019

Preprint

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“…This implies that the expectation of H 0 along the exact solution of (6.2) (the second moment of the exact solution in this case) has linear growth. Recently, [31] gives a review on numerical schemes for solving this kind of linear stochastic oscillator, but it does not contain high mean-square order methods like SSRK-α 1 in this paper. Firstly, we check the mean-square convergence of SSRK-0.5 scheme in Section 4.…”

Section: Stochastic Harmonic Oscillator With Additive Noisementioning

confidence: 99%

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Zhou

Zhang

Hong

et al. 2017

Journal of Computational and Applied Mathematics

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In this paper, we construct stochastic symplectic Runge-Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes. We also achieve mean-square order 2.0 symplectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis. Finally, linear and non-linear systems are solved numerically, which verifies the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, the linear growth property can be preserved exactly over long-time simulation.

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“…For the linear part, [2] shows that symplecticity, linear growth of its second moment and asymptotic oscillation around zero are valid for the numerical schemes for the linear oscillation problem. [3] analyzes the stochastic resonance in a single-well anharmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

Numerical discretization of stochastic oscillators with generalized numerical integrators

2021

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In this study, we propose a numerical scheme for stochastic oscillators with additive noise obtained by the method of variation of constants formula using generalized numerical integrators. For both of the displacement and the velocity components, we show that the scheme has an order of 3/2 in one step convergence and a first order in overall convergence. Theoretical statements are supported by numerical experiments.

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A review on numerical schemes for solving a linear stochastic oscillator

Cited by 21 publications

References 9 publications

The probabilistic superiority of stochastic symplectic methods via large deviations principles

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Numerical discretization of stochastic oscillators with generalized numerical integrators

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