Sven Wang, University of Cambridge
Convergence rates for penalised least squares 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. Under some 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, e.g. on the non-parametric Bayesian approach.