Numerical Methods for Stochastic Systems Preserving Symplectic Structure

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

doi:10.1137/s0036142901395588

Cited by 159 publications

(234 citation statements)

References 14 publications

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“…The particularity here lies in a systematic use of differential geometry to understand the properties of the Kolmogorov operator on a hypersurface of the ambient space. Let us also mention the analysis made in [17,16] where constrained symplectic SDEs and appropriate discretization schemes are introduced.…”

Section: This Condition Is Not Necessary (See the Examples In Sectionmentioning

confidence: 99%

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Conservative stochastic differential equations: Mathematical and numerical analysis

Faou

Lelièvre

2009

Math. Comp.

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Abstract. We consider stochastic differential equations on the whole Euclidean space possessing a scalar invariant along their solutions. The stochastic dynamics therefore evolves on a hypersurface of the ambient space. Using orthogonal coordinate systems, we show the existence and uniqueness of smooth solutions of the Kolmogorov equation under some ellipticity conditions over the invariant hypersurfaces. If we assume, moreover, the existence of an invariant measure, we show the exponential convergence of the solution towards its average. In the second part, we consider numerical approximation of the stochastic differential equation, and show the convergence and numerical ergodicity of a class of projected schemes. In the context of molecular dynamics, this yields numerical schemes that are ergodic with respect to the microcanonical measure over isoenergy surfaces.

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Section: This Condition Is Not Necessary (See the Examples In Sectionmentioning

confidence: 99%

“…This fact is not necessary for the analysis of the properties of the projected numerical schemes defined below, which require only the previous estimates on the manifold Σ 0 . However, we believe that these uniform bounds are necessary to understand the good behaviour of nonprojected schemes; see for instance [16].…”

Section: Erwan Faou and Tony Lelièvrementioning

confidence: 99%

Conservative stochastic differential equations: Mathematical and numerical analysis

Faou

Lelièvre

2009

Math. Comp.

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“…Note that in practise, during a computation in the critical or supercritical case, if u k gets very large it is wise to refine in time and choose a smaller time step. We could also use a truncation of the noise as in [24], [25]. In that way, we would not have to useũ…”

Section: 2mentioning

confidence: 99%

Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation

2006

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Abstract.In this article, we analyze the error of a semi discrete scheme for the stochastic non linear Schrödinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing of defocusing. Allowing sufficient spatial regularity we prove that the numerical scheme has strong order 1/2 in general and order 1 if the noise is additive. Furthermore, we also prove that the weak order is always 1.

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“…A major problem in constructing stochastic implicit methods is the possibility of obtaining unrealized (unbounded) solutions due to the nature of the Wiener process. To avoid unboundedness Milstein et al [6] introduced a modified Wiener increment ∆W =β √ h to approximate W (t) , wherē…”

Section: C580mentioning

confidence: 99%

“…The parameter A h is determined based on a strong order inequality requirement (Milstein et al, 2002) …”

Section: Introductionmentioning

confidence: 99%

Adaptive stepsize implementations of implicit stochastic Runge Kutta methods with modified Wiener increment

Herdiana

Burrage

2004

ANZIAMJ

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An adaptive stepsize algorithm is implemented on a stochastic implicit strong order 1 method, namely a stiffly accurate diagonal implicit stochastic Runge-Kutta method where a modified Wiener increment ∆W n is involved instead of a regular ∆W n = β √ h, β ∼ N (0, 1) to avoid unboundedness. The modified Wiener increment is equal to the regular one only ifThe parameter A h is determined based on a strong order inequality requirement (Milstein et al., 2002) C579The variable stepsize algorithm is based on Richardson's extrapolation. For every step change additional work is required to compute ∆W n so that the correct Brownian path is maintained from simulation to simulation. Numerical experiments in solving nonlinear and linear stochastic differential equation problems demonstrate that better approximations are obtained with the modified Wiener increment compared to using the regular Wiener increment.

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Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Cited by 159 publications

References 14 publications

Conservative stochastic differential equations: Mathematical and numerical analysis

Conservative stochastic differential equations: Mathematical and numerical analysis

Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation

Adaptive stepsize implementations of implicit stochastic Runge Kutta methods with modified Wiener increment

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