Approximate Minimax Estimation of Functionals of Solutions to the Wave Equation under Nonlinear Observations

The authors consider the problem of minimax estimation of a functional of the solution to the wave equation with rapidly oscillating coefficients. The observation (output signal) is nonlinear (has the operator of superposition type). For the small parameter epsilon > 0, the existence of the solution of the original problem is proved using the traditional minimax approach. Transition to a homogenized parameter problem allows us to remove the nonlinearity in the observation. The main result of the paper is that the minimax estimate of the problem with homogenized parameters is an approximate minimax estimate of the original problem.