Low-variance direct Monte Carlo simulations using importance weights

Husain A. Al-Mohssen; Nicolas G. Hadjiconstantinou

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 1069-1083
  • ISSN: 0764-583X

Abstract

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We present an efficient approach for reducing the statistical uncertainty associated with direct Monte Carlo simulations of the Boltzmann equation. As with previous variance-reduction approaches, the resulting relative statistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by the characteristic value of quantity of interest) is small and independent of the magnitude of the deviation from equilibrium, making the simulation of arbitrarily small deviations from equilibrium possible. In contrast to previous variance-reduction methods, the method presented here is able to substantially reduce variance with very little modification to the standard DSMC algorithm. This is achieved by introducing an auxiliary equilibrium simulation which, via an importance weight formulation, uses the same particle data as the non-equilibrium (DSMC) calculation; subtracting the equilibrium from the non-equilibrium hydrodynamic fields drastically reduces the statistical uncertainty of the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steady and unsteady, isothermal and non-isothermal problems; in all cases very good agreement between the two methods is found.

How to cite

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Al-Mohssen, Husain A., and Hadjiconstantinou, Nicolas G.. "Low-variance direct Monte Carlo simulations using importance weights." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 1069-1083. <http://eudml.org/doc/250772>.

@article{Al2010,
abstract = { We present an efficient approach for reducing the statistical uncertainty associated with direct Monte Carlo simulations of the Boltzmann equation. As with previous variance-reduction approaches, the resulting relative statistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by the characteristic value of quantity of interest) is small and independent of the magnitude of the deviation from equilibrium, making the simulation of arbitrarily small deviations from equilibrium possible. In contrast to previous variance-reduction methods, the method presented here is able to substantially reduce variance with very little modification to the standard DSMC algorithm. This is achieved by introducing an auxiliary equilibrium simulation which, via an importance weight formulation, uses the same particle data as the non-equilibrium (DSMC) calculation; subtracting the equilibrium from the non-equilibrium hydrodynamic fields drastically reduces the statistical uncertainty of the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steady and unsteady, isothermal and non-isothermal problems; in all cases very good agreement between the two methods is found. },
author = {Al-Mohssen, Husain A., Hadjiconstantinou, Nicolas G.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {DSMC; variance reduction; microscale gas flow; direct simulation Monte Carlo (DSMC); Monte Carlo; Boltzmann equation},
language = {eng},
month = {8},
number = {5},
pages = {1069-1083},
publisher = {EDP Sciences},
title = {Low-variance direct Monte Carlo simulations using importance weights},
url = {http://eudml.org/doc/250772},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Al-Mohssen, Husain A.
AU - Hadjiconstantinou, Nicolas G.
TI - Low-variance direct Monte Carlo simulations using importance weights
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 1069
EP - 1083
AB - We present an efficient approach for reducing the statistical uncertainty associated with direct Monte Carlo simulations of the Boltzmann equation. As with previous variance-reduction approaches, the resulting relative statistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by the characteristic value of quantity of interest) is small and independent of the magnitude of the deviation from equilibrium, making the simulation of arbitrarily small deviations from equilibrium possible. In contrast to previous variance-reduction methods, the method presented here is able to substantially reduce variance with very little modification to the standard DSMC algorithm. This is achieved by introducing an auxiliary equilibrium simulation which, via an importance weight formulation, uses the same particle data as the non-equilibrium (DSMC) calculation; subtracting the equilibrium from the non-equilibrium hydrodynamic fields drastically reduces the statistical uncertainty of the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steady and unsteady, isothermal and non-isothermal problems; in all cases very good agreement between the two methods is found.
LA - eng
KW - DSMC; variance reduction; microscale gas flow; direct simulation Monte Carlo (DSMC); Monte Carlo; Boltzmann equation
UR - http://eudml.org/doc/250772
ER -

References

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