Statistical modelling, or Monte Carlo methods, is the only possible way to solve a wide range of multivariate problems in the radiation transfer theory with due account for the great variety of geometrical and physical assumptions. This permits numerical handling of practical problems in atmospheric and oceanic optics, reactor physics and engineering, diffusion of impurities in probability fields, the theory of rarefied gases, and so on.
The weighted Monte Carlo estimates are constructed in the Laboratory to diminish the errors and to obtain depended estimates for the calculated functionals for different values of parameters of the mentioned problems, i.e. to improve the functional dependence under study. In addition, the weighted estimates make it possible to evaluate special functionals, for example, the derivatives with respect to the parameters.
Weighted Monte Carlo methods are constructed and optimised on the basis of the recurrent representations and Bellman principle. This approach is specially effective if the investigated equation is non-linear.
It is explained how to use asymptotics of the radiative transfer problems to improve the corresponding weighted estimates. The non-linear and minimax theories of weighted Monte Carlo estimates, the vector weighted estimates and the randomized algorithms are developed.
A lot of simulation methods for sampling random variables and vectors are constructed and investigated. The theory of discrete stochastic procedures for global estimation of functions presented in the integral form is developed.
New Monte Carlo algorithms for solving Helmholtz equation with nonconstant and especially the positive parameters are constructed and investigated. These algorithms are connected with formal representation of the solution on the basis of the Green function in the centre of the inscribed sphere. Additional investigation of known algorithms are performed to generalize their applications, especially for solving nonhomogeneous metagarmonic equations. New estimates for metagarmonic equations are constructed as the estimates of the parametric derivatives for the solution of the corresponding Helmholtz equation. The new Monte Carlo method is also constructed for solving the Dirichlet problem for the non-linear elliptic equation.
The investigations of the laboratory were supported by the Russian Foundation of Fundamental Research under grants N 94-01-00500, 94-05-16529 , 99-07-90422, 02-01-00958, 06-01-00046a, 09-01-0035, 12-01-0034, 12-01-00727; the program “Leading Scientific Schools” (project no. NSh-4774.2006.1).