Yet Another Variance Reduction Method for Direct Monte Carlo Simulations of Low-Signal Flows

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

doi:10.1063/1.3076482

Cited by 4 publications

(7 citation statements)

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“…(6). This equality between ν(y) and the integral of the kernel over y is necessary to maintain a constant norm for the distribution function.…”

Section: Methodsmentioning

confidence: 99%

“…Particle-based methods include the direct simulation Monte Carlo (DSMC) method 2-5 and the lattice Boltzmann method (LBM). [6][7][8][9][10][11] With DSMC, the system is coarsegrained into simulation particles representing groups of real particle. These coarse-grained particles are assigned positions and velocities and moved in space in time.…”

Section: Introductionmentioning

confidence: 99%

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A numerical solution of the linear Boltzmann equation using cubic B-splines

Khurana

Thachuk

2012

The Journal of Chemical Physics

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A numerical method using cubic B-splines is presented for solving the linear Boltzmann equation. The collision kernel for the system is chosen as the Wigner-Wilkins kernel. A total of three different representations for the distribution function are presented. Eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. Distribution functions, along with first and second moments, are evaluated for different mass and temperature ratios. Overall it is shown that the method is accurate and well behaved. In particular, moments can be predicted with very few points if the representation is chosen well. This method produces sparse matrices, can be easily generalized to higher dimensions, and can be cast into efficient parallel algorithms.

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“…(6). This equality between ν(y) and the integral of the kernel over y is necessary to maintain a constant norm for the distribution function.…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A numerical solution of the linear Boltzmann equation using cubic B-splines

Khurana

Thachuk

2012

The Journal of Chemical Physics

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“…This concept, which has been used in polymer simulation for a number of years [21], is illustrated in Figure 1 for a gas relaxation problem; the figure shows Figure 1. Illustration of the variance reduction principle for a molecular relaxation problem [3]. The variance of R V R is significantly reduced by replacing the "noisy" estimate R eq with its exact expected value R eq .…”

Section: Variance Reduction Using Importance Weights: Basic Conceptsmentioning

confidence: 99%

“…how actual simulation data [3] of R, R eq , and R eq , with R ≡ c 4 x , can be combined to yield the low-uncertainty estimator R V R .…”

Section: Variance Reduction Using Importance Weights: Basic Conceptsmentioning

confidence: 99%

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Low-variance direct Monte Carlo simulations using importance weights

Al-Mohssen¹,

Hadjiconstantinou²

2010

ESAIM: M2AN

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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.Mathematics Subject Classification. 60H30, 76P05.

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A Practical Variance Reduced DSMC Method

Al-Mohssen

Hadjiconstantinou

2011

AIP Conference Proceedings

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The goal of this work is to develop a practical method for simulating low-signal kinetic (and small-scale) gaseous flows. These flows have recently received renewed attention in connection with the design, manufacturing, and optimization of MEMS/NEMS devices operating in gaseous environments; they are typically described using the Boltzmann equation which is most efficiently solved using the direct simulation Monte Carlo (DSMC) method. DSMC is a simple and versatile simulation method which is very efficient in producing samples of the single particle distribution function used for estimating hydrodynamic properties. Unfortunately, in the case of low-speed flows the computational cost associated with reducing the statistical uncertainty of simulation outputs becomes overwhelming. We will present a variance reduction approach for reducing the statistical uncertainty associated with low-signal flows making their simulation not only possible but also efficient. Variance reduction is achieved using a control variate approach based on the observation that low-signal flows are typically close to an equilibrium state. As with previous variance reduction methods, significant variance reduction is achieved making the simulation of arbitrarily small deviations from equilibrium possible. However, in contrast to previous variancereduction methods, the method proposed, which we will refer to as the VRDSMC method, is able to reduce the variance with virtually no 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. By retaining the basic DSMC formulation, in contrast to previous approaches, the VRDSMC approach combines ease of implementation with computational efficiency and the ability to simulate all molecular interaction models available within the DSMC formulation. The work presented here represents a substantial improvement from the work presented in the previous symposium in two important ways. First, the kernel density estimation stabilization scheme has been further refined to allow substantially less bias without dramatically affecting stability. The second major improvement is the use of local cell reference states when performing particle collisions which results in a substantial reduction in the number of particles per cell required for stability, especially for small Knudsen numbers. Our validation tests show that the proposed VRDSMC method provides considerable variance reduction for only a small increase in computational cost and approximation error compared to equivalent DSMC simulations. In other words, by addressing the major weakness associated with DSMC, VRDSMC is well suited to the solution of low-signal kinetic problems of practical interest.

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Yet Another Variance Reduction Method for Direct Monte Carlo Simulations of Low-Signal Flows

Cited by 4 publications

References 0 publications

A numerical solution of the linear Boltzmann equation using cubic B-splines

A numerical solution of the linear Boltzmann equation using cubic B-splines

Low-variance direct Monte Carlo simulations using importance weights

A Practical Variance Reduced DSMC Method

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