skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Stochastic method for accommodation of equilibrating basins in kinetic Monte Carlo simulations

Abstract

A computationally simple way to accommodate "basins" of trapping states in standard kinetic Monte Carlo simulations is presented. By assuming the system is effectively equilibrated in the basin, the residence time (time spent in the basin before escape) and the probabilities for transition to states outside the basin may be calculated. This is demonstrated for point defect diffusion over a periodic grid of sites containing a complex basin.

Authors:
Publication Date:
Research Org.:
Idaho National Laboratory (INL)
Sponsoring Org.:
USDOE
OSTI Identifier:
912482
Report Number(s):
INL/JOU-06-12029
TRN: US200801%%1043
DOE Contract Number:
DE-AC07-99ID-13727
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Physics: Condensed Matter; Journal Volume: 19; Journal Issue: 7
Country of Publication:
United States
Language:
English
Subject:
36 - MATERIALS SCIENCE, 75 - CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; DIFFUSION; KINETICS; POINT DEFECTS; TRAPPING; diffusion; kinetic Monte Carlo; residence time

Citation Formats

Van Siclen, Clinton D. Stochastic method for accommodation of equilibrating basins in kinetic Monte Carlo simulations. United States: N. p., 2007. Web. doi:10.1088/0953-8984/19/7/072201.
Van Siclen, Clinton D. Stochastic method for accommodation of equilibrating basins in kinetic Monte Carlo simulations. United States. doi:10.1088/0953-8984/19/7/072201.
Van Siclen, Clinton D. Thu . "Stochastic method for accommodation of equilibrating basins in kinetic Monte Carlo simulations". United States. doi:10.1088/0953-8984/19/7/072201.
@article{osti_912482,
title = {Stochastic method for accommodation of equilibrating basins in kinetic Monte Carlo simulations},
author = {Van Siclen, Clinton D},
abstractNote = {A computationally simple way to accommodate "basins" of trapping states in standard kinetic Monte Carlo simulations is presented. By assuming the system is effectively equilibrated in the basin, the residence time (time spent in the basin before escape) and the probabilities for transition to states outside the basin may be calculated. This is demonstrated for point defect diffusion over a periodic grid of sites containing a complex basin.},
doi = {10.1088/0953-8984/19/7/072201},
journal = {Journal of Physics: Condensed Matter},
number = 7,
volume = 19,
place = {United States},
year = {Thu Feb 01 00:00:00 EST 2007},
month = {Thu Feb 01 00:00:00 EST 2007}
}
  • The previously developed ab initio model and the kinetic Monte Carlo method (KMCM) are used to simulate precipitation in a number of iron-copper alloys with different copper concentrations x and temperatures T. The same simulations are also made using an improved version of the previously suggested stochastic statistical method (SSM). The results obtained enable us to make a number of general conclusions about the dependences of the decomposition kinetics in Fe-Cu alloys on x and T. We also show that the SSM usually describes the precipitation kinetics in good agreement with the KMCM, and using the SSM in conjunction withmore » the KMCM allows extending the KMC simulations to the longer evolution times. The results of simulations seem to agree with available experimental data for Fe-Cu alloys within statistical errors of simulations and the scatter of experimental results. Comparison of simulation results with experiments for some multicomponent Fe-Cu-based alloys allows making certain conclusions about the influence of alloying elements in these alloys on the precipitation kinetics at different stages of evolution.« less
  • We develop an improved multilevel Monte Carlo (MLMC) method for estimating cumulative distribution functions (CDFs) of a quantity of interest, coming from numerical approximation of large-scale stochastic subsurface simulations. Compared with Monte Carlo (MC) methods, that require a significantly large number of high-fidelity model executions to achieve a prescribed accuracy when computing statistical expectations, MLMC methods were originally proposed to significantly reduce the computational cost with the use of multifidelity approximations. The improved performance of the MLMC methods depends strongly on the decay of the variance of the integrand as the level increases. However, the main challenge in estimating CDFsmore » is that the integrand is a discontinuous indicator function whose variance decays slowly. To address this difficult task, we approximate the integrand using a smoothing function that accelerates the decay of the variance. Additionally, we design a novel a posteriori optimization strategy to calibrate the smoothing function, so as to balance the computational gain and the approximation error. The combined proposed techniques are integrated into a very general and practical algorithm that can be applied to a wide range of subsurface problems for high-dimensional uncertainty quantification, such as a fine-grid oil reservoir model considered in this effort. Furthermore, the numerical results reveal that with the use of the calibrated smoothing function, the improved MLMC technique significantly reduces the computational complexity compared to the standard MC approach. Finally, we discuss several factors that affect the performance of the MLMC method and provide guidance for effective and efficient usage in practice.« less
  • In this paper, we develop an improved multilevel Monte Carlo (MLMC) method for estimating cumulative distribution functions (CDFs) of a quantity of interest, coming from numerical approximation of large-scale stochastic subsurface simulations. Compared with Monte Carlo (MC) methods, that require a significantly large number of high-fidelity model executions to achieve a prescribed accuracy when computing statistical expectations, MLMC methods were originally proposed to significantly reduce the computational cost with the use of multifidelity approximations. The improved performance of the MLMC methods depends strongly on the decay of the variance of the integrand as the level increases. However, the main challengemore » in estimating CDFs is that the integrand is a discontinuous indicator function whose variance decays slowly. To address this difficult task, we approximate the integrand using a smoothing function that accelerates the decay of the variance. In addition, we design a novel a posteriori optimization strategy to calibrate the smoothing function, so as to balance the computational gain and the approximation error. The combined proposed techniques are integrated into a very general and practical algorithm that can be applied to a wide range of subsurface problems for high-dimensional uncertainty quantification, such as a fine-grid oil reservoir model considered in this effort. The numerical results reveal that with the use of the calibrated smoothing function, the improved MLMC technique significantly reduces the computational complexity compared to the standard MC approach. Finally, we discuss several factors that affect the performance of the MLMC method and provide guidance for effective and efficient usage in practice.« less
  • Cited by 10
  • We present a higher order kinetic Monte Carlo methodology suitable to model the evolution of systems in which the transition rates are non- trivial to calculate or in which Monte Carlo moves are likely to be non- productive flicker events. The second order residence time algorithm first introduced by Athenes et al.[1] is rederived from the n-fold way algorithm of Bortz et al.[2] as a fully stochastic algorithm. The second order algorithm can be dynamically called when necessary to eliminate unproductive flickering between a metastable state and its neighbors. An algorithm combining elements of the first order and second ordermore » methods is shown to be more efficient, in terms of the number of rate calculations, than the first order or second order methods alone while remaining statistically identical. This efficiency is of prime importance when dealing with computationally expensive rate functions such as those arising from long- range Hamiltonians. Our algorithm has been developed for use when considering simulations of vacancy diffusion under the influence of elastic stress fields. We demonstrate the improved efficiency of the method over that of the n-fold way in simulations of vacancy diffusion in alloys. Our algorithm is seen to be an order of magnitude more efficient than the n-fold way in these simulations. We show that when magnesium is added to an Al-2at.%Cu alloy, this has the effect of trapping vacancies. When trapping occurs, we see that our algorithm performs thousands of events for each rate calculation performed.« less