Variance reduction for Fokker–Planck based particle Monte Carlo schemes

Gorji, M. Hossein; Andric, Nemanja; Jenny, Patrick

doi:10.1016/j.jcp.2015.04.008

Cited by 15 publications

(11 citation statements)

References 20 publications

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“…In applications such as flow in porous media, it is common to have a trend in the log-conductivity field. Based on the work in [17], the algorithm proposed for mean calculation can be modified to use the trend in the conductivity field to reduce the variance of the estimated mean. In short, for every realization of K, one could track a particle in that realization along with a shadow particle in the trend conductivity field using the same random numbers and store the boundary conditions at the hitting points of the boundary for both particles.…”

Section: Variance Reduction For Mean Calculationmentioning

confidence: 99%

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: Application to deterministic problems and uncertainty quantification

Delgoshaie

Jenny

2019

Journal of Computational Physics: X

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Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the presence of uncertainty in the conductivity field. Based on the relation between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods are available to solve these PDEs. These methods are especially relevant for cases where we are interested in the solution in a small subset of the domain. The existing MC methods based on the stochastic formulation require restrictively small time steps for high variance conductivity fields. Moreover, in many applications the conductivity is piecewise constant and the existing methods are not readily applicable in these cases. Here we provide an algorithm to solve onedimensional elliptic problems that bypasses these two limitations. The methodology is demonstrated using problems governed by deterministic and stochastic PDEs. It is shown that the method provides an efficient alternative to compute the statistical moments of the solution to a stochastic PDE at any point in the domain. A variance reduction scheme is proposed for applying the method for efficient mean calculations.

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Section: Variance Reduction For Mean Calculationmentioning

confidence: 99%

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: Application to deterministic problems and uncertainty quantification

Delgoshaie

Jenny

2019

Journal of Computational Physics: X

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“…We now briefly describe the method proposed by [9]. The objective of the method is to create a stochastic process Z t that is able to run in parallel to the original particle scheme, where crucially, the macroscopic fields are already known.…”

Section: Parallel Process Variance Reductionmentioning

confidence: 99%

“…We thank Dr. Hossein Gorji and coworkers for sending us their paper [9] prior to its finalized publication. This research is financially supported by EPSRC Programme Grant EP/I011927/1.…”

Section: Aknowledgementsmentioning

confidence: 99%

“…Like DSMC, it is solved stochastically using notional particles that represent a certain number of real particles in the gas to be simulated, and as such, the basic algorithm suffers from the same inhibitive scaling with the Mach number. Recently Gorji et al [9] have proposed a method to reduce the variance of the Fokker-Planck solution algorithm that relies on creating a correlated equilibrium solution using the same set of random numbers that are used in the stochastic solution of the non-equilibrium process. The parallel correlated equilibrium process is in effect a control variate for the non-equilibrium process.…”

Section: Introductionmentioning

confidence: 99%

“…In section 3 we describe the variance reduction scheme proposed by Gorji el al. [9], which creates a correlated equilibrium scheme. In section 4 we propose our importance sampling scheme, which we test in section 5 on a homogeneous relaxation, Couette flow and a lid-driven cavity flow.…”

Section: Introductionmentioning

confidence: 99%

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Importance sampling variance reduction for the Fokker–Planck rarefied gas particle method

Collyer

Connaughton

Lockerby

2016

Journal of Computational Physics

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Models and methods that are able to accurately and efficiently predict the flows of low-speed rarefied gases are in high demand, due to the increasing ability to manufacture devices at micro and nano scales. One such model and method is a Fokker-Planck approximation to the Boltzmann equation, which can be solved numerically by a stochastic particle method. The stochastic nature of this method leads to noisy estimates of the thermodynamic quantities one wishes to sample when the signal is small in comparison to the thermal velocity of the gas. Recently, Gorji et al have proposed a method which is able to greatly reduce the variance of the estimators, by creating a correlated stochastic process which acts as a control variate for the noisy estimates. However, there are potential difficulties involved when the geometry of the problem is complex, as the method requires the density to be solved for independently. Importance sampling is a variance reduction technique that has already been shown to successfully reduce the noise in direct simulation Monte Carlo calculations. In this paper we propose an importance sampling method for the Fokker-Planck stochastic particle scheme. The method requires minimal change to the original algorithm, and dramatically reduces the variance of the estimates. We test the importance sampling scheme on a homogeneous relaxation, planar Couette flow and a lid-driven-cavity flow, and find that our method is able to greatly reduce the noise of estimated quantities. Significantly, we find that as the characteristic speed of the flow decreases, the variance of the noisy estimators becomes independent of the characteristic speed.

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Particle simulation of nonequilibrium gas flows based on ellipsoidal statistical Fokker–Planck model

2018

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Variance reduction for Fokker–Planck based particle Monte Carlo schemes

Cited by 15 publications

References 20 publications

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: Application to deterministic problems and uncertainty quantification

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: Application to deterministic problems and uncertainty quantification

Importance sampling variance reduction for the Fokker–Planck rarefied gas particle method

Particle simulation of nonequilibrium gas flows based on ellipsoidal statistical Fokker–Planck model

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