13 research outputs found
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
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
Goal-oriented sensitivity analysis for lattice kinetic Monte Carlo simulations
In this paper we propose a new class of coupling methods for the sensitivity
analysis of high dimensional stochastic systems and in particular for lattice
Kinetic Monte Carlo. Sensitivity analysis for stochastic systems is typically
based on approximating continuous derivatives with respect to model parameters
by the mean value of samples from a finite difference scheme. Instead of using
independent samples the proposed algorithm reduces the variance of the
estimator by developing a strongly correlated-"coupled"- stochastic process for
both the perturbed and unperturbed stochastic processes, defined in a common
state space. The novelty of our construction is that the new coupled process
depends on the targeted observables, e.g. coverage, Hamiltonian, spatial
correlations, surface roughness, etc., hence we refer to the proposed method as
em goal-oriented sensitivity analysis. In particular, the rates of the coupled
Continuous Time Markov Chain are obtained as solutions to a goal-oriented
optimization problem, depending on the observable of interest, by considering
the minimization functional of the corresponding variance. We show that this
functional can be used as a diagnostic tool for the design and evaluation of
different classes of couplings. Furthermore the resulting KMC sensitivity
algorithm has an easy implementation that is based on the Bortz-Kalos-Lebowitz
algorithm's philosophy, where here events are divided in classes depending on
level sets of the observable of interest. Finally, we demonstrate in several
examples including adsorption, desorption and diffusion Kinetic Monte Carlo
that for the same confidence interval and observable, the proposed
goal-oriented algorithm can be two orders of magnitude faster than existing
coupling algorithms for spatial KMC such as the Common Random Number approach
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
We present a mathematical framework for constructing and analyzing parallel
algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting
algorithms have the capacity to simulate a wide range of spatio-temporal scales
in spatially distributed, non-equilibrium physiochemical processes with complex
chemistry and transport micro-mechanisms. The algorithms can be tailored to
specific hierarchical parallel architectures such as multi-core processors or
clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms
are controlled-error approximations of kinetic Monte Carlo algorithms,
departing from the predominant paradigm of creating parallel KMC algorithms
with exactly the same master equation as the serial one.
Our methodology relies on a spatial decomposition of the Markov operator
underlying the KMC algorithm into a hierarchy of operators corresponding to the
processors' structure in the parallel architecture. Based on this operator
decomposition, we formulate Fractional Step Approximation schemes by employing
the Trotter Theorem and its random variants; these schemes, (a) determine the
communication schedule} between processors, and (b) are run independently on
each processor through a serial KMC simulation, called a kernel, on each
fractional step time-window.
Furthermore, the proposed mathematical framework allows us to rigorously
justify the numerical and statistical consistency of the proposed algorithms,
showing the convergence of our approximating schemes to the original serial
KMC. The approach also provides a systematic evaluation of different processor
communicating schedules.Comment: 34 pages, 9 figure
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
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
From mesoscale back to microscale: reconstruction schemes for coarse-grained stochastic lattice systems
Starting from a microscopic stochastic lattice spin system and the corresponding coarse-grained model we introduce a mathematical strategy to recover microscopic information given the coarse-grained data. We define “reconstructed” microscopic measures satisfying two conditions: (i) they are close in specific relative entropy to the initial microscopic equilibrium measure conditioned on the coarse-grained, data, and (ii) their sampling is computationally advantageous when compared to sampling directly from the conditioned microscopic equilibrium measure. By using different techniques we consider the cases of both short and long range microscopic models