An infinite swapping approach to the rare-event sampling problem

In the present paper we identify a rigorous property of a number of tempering-based Monte Carlo sampling methods, including parallel tempering as well as partial and infinite swapping. Based on this property we develop a variety of performance measures for such rare-event sampling methods that are broadly applicable, informative, and straightforward to implement. We illustrate the use of these performance measures with a series of applications involving the equilibrium properties of simple Lennard-Jones clusters, applications for which the performance levels of partial and infinite swapping approaches are found to be higher than those of conventional parallel tempering.

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“…(3.6) of Ref. 21).) It follows from the explicit form of μ that the marginal distribution on S is uniform on P{N t }.…”

Section: Appendix B: Equal Occupancy Demonstrationmentioning

confidence: 97%

Rare-event sampling: Occupation-based performance measures for parallel tempering and infinite swapping Monte Carlo methods

Doll

Plattner

Freeman

et al. 2012

The Journal of Chemical Physics

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“…A possible remedy for this issues is to use parallel computing in order to accelerate sampling of the state space. For instance, the parallel tempering method (also known as the replica exchange) [12,10,9,17] has been successfully applied to many problems by simulating multiple replicas of the original systems, each replica at a different temperature. However, the method requires the time reversibility of the underlying processes, which is typically not true for processes that model chemical reaction networks or systems with non-equilibrium steady states.…”

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confidence: 99%

Stationary Averaging for Multiscale Continuous Time Markov Chains Using Parallel Replica Dynamics

Wang¹,

Plecháč²,

Aristoff³

2018

Multiscale Model. Simul.

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Abstract. We propose two algorithms for simulating continuous time Markov chains in the presence of metastability. We show that the algorithms correctly estimate, under the ergodicity assumption, stationary averages of the process. Both algorithms, based on the idea of the parallel replica method, use parallel computing in order to explore metastable sets more efficiently. The algorithms require no assumptions on the Markov chains beyond ergodicity and the presence of identifiable metastability. In particular, there is no assumption on reversibility. For simpler illustration of the algorithms, we assume that a synchronous architecture is used throughout of the paper. We present error analyses, as well as numerical simulations on multi-scale stochastic reaction network models in order to demonstrate consistency of the method and its efficiency.Key words. Markov chains, Monte Carlo, reversibility, stationary distribution, metastability, parallel replica, stochastic reaction networks, multi-scale dynamics, coarse graining AMS subject classifications. 60J22, 65C05, 65Z05, 82B31, 92E201. Introduction. We focus on computing stationary averages of continuous time Markov chains in a synchronous architecture. More precisely, if π is the stationary distribution of a continuous time Markov chain (CTMC) and f is a function on the state space, we aim at estimating the average π(f ) ≡ E π [f ] by taking a time average on a long trajectory of the CTMC. There are many methods for computing stationary averages of stochastic processes, however, the vast majority of them rely on reversibility of the process, e.g., as in Markov chain Monte Carlo [19]. Computational cost of the ergodic (trajectory) averaging becomes prohibitive when the convergence to the stationary distribution is slow due to metastability of the dynamics, for example in the presence of rare events or large time scale disparities (multi-scale dynamics), [20]. A possible remedy for this issues is to use parallel computing in order to accelerate sampling of the state space. For instance, the parallel tempering method (also known as the replica exchange) [12,10,9,17] has been successfully applied to many problems by simulating multiple replicas of the original systems, each replica at a different temperature. However, the method requires the time reversibility of the underlying processes, which is typically not true for processes that model chemical reaction networks or systems with non-equilibrium steady states. In fact, there are not many methods that parallelize Monte Carlo simulation for irreversible processes with metastability, in particular if long-time sampling such as ergodic averaging, is required. We present a parallel computing approach for CTMCs without time reversibility. One advantage of the proposed algorithms is that they may be used, in principle, on arbitrary CTMCs. The same idea applies to the continuous state space Markov processes. However, gains in efficiency can occur only if the process is metastable.In this contribution we consider only model...

