Coarse Graining of Nonreversible Stochastic Differential Equations: Quantitative Results and Connections to Averaging

Hartmann, Carsten; Neureither, Lara; Sharma, Upanshu

doi:10.1137/19m1299852

Cited by 11 publications

(4 citation statements)

References 42 publications

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“…Furthermore, our analysis predominantly exploits entropy techniques and functional inequalities, which are traditionally used to study long-time behaviour of jump processes and not for multiscale problems. A key outcome of our results in the multiscale setting is that the effective dynamics converges to the averaged dynamics in a fairly general setting -such results only hold in a limited setting for diffusions [HNS20].…”

Section: Main Results Novelty and Outline Of The Articlementioning

confidence: 91%

Quantitative coarse-graining of Markov chains

Hilder¹,

Sharma²

2022

Preprint

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Coarse-graining techniques play a central role in reducing the complexity of stochastic models, and are typically characterised by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyse an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.

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Section: Main Results Novelty and Outline Of The Articlementioning

confidence: 91%

Quantitative coarse-graining of Markov chains

Hilder¹,

Sharma²

2022

Preprint

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“…For reversible fast-slow processes, such as (3.52), the Markovian effective dynamics (3.57) is identical to the averaged process given by (3.53). See Hartmann, Neureither and Sharma (2020) is small for all dominant eigenfunctions.…”

Section: Remark 323 (Fast-slow Process)mentioning

confidence: 95%

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

Schütte¹,

Klus²,

Hartmann³

2023

Acta Numerica

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One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.

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“…We should stress, that the above argument is strongly tied to the specific form of the dynamics (36) and the coordinate choice, in that the situation that the projection of the drift generates an invariant measure that is related to the original invariant measure by the same projection is not a triviality; even if an SDE like (9) has a unique invariant measure with a well-defined marginal, its projection may not even have an invariant measure; see Neureither (2019); Hartmann et al (2020).…”

Section: Some Notationmentioning

confidence: 99%

Reachability Analysis of Randomly Perturbed Hamiltonian Systems

Hartmann¹,

Neureither²,

Strehlau³

2021

Preprint

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In this paper, we revisit energy-based concepts of controllability and reformulate them for control-affine nonlinear systems perturbed by white noise. Specifically, we discuss the relation between controllability of deterministic systems and the corresponding stochastic control systems in the limit of small noise and in the case in which the target state is a measurable subset of the state space. We derive computable expression for hitting probabilities and mean first hitting times in terms of empirical Gramians, when the dynamics is given by a Hamiltonian system perturbed by dissipation and noise, and provide an easily computable expression for the corresponding controllability function as a function of a subset of the state variables.

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Coarse Graining of Nonreversible Stochastic Differential Equations: Quantitative Results and Connections to Averaging

Cited by 11 publications

References 42 publications

Quantitative coarse-graining of Markov chains

Quantitative coarse-graining of Markov chains

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

Reachability Analysis of Randomly Perturbed Hamiltonian Systems

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