Predicting the statistical error of analog particle tracing Monte Carlo

TL;DR

This paper develops a method to accurately predict the statistical error of analog particle tracing Monte Carlo simulations, enabling optimization and efficient resource allocation for high-dimensional kinetic equations.

Contribution

It introduces a novel approach to calculate and predict the statistical error a priori for Monte Carlo methods estimating quantities of interest in high-dimensional systems.

Findings

Error predictors are effective for different QoI estimators.

Predictors enable optimization of Monte Carlo and hybrid methods.

Numerical experiments validate the theoretical predictions.

Abstract

Large particle systems are often described by high-dimensional (linear) kinetic equations that are simulated using Monte Carlo methods for which the asymptotic convergence rate is independent of the dimensionality. Even though the asymptotic convergence rate is known, predicting the actual value of the statistical error remains a challenging problem. In this paper, we show how the statistical error of an analog particle tracing Monte Carlo method can be calculated (expensive) and predicted a priori (cheap) when estimating quantities of interest (QoI) on a histogram. We consider two types of QoI estimators: point estimators for which each particle provides one independent contribution to the QoI estimates, and analog estimators for which each particle provides multiple correlated contributions to the QoI estimates. The developed statistical error predictors can be applied to other QoI estimators and nonanalog simulation routines as well. The error analysis is based on interpreting the number of particle visits to a histogram bin as the result of a (correlated) binomial experiment. The resulting expressions can be used to optimize (non)analog particle tracing Monte Carlo methods and hybrid simulation methods involving a Monte Carlo component, as well as to select an optimal particle tracing Monte Carlo method from several available options. Additionally, the cheap statistical error predictors can be used to determine a priori the number of particles N that is needed to reach a desired accuracy. We illustrate the theory using a linear kinetic equation describing neutral particles in the plasma edge of a fusion device and show numerical results. The code used to perform the numerical experiments is openly available.