A Mathematical Framework for Exact Milestoning

Aristoff, David; Bello-Rivas, Juan M.; Elber, Ron

doi:10.1137/15m102157x

Cited by 24 publications

(37 citation statements)

References 56 publications

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“…We conclude by briefly connecting the discussion above to Exact Milestoning [2,4], an algorithm mentioned in the Introduction for sampling dynamical quantities like mean hitting times. Consider the following seemingly more general framework.…”

Section: Algorithm 4 a We Sampler For Stationary Averagesmentioning

confidence: 96%

“…Martingale framework and variance. Recall that K is the transition kernel of (X p ) p≥0 , and recall the definitions of η p andη p from (2). In this section and below, n ≥ 0 and a bounded function f :…”

Section: Assign the Weightωmentioning

confidence: 99%

“…Methods that are related in scope and design include Exact Milestoning [4,15], Non-Equilibrium Umbrella Sampling [28,25], Trajectory Tilting [26], Transition Interface Sampling [27], Forward Flux Sampling [1], and Boxed Molecular Dynamics [16]. See for instance [2,9] for review and comparison of these methods. We will comment on Exact Milestoning in the Appendix below.…”

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confidence: 99%

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Analysis and optimization of weighted ensemble sampling

Aristoff

2018

ESAIM: M2AN

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We give a mathematical framework for weighted ensemble (WE) sampling, a binning and resampling technique for efficiently computing probabilities in molecular dynamics. We prove that WE sampling is unbiased in a very general setting that includes adaptive binning. We show that when WE is used for stationary calculations in tandem with a coarse model, the coarse model can be used to optimize the allocation of replicas in the bins.

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Section: Algorithm 4 a We Sampler For Stationary Averagesmentioning

confidence: 96%

Section: Assign the Weightωmentioning

confidence: 99%

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confidence: 99%

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Analysis and optimization of weighted ensemble sampling

Aristoff

2018

ESAIM: M2AN

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“…The shell method is closely related to milestoning [30,35,36] and Markov state models [26][27][28], though more tailored to the problem at hand. In particular, our interest here is in computing the first-passage probabilities rather than in approximating the underlying process.…”

Section: Though the Main Purpose Is To Estimate H Rr C On ∂S A Byprmentioning

confidence: 99%

Capacities and efficient computation of first-passage probabilities

2020

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A reversible diffusion process is initialized at position x 0 and run until it first hits any of several targets. What is the probability that it terminates at a particular target? We propose a computationally efficient approach for estimating this probability, focused on those situations in which it takes a long time to hit any target. In these cases, direct simulation of the hitting probabilities becomes prohibitively expensive. On the other hand, if the timescales are sufficiently long, then the system will essentially "forget" its initial condition before it encounters a target. In these cases the hitting probabilities can be accurately approximated using only local simulations around each target, obviating the need for direct simulations. In empirical tests, we find that these local estimates can be computed in the same time it would take to compute a single direct simulation, but that they achieve an accuracy that would require thousands of direct simulation runs.

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“…To understand better what the validity of (6) entails, it is useful to consider first a variant of milestoning, termed exact milestoning [2,3], in which more information about the process is kept than in optimal milestoning. Specifically, exact milestoning uses the first hitting chain defined as follows: Thus, the index chain {ξ n : n ∈ N 0 } can be viewed as a coarse-grained sequence of the first hitting chain {Y n : n ∈ N 0 } in which one reduces the exact positions on the milestones to the indices of these milestones.…”

Section: Warm-up: Exact Milestoningmentioning

confidence: 99%

A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

Vanden‐Eijnden

2017

Comm Pure Appl Math

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ABSTRACT. Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.

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A Mathematical Framework for Exact Milestoning

Cited by 24 publications

References 56 publications

Analysis and optimization of weighted ensemble sampling

Analysis and optimization of weighted ensemble sampling

Capacities and efficient computation of first-passage probabilities

A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

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