Coupled coarse graining and Markov Chain Monte Carlo for lattice systems

We propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models, capable of handling correctly long and short-range particle interactions. The proposed method is a Metropolis-type algorithm with the proposal probability transition matrix based on the coarse-grained approximating measures introduced in a series of works of M. Katsoulakis, A. Majda, D. Vlachos and P. Plechac, L. Rey-Bellet and D.Tsagkarogiannis,. We prove that the proposed algorithm reduces the computational cost due to energy differences and has comparable mixing properties with the classical microscopic Metropolis algorithm, controlled by the level of coarsening and reconstruction procedure. The properties and effectiveness of the algorithm are demonstrated with an exactly solvable example of a one dimensional Ising-type model, comparing efficiency of the single spin-flip Metropolis dynamics and the proposed coupled Metropolis algorithm.Comment: 20 pages, 4 figure

Error analysis of coarse-grained kinetic Monte Carlo method

In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and we demonstrate a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions and metastability.Comment: 30 page

Spatial multi-level interacting particle simulations and information theory-based error quantification

We propose a hierarchy of multi-level kinetic Monte Carlo methods for sampling high-dimensional, stochastic lattice particle dynamics with complex interactions. The method is based on the efficient coupling of different spatial resolution levels, taking advantage of the low sampling cost in a coarse space and by developing local reconstruction strategies from coarse-grained dynamics. Microscopic reconstruction corrects possibly significant errors introduced through coarse-graining, leading to the controlled-error approximation of the sampled stochastic process. In this manner, the proposed multi-level algorithm overcomes known shortcomings of coarse-graining of particle systems with complex interactions such as combined long and short-range particle interactions and/or complex lattice geometries. Specifically, we provide error analysis for the approximation of long-time stationary dynamics in terms of relative entropy and prove that information loss in the multi-level methods is growing linearly in time, which in turn implies that an appropriate observable in the stationary regime is the information loss of the path measures per unit time. We show that this observable can be either estimated a priori, or it can be tracked computationally a posteriori in the course of a simulation. The stationary regime is of critical importance to molecular simulations as it is relevant to long-time sampling, obtaining phase diagrams and in studying metastability properties of high-dimensional complex systems. Finally, the multi-level nature of the method provides flexibility in combining rejection-free and null-event implementations, generating a hierarchy of algorithms with an adjustable number of rejections that includes well-known rejection-free and null-event algorithms.Comment: 34 page

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions

We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a d-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarse-grained lattice systems with short and long-range interactions.Comment: 31 pages, 2 figure

Multilevel coarse graining and nano--pattern discovery in many particle stochastic systems

In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of stochastic lattice systems with competing short and long range interactions. Each Monte Carlo step is composed by two or more sub - steps efficiently coupling coarse and microscopic state spaces. The method can be designed to sample the exact or controlled-error approximations of the target distribution, providing information on levels of different resolutions, as well as at the microscopic level. In both strategies the method achieves significant reduction of the computational cost compared to conventional Markov Chain Monte Carlo methods. Applications in phase transition and pattern formation problems confirm the efficiency of the proposed methods.Comment: 37 page

A new framework for extracting coarse-grained models from time series with multiscale structure

In many applications it is desirable to infer coarse-grained models from observational data. The observed process often corresponds only to a few selected degrees of freedom of a high-dimensional dynamical system with multiple time scales. In this work we consider the inference problem of identifying an appropriate coarse-grained model from a single time series of a multiscale system. It is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting. Here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback. Furthermore, we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse-grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first-and often inadequate-approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.Comment: 37 pages, 8 figure