Approximate minimax estimation of functional from the solution of parabolic boundary-value problem with rapidly oscillating coefficients under nonlinear observations
Keywords:minimax estimation, parabolic boundary-value problem, rapidly oscillating coefficients, averaged problem, uncertainty, approximate estimate
AbstractThe article deals with the problem of the minimax estimation of a functional from the solution of a parabolic boundary-value problem with rapidly oscillating coefficients. We measure not the value that describes the investigated process, but some value from the solution with an operator, which determines the way of measurement. The problem is complicated not only due to the rapidly fluctuating coefficients and unknown functions that are included in the equation and initial conditions, but also due to the observation being nonlinear (it has a superposition type operator). At a value of the small parameter, the solution existence for the original problem is established using the traditional minimax approach. The transition to a problem with averaged parameters allows us to get rid of nonlinearity in the observation. The main result of the work is to prove that the minimax estimate of the problem with averaged coefficients is an approximate minimax estimate of the original problem.
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