Investigation and improvement of biased Monte-Carlo estimates

The numerical statistical modeling of the free path of a particle in a collision transport model with allowance for an external force acceleration is based on time stepping. For the corresponding deterministic relative error, a new constructive estimate is obtained, which is used to choose a suitable step size. The standard statistical “local estimates” of the particle flux density are biased because the contributions made by the collisions in a “local ball” of small radius are set to zero to make the variance bounded. Practically effective estimates of the corresponding relative error are presented. Additionally, a uniform optimization of a histogram-type functional estimate of the particle distribution density is presented assuming that the corresponding statistical ensemble is Poisson distributed. It turns out that the deterministic error in optimal (in terms of time complexity) versions of the considered algorithms is close to the statistical error.