CMX Lunch Seminar
Sven Wang is a final year PhD student at the University of Cambridge. His advisor is Prof Richard Nickl. Sven is affiliated to the Department of Pure Mathematics and Mathematical Statistics (DPMMS), Trinity College Cambridge, the Cantab Capital Institute for the Mathematics of Information (CCIMI) and the Cambridge Centre for Analysis (CCA). Sven's research interests are: - statistical and computational aspects of inverse problems, especially non-linear ones. - PDE-related statistical inverse 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.