Random fields estimation and related numerical problems

Alexander Ramm Kansas State University Mathematics

Random fields estimation and related numerical problems Alexander G.Ramm email: ramm@math.ksu.edu http://www.math.ksu.edu/~ramm Abstract 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 D\in 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), x\in D. $ An analytical method for solving the basic integral equation (*) of estimation theory is given, numerical methods for solving this equation are proposed. Singular perturbation theory is given for the basic equation (*).