Institute of Computational Mathematics and Mathematical Geophysics (Novosibirsk, Russia)

Address: Russia, 630090, Novosibirsk-90, Lavrentiev Avenue, 6. Telephones: (383) 3307721. Fax: (383) 3308783

- Corresponding member of RAS Mikhailov Gennadii Alekseevich
- Prof. Antyufeev Victor Stepanovich,
- Prof. Ukhinov Sergey Anatolyevich,
- Dr. Lotova Galiya Zufarovna,
- Dr. Marchenko Mikhail Aleksandrovich,
- Dr. Shkarupa Elena Valerevna,
- Dr. Medvedev Iliya Nikolaevich,
- Dr. Korotchenko Mariya Andreevna
- Dr. Tracheva Natalia Valerevna,
- Dr. Chimaeva Anna Sergeevna,
- Usov Alexander Georgievich,
- Gurova Zoya Vasilievna

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).