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 (*).
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