Parallelization, Processor Communication and Error Analysis in Lattice Kinetic Monte Carlo

Abstract

In this paper we study from a numerical analysis perspective the Fractional Step Kinetic Monte Carlo (FS-KMC) algorithms proposed in [1] for the parallel simulation of spatially distributed particle systems on a lattice. FS-KMC are fractional step algorithms with a time-stepping window ∆t, and as such they are inherently partially asynchronous since there is no processor communication during the period ∆t. In this contribution we primarily focus on the error analysis of FS-KMC algorithms as approximations of conventional, serial kinetic Monte Carlo (KMC). A key aspect of our analysis relies on emphasizing a goal-oriented approach for suitably defined macroscopic observables (e.g., density, energy, correlations, surface roughness), rather than focusing on strong topology estimates for individual trajectories.One of the key implications of our error analysis is that it allows us to address systematically the processor communication of different parallelization strategies for KMC by comparing their (partial) asynchrony, which in turn is measured by their respective fractional time step ∆t for a prescribed error tolerance.

Information criteria for quantifying loss of reversibility in parallelized KMC

“…A new insight provided in [10] was that parallel algorithms, such as the one described in Figure 1, can be formulated as operator splitting schemes. This connection allows for the design, error quantification, and performance analysis of such algorithms [11]. Specifically, this approach allows for an observablefocused error analysis, through which a practitioner can pick both the scheme class and specific parameters that fit the computational needs.…”

“…In the context of Parallel KMC, the commutator term C = C(σ, σ ′ ) captures the error due to mismatches on the boundary regions between the different sublattices [11].…”

“…However, the present work could also be extended to the case of unbounded operators [23,24], where alternative representations for the semigroups could be used for the error analysis. We are not handling such cases here, as the Markov generators of stochastic particle systems are bounded operators [11].…”

“…For the adjustment to the correct timescale, computation of an equilibrium autocorrelation function is also required. Although such schemes resemble the random Lie-Trotter splitting [11] and they can be numerically analyzed in a similar manner, we will not discuss them here, mainly due to the technical differences with schemes that employ a fixed computational schedule [10]. More specificially, our focus is on general partially asynchronous parallel algorithms, like the one in SPPARKS.…”

Pathwise Error Bounds in Multiscale Variable Splitting Methods for Spatial Stochastic Kinetics

“…With few exceptions [3,25], the main body of work has been done in the wellstirred (or 0-dimensional) setting. Since the work [12] and the software described in [8], however, it is fairly well understood how spatial models are to be developed.…”