Institute of Computational Mathematics and Mathematical Geophysics

Тезисы докладов

Пленарные доклады

An analytical theory of random fields estimation by criterion of minimum of variance of the error of the estimate is developed. This theory does not assume a Markovian or Gaussian nature of the random field. The data are the covariance functions of the observed random field of the form u(x)=s(x)+n(x), where s(x) is the "useful signal" and n(x) is noise, and u(x) is observed in a bounded domain Din R^r of an r-dimensional Euclidean space, r>1. One wants to estimate a linear operator As acting on s. For example, if A=I, the identity operator, then one has the filtering problem, etc. Estimation theory seeks an optimal linear estimate Lu, "filter", for which $overline |Lu-As|^2=min$, where the overline stands for the variance and $Lu:=int_D h(x,y)u(y)dy$. For $h$ one gets a multidimensional integral equation of the type (*) $ Rh:=int_D R(x,y)h(y)dy=f(x), xin D. $

An analytical method for solving the basic integral equation (*) of estimation theory is given, numerical methods for solving this equation are proposed and their efficiency is demonstrated by examples, a singular perturbation theory for (*) is developed, that is, a study of the limiting behavior of the equation (**) $ (epsilon I+R)h_{epsilon}=f, $ as $epsilonto 0$ is given. Statistically this corresponds to the case when the white component in noise tends to zero. The random fields are not assumed to be homogeneous.

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