Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics

Duong, Manh Hong; Lamacz, Agnes; Peletier, Mark A.; Schlichting, André; Sharma, Upanshu

doi:10.1088/1361-6544/aaced5

Cited by 28 publications

(29 citation statements)

References 32 publications

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“…Since we are able to obtain this inequality from the duality formulation of the rate functionals, our method would offer an alternative technique for obtaining quantitative estimate of the coarse-graining error for both dissipative and non-dissipative systems. We address this issue in detail in a companion article [23].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…This inequality provides an upper bound on the free energy and the integral of the Fisher information by the rate functional and initial free energy. To keep the paper to a reasonable length, we address this issue in details separately in a companion article [23].…”

Section: In Comparison With Recently Developed Variational-evolutionamentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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“…We start by discussing the projection of the underdamped Langevin process by a map

depending only on the position coordinates. Oftentimes, one is not interested in quantities that explicitly depend on the momenta, which renders this a realistic setting (see [ 35 ] for an approach to model reduction which includes the momenta). Unfortunately, the coefficients of the effective dynamics ( 13 ) are identically zero in this case, see also [ 34 , 43 ].…”

Section: Underdamped Langevin Dynamicsmentioning

confidence: 99%

“…As the momentum variables of these dynamics often play just an auxiliary role, an interesting question to address is how to define a reduced dynamics that only involves the position state variables. As the conditioning approach does not provide meaningful answers in this case, finding meaningful effective equations remains an open problem in this setting [ 33 , 34 , 35 ].…”

Section: Introductionmentioning

confidence: 99%

Spectral Properties of Effective Dynamics from Conditional Expectations

Nüske

Koltai

Boninsegna

et al. 2021

Entropy

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The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.

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“…This approach has not been applied before for porous media flows although it can be regarded as a special case of coarse-graining or model reduction (Givon et al 2004) of the high-dimensional Langevin governing equations for particle position and velocity. While this has been extensively studied for Hamiltonian-type systems (Hijón et al 2006;Di Pasquale et al 2019), the coarse-graining of general advection-diffusion models has been only recently studied for the overdamped Langevin equation (Duong et al 2018;Legoll and Lelièvre 2010;Hudson and Li 2018).…”

Section: Introductionmentioning

confidence: 99%

Probability density function (PDF) models for particle transport in porous media

Icardi

Dentz

2020

Int J Geomath

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Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous medium, in fact, spatial PDFs can be defined to characterise local fluctuations and improve or better understand the closures performed by classical upscaling methods. In the study of hydrodynamical dispersion, for example, PDE-based PDF approach can replace expensive and noisy Lagrangian simulations (e.g., trajectories of drift-diffusion stochastic processes). In this work we derive a joint position-velocity Fokker–Planck equation to model the motion of particles undergoing advection and diffusion in in deterministic or stochastic heterogeneous velocity fields. After appropriate closure assumptions, this description can help deriving rigorously stochastic models for the statistics of Lagrangian velocities. This is very important to be able to characterise the dispersion properties and can, for example, inform velocity evolution processes in continuous time random walk dispersion models. The closure problem that arises when averaging the Fokker–Planck equation shows also interesting similarities with the mixing problem and can be used to propose alternative closures for anomalous dispersion.

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Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics

Cited by 28 publications

References 32 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

Spectral Properties of Effective Dynamics from Conditional Expectations

Probability density function (PDF) models for particle transport in porous media

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