Low-variance direct Monte Carlo simulations using importance weights

Al-Mohssen, Husain A.; Hadjiconstantinou, Nicolas G.

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 1069-1083.

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.

Classification : 60H30, 76P05
Mots clés : DSMC, variance reduction, microscale gas flow

[1] F.J. Alexander, A.L. Garcia and B.J. Alder, Cell size dependence of transport coefficients in stochastic particle algorithms. Phys. Fluids 10 (1998) 1540-1542.

[2] H.A. Al-Mohssen, An Excursion with the Boltzmann Equation at Low Speeds: Variance-Reduced DSMC. Ph.D. Thesis, Massachusetts Institute of Technology, Dept. of Mechanical Engineering, Cambridge (2010).

[3] H.A. Al-Mohssen and N.G. Hadjiconstantinou, Yet Another Variance Reduction Method for Direct Monte Carlo Simulations of Low-Signal Flows, in 26th International Symposium on Rarefied Gas Dynamics, T. Abe Ed., AIP, Kyoto (2008) 257-262.

[4] L.L. Baker and N.G. Hadjiconstantinou, Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17 (2005) 051703. | Zbl

[5] L.L. Baker and N.G. Hadjiconstantinou, Variance-reduced particle methods for solving the Boltzmann equation. J. Comput. Theor. Nanosci. 5 (2008) 165-174.

[6] L.L. Baker and N.G. Hadjiconstantinou, Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: A discontinuous Galerkin formulation. Int. J. Numer. Methods Fluids 58 (2008) 381-402.

[7] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press (1994).

[8] C. Cercignani, The Boltzmann equation and its applications. Springer-Verlag (1988). | Zbl

[9] C. Cercignani, Mathematical Methods in Kinetic Theory. Plenum Press (1990). | Zbl

[10] C. Cercignani, Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems. Springer (2006). | Zbl

[11] J. Chun and D.L. Koch, A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number. Phys. Fluids 17 (2005) 107107. | Zbl

[12] A. Doucet and X. Wang, Monte Carlo methods for signal processing: a review in the statistical signal processing context. IEEE Signal Process. Mag. 22 (2005) 152-170.

[13] A.L. Garcia and W. Wagner, Time step truncation error in direct simulation Monte Carlo. Phys. Fluids 12 (2000) 2621-2633. | Zbl

[14] P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer (2004). | Zbl

[15] N.G. Hadjiconstantinou, Analysis of discretization in the direct simulation Monte Carlo. Phys. Fluids 12 (2000) 2634-2638. | Zbl

[16] N.G. Hadjiconstantinou, The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluids 18 (2006) 111301. | Zbl

[17] N.G. Hadjiconstantinou, A.L. Garcia, M.Z. Bazant and G. He, Statistical error in particle simulations of hydrodynamic phenomena. J. Comput. Phys. 187 (2003) 274-297. | Zbl

[18] T.M.M. Homolle and N.G. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo. Phys. Fluids 19 (2007) 041701. | Zbl

[19] T.M.M. Homolle and N.G. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation. J. Comput. Phys. 226 (2007) 2341-2358.

[20] C.D. Landon, Weighted Particle Variance Reduction of Direct Simulation Monte Carlo for the Bhatnagar-Gross-Krook Collision Operator. M.S. Thesis, Massachusetts Institute of Technology, Dept. of Mechanical Engineering, Cambridge (2010).

[21] H.C. Ottinger, B.H.A.A. Van Den Brule and M.A. Hulsen, Brownian configuration fields and variance reduced CONNFFESSIT. J. Non-Newton. Fluid Mech. 70 (1997) 255-261.

[22] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes. Cambridge University Press (2007). | Zbl

[23] G.A. Radtke and N.G. Hadjiconstantinou, Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation. Phys. Rev. E 79 (2009) 056711.

[24] R.Y. Rubinstein, Simulation and the Monte Carlo Method. Wiley (1981). | Zbl

[25] D.W. Scott, Multivariate Density Estimation. John Wiley & Sons (1992). | Zbl

[26] Y. Sone, Kinetic Theory and Fluid Dynamics. Birkhauser (2002). | Zbl

[27] W. Wagner, A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation. J. Stat. Phys. 66 (1992) 1011-1044. | Zbl

[28] W. Wagner, Deviational Particle Monte Carlo for the Boltzmann Equation. Monte Carlo Methods Appl. 14 (2008) 191-268. | Zbl