Some Remarks on Preconditioning Molecular Dynamics

Alrachid, Houssam; Mones, Letif; Ortner, Christoph

doi:10.5802/smai-jcm.29

Cited by 11 publications

(8 citation statements)

References 33 publications

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“…2) and the nonreversible Langevin samplers [20,21]. As a particular example of the general framework that was introduced in [18], we mention the preconditioned overdamped Langevin dynamics dX t = −P∇V (X t ) dt + √ 2P dW t , that was presented in [4]. There, the long-time behaviour of as well as the asymptotic variance of the corresponding estimator π T ( f ) are studied and applied to equilibrium sampling in molecular dynamics.…”

Section: Introductionmentioning

confidence: 99%

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

2017

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In this paper we introduce and analyse Langevin samplers that consist of perturbations of the standard underdamped Langevin dynamics. The perturbed dynamics is such that its invariant measure is the same as that of the unperturbed dynamics. We show that appropriate choices of the perturbations can lead to samplers that have improved properties, at least in terms of reducing the asymptotic variance. We present a detailed analysis of the new Langevin sampler for Gaussian target distributions. Our theoretical results are supported by numerical experiments with non-Gaussian target measures.

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Section: Introductionmentioning

confidence: 99%

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

2017

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“…To do this, there is a variety of techniques based on Monte Carlo methods [ 35 ] and corresponding multilevel approaches. [ 36 ] A particularly prominent variant is the so‐called Langevin sampler, [ 37,38 ] or its pre‐conditioned or underdamped versions, [ 39,40 ] that generates a sequence of parameter points

(θ_{1}, \dots, θ_{n})

according to the following iterative scheme: In each step, one first computes the proposal

{\tilde{θ}}_{k + 1}

for the next parameter point via

{\overset{θ}{true}}_{k + 1} = θ_{k} - Δ t grad S_{X} false(θ_{k} false) + \sqrt{2 Δ t} r_{k}

where

r_{j}

is a random number generated from the standard

m

‐dimensional normal distribution with mean 0 and variance 1,

Δ t

some sufficiently small stepsize, and

grad S_{X}

the gradient of the function

S_{X}

from Equation (37). Next, one determines the acceptance probability

α

according to the Metropolis–Hastings algorithm [ 41 ] :

α = \min \{1, \frac{e^{- S_{X} false({\tilde{θ}}_{k + 1} false)}}{e^{- S_{X} false(θ_{k} false)}} \frac{q false(θ_{k}, {\tilde{θ}}_{k + 1} false)}{q false({\tilde{θ}}_{k + 1}, θ_{k} false)}\}

with …”

Section: Parameter Estimationmentioning

confidence: 99%

Deterministic and Stochastic Parameter Estimation for Polymer Reaction Kinetics I: Theory and Simple Examples

Wulkow

Telgmann²,

Hungenberg³

et al. 2021

Macro Theory & Simulations

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Two different approaches to parameter estimation (PE) in the context of polymerization are introduced, refined, combined, and applied. The first is classical PE where one is interested in finding parameters which minimize the distance between the output of a chemical model and experimental data. The second is Bayesian PE allowing for quantifying parameter uncertainty caused by experimental measurement error and model imperfection. Based on detailed descriptions of motivation, theoretical background, and methodological aspects for both approaches, their relation are outlined. The main aim of this article is to show how the two approaches complement each other and can be used together to generate strong information gain regarding the model and its parameters. Both approaches and their interplay in application to polymerization reaction systems are illustrated. This is the first part in a two‐article series on parameter estimation for polymer reaction kinetics with a focus on theory and methodology while in the second part a more complex example will be considered.

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“…To do this, there is a variety of techniques based on Monte Carlo methods [35] and corresponding multilevel approaches. [36] A particularly prominent variant is the so-called Langevin sampler, [37,38] or its pre-conditioned or underdamped versions, [39,40] that generates a sequence of parameter points (𝜃 1 , … , 𝜃 n ) according to the following iterative scheme: In each step, one first computes the proposal θk+1 for the next parameter point via…”

Section: Computing the Posterior And Related Expectation Valuesmentioning

confidence: 99%

Deterministic and Stochastic Parameter Estimation for Polymer Reaction Kinetics I: Theory and Simple Examples

Wulkow¹,

Telgmann²,

Hungenberg³

et al. 2021

Macro Theory & Simulations

View full text Add to dashboard Cite

Two different approaches to parameter estimation (PE) in the context of polymerization are introduced, refined, combined, and applied. The first is classical PE where one is interested in finding parameters which minimize the distance between the output of a chemical model and experimental data. The second is Bayesian PE allowing for quantifying parameter uncertainty caused by experimental measurement error and model imperfection. Based on detailed descriptions of motivation, theoretical background, and methodological aspects for both approaches, their relation are outlined. The main aim of this article is to show how the two approaches complement each other and can be used together to generate strong information gain regarding the model and its parameters. Both approaches and their interplay in application to polymerization reaction systems are illustrated. This is the first part in a two-article series on parameter estimation for polymer reaction kinetics with a focus on theory and methodology while in the second part a more complex example will be considered.

show abstract

Some Remarks on Preconditioning Molecular Dynamics

Cited by 11 publications

References 33 publications

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

Deterministic and Stochastic Parameter Estimation for Polymer Reaction Kinetics I: Theory and Simple Examples

Deterministic and Stochastic Parameter Estimation for Polymer Reaction Kinetics I: Theory and Simple Examples

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