Abstract:
We aim to consider PDE-constrained nonparametric regression problems in which the unknown parameter is the coefficient function of a second order elliptic partial differential equation. The corresponding unique solution of a Dirichlet boundary problem is observed contaminated by a Gaussian White Noise. Based on a penalized least squares estimator, we want to recover the unknown coefficient function and derive statistical properties. To do this, we repeat the concept of Tikhonov regularization of inverse problems and describe a generalized statistical model to which the PDE-model described above can later be applied.