Long Time Accuracy of Lie--Trotter Splitting Methods for Langevin Dynamics

Abdulle, Assyr; Vilmart, Gilles; Zygalakis, Konstantinos C.

doi:10.1137/140962644

Cited by 48 publications

(64 citation statements)

References 31 publications

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“…We now turn to the discretization of the Feynman-Kac semigroup (3). We first define discretization schemes, and show that they are ergodic for some limiting measure under mild assumptions.…”

Section: Discretizationmentioning

confidence: 99%

Error estimates on ergodic properties of discretized Feynman–Kac semigroups

Ferré

Stoltz

2019

Numer. Math.

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We consider the numerical analysis of the time discretization of Feynman-Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present error estimates à la Talay-Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman-Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

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Section: Discretizationmentioning

confidence: 99%

Error estimates on ergodic properties of discretized Feynman–Kac semigroups

Ferré

Stoltz

2019

Numer. Math.

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“…We also point out that ∂/∂x in (1) simply implies the partial derivative although it typically denotes the functional derivative when x is a function/field. Moreover, (1) is supplemented by two degeneracy conditions:…”

Section: Generic Formulationmentioning

confidence: 99%

Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

Shang

Öttinger

2020

Proc. R. Soc. A.

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We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

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“…Also from the viewpoint of structure preservation, symplectic methods have, in general, good performance, although it is interesting to observe some exceptions such as the example in [2] for the simulation of the invariant measure of Langevin dynamics.…”

Section: Generating Functions For Weakly Convergent Stochastic Symplementioning

confidence: 99%

“…(1,0,0) + 2G 1 [1] (0,1) + 2G 1 [2] (1) = 2G 1 [1] (1,0) + 2H [2] 1 , 1,1,1) + G 1 [1] (1,1,1) = G 1 [1] (1,1,1) , G 1 (1,0,1,1…”

Section: Appendix 1: An Illustrating Explanation Regarding Influence mentioning

confidence: 99%

“…Pavliotis et al [25] uses modified equations to exhibit the poor behavior of the Euler methods for small random perturbations of Hamiltonian flows, and [1] proposes a new method for constructing numerical integrators with high weak order for the time integration of SDEs, inspired by the theory of stochastic modified equations. The recent work [2] analyses sufficient conditions for the Lie-Trotter splitting to preserve the invariant measure of nonlinear ergodic Langevin dynamics, with the application of the stochastic backward error analysis.…”

Section: Introductionmentioning

confidence: 99%

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Modified equations for weakly convergent stochastic symplectic schemes via their generating functions

2015

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In this paper, an approach of constructing modified equations of weak k +k order (k ≥ 1) apart from the k-th order weakly convergent stochastic symplectic methods, i.e., stochastic symplectic methods with respect to weak convergence and of weak order k, is given using the underlying generating functions of them. This approach is valid for stochastic Hamiltonian systems with additive noises, and those with multiplicative noises but for which the Hamiltonian functions H r ( p, q), r ≥ 1 associated to the diffusion parts depend only on p or only on q. In such cases, we find that the modified equations of the weakly convergent stochastic symplectic methods are perturbed stochastic Hamiltonian systems of the original systems.

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Long Time Accuracy of Lie--Trotter Splitting Methods for Langevin Dynamics

Cited by 48 publications

References 31 publications

Error estimates on ergodic properties of discretized Feynman–Kac semigroups

Error estimates on ergodic properties of discretized Feynman–Kac semigroups

Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

Modified equations for weakly convergent stochastic symplectic schemes via their generating functions

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