‘Path’ and ‘time’ estimates in the Monte Carlo method

Abstract

We propose new weight modifications of a 'path* estimate for the calculation of linear func- tionals (Φ,/ι) of radiation intensity Φ. In the framework of the 'collision* model of a transport process we mathematicall y justify the estimate, including variance finiteness, in the case of the function /ι with alternating signs. We construct and justify the analogous 'time' estimate for the calculation of linear functionals of the con- centration of particles moving along trajectories of multidimensional diffusion processes (the concentration of trajectories for brevity). In this paper we consider a problem of estimating the linear functionals (Φ, h) of radia- tion intensity Φ by new weight modifications of a 'path' estimate, which implies that the mean path of a particle in a domain is equal to the integral of radiation intensity in this domain. We use the 'collision' model of a transport process, i.e. the Markov homoge- neous chain terminating with probability one. Its states are the points in the phase space of coordinate velocities at which instantaneous velocity changes occur along a random particle trajectory. In the framework of this model we mathematicall y justify 'path es- tiamates' in the case of the function h with alternating signs. Due to this justification we can study the problem of variance finiteness for the new weight modifications of the estimate as well, i.e. when an auxiliary 'nonphysicaF Markov chain is modelled and the estimate is multiplied by a special weight multiplier. We construct and justify the analogous 'time estimate' for the calculation of linear functionals of the concentration of particles moving along trajectories of multidimen- sional diffusion processes (the concentration of trajectories for brevity). This estimate is the trajectory time integral of a given weight function. We study the problem of the variance finiteness of this estimate and a possibility of its approximate numerical implementation by the Euler scheme. We establish a relation between the time es- timate and the probabilistic representations of solutions of boundary value problems. Under certain conditions this allows us to justify the variance finiteness of the estimate if there is particle multiplication. It is shown that we can approximately calculate the first eigenvalue of a diffusion operator by the easily realizable parametric differentiation of a special weight estimate. 1. STANDARD WEIGHT ESTIMATES OF THE MONTE CARLO METHOD AND THEIR PARAMETRIC DIFFERENTIATION