Analysing ill-conditioned Markov chains

Abstract

Discrete state Markov chains in discrete or continuous time are widely used to model phenomena in the social, physical and life sciences. In many cases, the model can feature a large state space, with extreme differences between the fastest and slowest transition timescales. Analysis of such ill-conditioned models is often intractable with finite precision linear algebra techniques. In this contribution, we propose a solution to this problem, namely partial graph transformation, to iteratively eliminate and renormalize states, producing a low-rank Markov chain from an ill-conditioned initial model. We show that the error induced by this procedure can be minimized by retaining both the renormalized nodes that represent metastable superbasins, and those through which reactive pathways concentrate, i.e. the dividing surface in the discrete state space. This procedure typically returns a much lower rank model, where trajectories can be efficiently generated with kinetic path sampling. We apply this approach to an ill-conditioned Markov chain for a model multi-community system, measuring the accuracy by direct comparison with trajectories and transition statistics. This article is part of a discussion meeting issue ‘Supercomputing simulations of advanced materials’.

Analysis and interpretation of first passage time distributions featuring rare events

“…However, they require a definition of metastable macrostates, and become computationally expensive when these states are large. Instead, we have recently introduced the partial graph transformation approach (pGT), 30 and we showed how judicious choice of the set of removed states can yield a reduced network that retains the full first passage time distribution quite well.…”

“…The equilibrium occupation probability distribution for a pGT network Ω where nodes in set have been removed, , can be obtained by applying the balance condition. 30 The result isThis new stationary distribution and the corresponding branching probabilities, and waiting times, can be used to define free energies for the retained minima, f s ( T ), and the transition states that connect them, aswhere the rate constant in is or with . The rate constants in the reduced network define a master equation for the retained states .…”

On the Mechanism of Polaritonic Rate Suppression from Quantum Transition Paths

“…The above formulation of the quantum master equation can be used to visualize the polariton energy landscape directly by translating the equilibrium occupation probabilities and transition matrix into the equivalent relative free energies . Hence, we obtain the free energy disconnectivity graph , in Figure d.…”

“…The above formulation of the quantum master equation can be used to visualize the polariton energy landscape directly by translating the equilibrium occupation probabilities and transition matrix into the equivalent relative free energies. 42 Hence, we obtain the free energy disconnectivity graph 32,33 in Figure 1d. In this representation, the vertical scale is the effective free energy, the bottom of each line corresponds to an eigenstate, and the eigenstates are connected together in a regular series of free energy thresholds when they can interconvert by any sequence of transition states that lies below the threshold.…”

Supercomputing modelling of advanced materials: preface

“…discuss developments in kinetic Monte Carlo techniques used in modelling chemical kinetics and of growing importance in computational catalysis. Woods et al [14] present developments in the field of discrete Markov chains of importance not just in physical but also in life and social sciences. Blake et al [15] describe the development of a consistent set of forcefields for modelling aqueous metal carbonates-a topic again of relevance to understanding crystallization-and they show how large-scale simulations are facilitated using GPU technology.…”