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“…9, and the cost of these calculations increases very rapidly with the number of replicas N, because the number of permutations grows as a factorial of N. On top of this, to estimate averages we also need to compute η i,j ðXÞ, which is costly too. One way to address this problem is to introduce additional approximations, for example, partial swapping (18,19). Next, we propose an alternative strategy that permits us to reach the infinite-swap limit while working directly with the original dynamics defined in Eqs.…”

Section: Significancementioning

confidence: 99%

“…How to implement high-frequency swapattempt REMD is not obvious, however, because increasing the attempt frequency of the swap also increases the computational cost, sometimes with little gain because this also leads to more rejections. One way to get around this difficulty is to take the so- (19,20).In this paper, we propose a different solution to this problem that involves no approximation. It is based on the observation that REMD can be reformulated in such a way that the temperature swaps satisfy a continuous-time Markov jump process (MJP) rather than a discrete-time Markov chain.…”

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Multiscale implementation of infinite-swap replica exchange molecular dynamics

Abrams

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

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Replica exchange molecular dynamics (REMD) is a popular method to accelerate conformational sampling of complex molecular systems. The idea is to run several replicas of the system in parallel at different temperatures that are swapped periodically. These swaps are typically attempted every few MD steps and accepted or rejected according to a Metropolis-Hastings criterion. This guarantees that the joint distribution of the composite system of replicas is the normalized sum of the symmetrized product of the canonical distributions of these replicas at the different temperatures. Here we propose a different implementation of REMD in which (i) the swaps obey a continuous-time Markov jump process implemented via Gillespie's stochastic simulation algorithm (SSA), which also samples exactly the aforementioned joint distribution and has the advantage of being rejection free, and (ii) this REMD-SSA is combined with the heterogeneous multiscale method to accelerate the rate of the swaps and reach the so-called infinite-swap limit that is known to optimize sampling efficiency. The method is easy to implement and can be trivially parallelized. Here we illustrate its accuracy and efficiency on the examples of alanine dipeptide in vacuum and C-terminal β-hairpin of protein G in explicit solvent. In this latter example, our results indicate that the landscape of the protein is a triple funnel with two folded structures and one misfolded structure that are stabilized by H-bonds.importance sampling | REMD | SSA | HMM | protein-folding A ll-atom molecular dynamics (MD) simulations are increasingly used as a tool to study the structure and function of large biomolecules and other complex molecular systems from first principles. By allowing us to scrutinize these systems at spatiotemporal resolutions that are typically beyond experimental reach, MD simulations are changing the way we understand chemical and biological processes, and their impact in areas such as chemical engineering, synthetic biology, drug design, and so on is bound to increase in the coming years (1-3). However, exploiting the full potential of MD simulations remains a major challenge, primarily due to approximations in all-atom force fields and the limitation of accessible timescales available to standard MD. Force fields continue to be improved (4-7), and hardware developments involving special-purpose high-performance computers (8), high-performance graphics processing units (9), or massively parallel simulations (10) also permit us to enlarge the scope of MD simulations. However, it has been argued (11) that algorithm developments will be the key to access biologically relevant timescales with MD. In particular, there is a great need for methods capable of accelerating the exploration and sampling of the conformational space of complex molecular systems.Among the various techniques that have been introduced to this end in recent years, replica exchange MD (REMD) (12) remains one of the most popular. The reasons for this success are simple: REMD does not requ...

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An infinite swapping approach to the rare-event sampling problem

Cited by 53 publications

References 36 publications

Rare-event sampling: Occupation-based performance measures for parallel tempering and infinite swapping Monte Carlo methods

Rare-event sampling: Occupation-based performance measures for parallel tempering and infinite swapping Monte Carlo methods

Stationary Averaging for Multiscale Continuous Time Markov Chains Using Parallel Replica Dynamics

Multiscale implementation of infinite-swap replica exchange molecular dynamics

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