Minmax estimation of functionals of solutions of initial-boundary-value problems for parabolic equations

Abstract

The minmax estimation theory of linear functionals of solutions of variational equations in equipped Hilbert spaces is presented in detail in [1]. Its application to particular boundary-value problems involves an appropriate choice of Sobolev spaces, bilinear forms, and observation operators. A particular boundary-value problems is not always representable in variational form, and such cases additional difficulties arise with estimation from observations. Below we consider these issues for minmax estimation with special observation operators and with unknown functions from special definition domains.Let f~ be a bounded domain in space R n with a sufficiently smooth boundary F, D T the cylinder f~ x (to, T), and ~:T its lateral surface I ~ X (t 0, T). The functions aij(x, t), ao(X, t) defined in the cylinder satisfy the conditions aij(x, t) = aji(x, t) almost everywhere in D T and there exists a constant c~ > 0 such that Iff 1 E L,z(DT), fo E L2(~), the coefficients of the operator A(t) are twice continuously differentiable in fl • [t o, T], and n _< 3, then problem (1) has a unique generalized solution in the space W~2'I(DT), which is defined at isolated points xj, j = 1 ..... m, of the domain ~ [2].Suppose that at the points ~, j = 1 ..... m, we observe the realization of the stochastic processes y.i(t) = ~(xj, t) + ~j(t), t E (to, T), where ,:(x, t) is the solution of problem (i) with some functionft(x, t) and a known functionf0(x), ~j(t) are stochastic processes with mean zero M~y(t) = 0 and unknown correlation function R(t, s) = (rij(t, s) k / = ,r-~, %(t, s) = M~i(t)~l(s),from the domain n T K={R: ~ f rii(t, t)qi2(t)dt ~ 1}. i=1 t o