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

Abstract

We propose new weight modifications of a 'path* estimate for the calculation of linear functionals (Φ,/ι) of radiation intensity Φ. In the framework of the 'collision* model of a transport process we mathematically 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 concentration 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 radiation 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 homogeneous 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 mathematically justify 'path estiamates' 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 multidimensional 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 estimate 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.

Estimation of the gradient of the solution of an adjoint diffusion equation by the Monte Carlo method

“…In view of this, it is possible to statistically estimate a probabilistic error. The above variance for ¼ remains finite for a sufficiently small domain [1].…”