A Statistical Approach to Some Inverse Problems for Partial Differential Equations

Some inverse problems for the Laplace equation and heat-conduction equation are considered. The solutions of these equations are assumed to be observed under the Gaussian white noise of low intensity. The problem consists of the renewal of the unknown smooth boundary conditions or initial conditions from the solution observed against a noise background. It is shown that the minimax estimates of the second order are linear for low spectral density of the noise.