CMX Lunch Seminar
Statistical Guarantees for MAP Estimators in PDE-Constrained Regression Problems
The main topic of the talk are convergence rates for penalised least squares (PLS) estimators in non-linear statistical inverse problems, which can also be interpreted as Maximum a Posteriori (MAP) estimators for certain Gaussian Priors. Under general conditions on the forward map, we prove convergence rates for PLS estimators.
In our main example, the parameter f is an unknown heat conductivity function in a steady state heat equation [a second order elliptic PDE]. The observations consist of a noisy version of the solution u[f] to the boundary value corresponding to f. The PDE-constrained regression problem is shown to be solved a minimax-optimal way.
This is joint work with S. van de Geer and R. Nickl. If time permits, we will mention some related work on the non-parametric Bayesian approach, as well as computational questions for the Bayesian posterior.