Volume 18, Issue 6

The methodology of investigation and solution of inverse problems based on the optimal control theory and adjoint equations is presented. Some applications are considered.

We establish the stochastic dependence of the frequency variation process of spontaneous chromosome aberrations in human blood lymphocytes on the deterministic periodic components of the state of the geomagnetic field and solar activity by methods of mathematical modelling using experimental data.We prove that the random process of the frequency variation of spontaneous chromosome aberrations has a deterministic periodic component with two 4.8- and 67.5-year periods. Each of them is specified by the corresponding dominating time variation periods of the geomagnetic field induction and the Wolf numbers.

We consider finite volume approximations in exponential form for multidimensional diffusion-convection equations of elliptic and parabolic types with various boundary conditions.We describe the algebraic properties of the arising systems of equations and the iteration algorithms of their solution by incomplete factorization, biconjugate and gradient squared. We study convergence of approximate solutions. The examples of numerical experiments are given.

In many cases, values of the problem data (coefficients of a differential equation, boundary conditions, and right–hand sides) are not given exactly. In practical problems, we only know that they belong to certain sets of possible data values. Therefore estimation of errors of the approximate solution must take into account not only the approximation error, but also those arising due to indeterminacy of the data. The objective of this paper is to introduce a general scheme for deriving a posteriori estimates of this type. The method is based upon using functional-type a posteriori estimates that have been earlier derived in [(S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory. Zap. Nauch. Semin. , St. Petersburg Branch of V. A. Steklov Institute of Mathematics (1997) 243 , 201–214.), (S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp. (2000) 69 , No. 230, 481–500.), (S. Repin, Two-sided estimatesof deviations from exact solutions for uniformly elliptic equations. Transactions of St. Petersburg Math. Soc. (2001) 9 , 148–179 (in Russian).)] and some other papers for boundary-value problems with operators of elliptic type. Estimates obtained in the paper are of two types. They show the errors in the worst- and best-case situations depending on the way the data error is combined with the approximation one.

In this paper, we consider Monte Carlo algorithms connected with function approximations on functional bases. These include procedures for numerical modelling of stochastic elements by using histograms, polygons of frequencies and the other approximations of probabilistic densities as well as rejection methods (including a two-sided technique). The function approximations are also used in stochastic algorithms of numerical integration (the importance sampling method, the main part selection method, the two-sided geometrical method and others). Finally, the function approximation forms a basis for the construction of discrete-stochastic procedures.We formulate additional (besides approximation) properties of functional bases used in the above procedures. They are: stability and the possibility to construct efficient algorithms for realizing the samples of random variables for the corresponding density approximations.We make the comparative analysis of the known functional bases from the viewpoint of their use in algorithms of numerical statistical simulation.