Lowell L. Baker

Variance reduction for Monte Carlo solutions of the Boltzmann equation

We show that by considering only the deviation from equilibrium, significant computational savings can be obtained in Monte Carlo evaluations of the Boltzmann collision integral for flow problems in the small Mach number (Ma) limit. The benefits of this variance reduction approach include a significantly reduced statistical uncertainty when the deviation from equilibrium is small, and a flow-velocity signal-to-noise ratio that remains approximately constant with Ma in the Ma⪡1 limit. This results in stochastic Boltzmann solution methods whose computational cost for a given signal-to-noise ratio is essentially independent of Ma for Ma⪡1; our numerical implementation demonstrates this for Mach numbers as low as 10−5. These features are in sharp contrast to current particle-based simulation techniques in which statistical sampling leads to computational cost that is proportional to Ma−2, making calculations at small Ma very expensive.

On Variance-Reduced Simulations of the Boltzmann Transport Equation for Small-Scale Heat Transfer Applications

We present and discuss a variance-reduced stochastic particle simulation method for solving the relaxation-time model of the Boltzmann transport equation. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared with traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods. The proposed method is developed and validated in the context of dilute gases; despite this, it is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is applicable.

Variance Reduction in Particle Methods for Solving the Boltzmann Equation

We present a new particle scheme for solving the Boltzmann equation; this scheme incorporates a recently developed variance reduction technique discussed in [L. L. Baker and N. G. Hadjiconstantinou, Physics of Fluids, vol. 17, art. no 051703, 2005] which exhibits a significant computational efficiency advantage for low speed flows, compared to traditional particle methods. This paper describes how this variance reduction approach, achieved by simulating only the deviation from equilibrium, can be implemented as a particle simulation method. The new scheme is validated using time dependent shear flow calculations.© 2006 ASME

On Efficient Particle Methods for Solving the Boltzmann Transport Equation in the Relaxation-Time Approximation

We present and discuss a variance-reduced stochastic particle simulation method for solving the relaxation-time model of the Boltzmann transport equation. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared to traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods. The proposed method is developed and validated in the context of dilute gases; despite this, it is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is applicable.Copyright

A Variance Reduction Approach for Monte Carlo Solutions of the Non-Linear Boltzmann Equation

We show that simple and efficient Monte Carlo-based solution methods for the Boltzmann equation for low-speed applications can be constructed by using appropriate variance reduction techniques. More specifically, we show that evaluation of the collision integral by sampling a representative number of collisions can be significantly accelerated by considering only the deviation from equilibrium, since this allows one to avoid considering a large number of collisions with no net effect. As the deviation from equilibrium decreases, the degree of variance reduction increases, leading to a signal to noise ratio that remains approximately constant. Thus, unlike particle techniques in which statistical sampling results in computational cost that is inversely proportional to the square of the Mach number, the approach presented here exhibits computational cost which is almost independent of the Mach number in the small Mach number limit. This is verified by numerical experiments at Mach numbers as low as O(10−5 ). We validate this approach by comparing its results with analytical and direct Monte Carlo simulations of the Boltzmann equation.Copyright