Study of 'value' modelling efficiency in the Monte Carlo method

Abstract

To construct weight modifications of numerical statistical simulation we include auxiliary variables whose values define transitions in the basic Markov chain in the number of the phase space coordinates. On realizing each auxiliary random variable, the auxiliary weight is multiplied by the ratio of the respective initial and modelled distribution densities. In this paper, we study the efficiency of partial 'value' modelling associated with the construction of modelled distribution of some auxiliary random variable by multiplying the initial density by a 'value function' that corresponds to the solution of the adjoint equation. We obtain conditions under which the value modelling of the initial distribution reduces its variance as compared to direct modelling. Using the conditions obtained, we analyse the efficiency of the value modelling of the initial distribution for some problems of the transport theory. We also study the efficiency of the value modelling of the free path when solving transport theory problems. Using the solution to a system of linear algebraic equations as an example, we show that, in principle, partial value modelling may increase the estimate variance.