Recurrent partial averaging in the theory of weighted Monte Carlo methods

Various weighted algorithms of numerical statistical modelling with trajectory branching are formulated and studied in the case where the next weight factor exceeds one. As the result, the 'weight' of a separate 'branch' does not exceed one and the variance of the estimate of the calculated functional is finite. The problems of unbiasedness and variance finiteness are solved based on the method of recurrent 'partial' averaging presented in this paper. As an application, estimates of the particle reproduction rate and the solutions to the Helmholtz equation are considered. The comparative computational cost of the method of 'exponential transformation' with branching is studied for the radiation transfer theory problems based on the Galton-Watson majorant process. In this connection, an expression for the second moment of the total number of particles in the Galton-Watson subcritical process is obtained. The minimum variance problem for an integer-valued random variable under a fixed mathematical expectation is also solved. In addition, variances of weighted algorithms with branching are considered for solution of integral equations with power nonlinearity.