A mathematical formalization of the parallel replica dynamics

The quasi-steady-state approximation (or stochastic averaging principle) is a useful tool in the study of multiscale stochastic systems, giving a practical method by which to reduce the number of degrees of freedom in a model. The method is extended here to slow-fast systems in which the fast variables exhibit metastable behaviour. The key parameter that determines the form of the reduced model is the ratio of the timescale for the switching of the fast variables between metastable states to the timescale for the evolution of the slow variables. The method is illustrated with two examples: one from biochemistry (a fast-species-mediated chemical switch coupled to a slower-varying species), and one from ecology (a predator-prey system). Numerical simulations of each model reduction are compared with those of the full system.

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“…(39), and the exact full system MTE (32). We use n = 30, N = 150 and the parameter values (29) and (31).…”

Section: Resultsmentioning

confidence: 99%

Model reduction for slow–fast stochastic systems with metastable behaviour

Bruna

Chapman

Smith

2014

The Journal of Chemical Physics

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“…However, we believe both the CTMC ParRep and the embedded ParRep can be implemented in asynchronous architectures as well. In particular, the idea for handling asynchronous processors discussed in [15] (Section 3) can, in principle, be applied to the embedded ParRep as well. We will focus on formalizing these synchronization ideas in our future work.…”

Section: Reactionmentioning

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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“…The following result is proven in 46 . Let (X t ) t≥0 satisfy (6) with X 0 ∈ S. Introduce the first two eigenvalues −λ 2 < −λ 1 < 0 of the operator L † on S with homogeneous Dirichlet boundary conditions on ∂ S (see Section 2.3).…”

Section: Error Estimate On the Exit Eventmentioning

confidence: 83%

Jump Markov models and transition state theory: the quasi-stationary distribution approach

et al. 2016

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We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.

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A mathematical formalization of the parallel replica dynamics

Cited by 97 publications

References 18 publications

Model reduction for slow–fast stochastic systems with metastable behaviour

Model reduction for slow–fast stochastic systems with metastable behaviour

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

Jump Markov models and transition state theory: the quasi-stationary distribution approach

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