Core Concepts
The core message of this article is to provide expressions that can accurately predict the statistical error (variance) of analog particle tracing Monte Carlo methods when estimating quantities of interest on a histogram.
Abstract
The article introduces an analog particle tracing Monte Carlo method for solving high-dimensional kinetic equations, such as those describing the behavior of neutral particles in the plasma edge of a fusion device. The method introduces a statistical error (noise) that depends on the number of particles N used in the simulation.
The key highlights and insights are:
The authors derive four expressions to describe the variance on the outcome of a (correlated) binomial experiment, which represents the estimation of quantities of interest on a histogram using the particle tracing Monte Carlo method.
The first expression provides a simple upper bound on the variance, which is quadratic in the number of Bernoulli trials L. This upper bound typically overestimates the actual variance.
The second expression assumes the Bernoulli trials are independent, leading to a variance that is linear in L.
The third expression assumes the Bernoulli trials have a Markov dependence, where the current trial depends on the outcome of the previous trial. This expression provides a cheap a priori predictor for the variance.
The fourth expression assumes the binomial experiment is driven by a hidden Markov process, which corresponds to the continuous particle dynamics being coarse-grained onto a histogram. This expression is more accurate but computationally expensive.
The authors verify the accuracy of these variance expressions through numerical experiments, where they vary the model parameters (collision rates, post-collisional velocity distribution) and compare the predicted variances to the actual variances observed in the simulations.
The cheap Markov process based variance predictor can be used to optimize particle tracing Monte Carlo methods, select the best simulation and estimation combinations, and determine the required number of particles a priori to reach a desired accuracy.
Stats
The number of particles N used in the simulation determines the statistical error.
The collision rates Ri and Rcx, as well as the post-collisional velocity distribution M(v|x,t), affect the variance on the quantity of interest estimates.
Quotes
"The precise value of the statistical error depends on the kinetic equation under consideration as well as on the choice of simulation and estimation method."
"Predicting the variance allows to determine the number of particles N needed to achieve a desired accuracy a priori and gives a good indication of the computational cost of a Monte Carlo simulation."