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Sven WANG (Université de Cambridge) – “Convergence rates for penalised least squares estimators in PDE-constrained regression problems”
November 18, 2019, 2:00 pm - 3:15 pm
The Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:15 pm
Date: 18th of November 2019
Place: Room 3001.
Sven WANG (Université de Cambridge) – “Convergence rates for penalised least squares estimators in PDE-constrained regression problems“
Abstract: In this talk, we study the convergence rates for Tikhonov-type penalised least squares estimators (with Sobolev-norm type penalty) in non-linear statistical inverse problems.
Our main example is a non-parametric statistical model where the parameter f is an unknown heat conductivity function in a steady state heat equation [which is second order elliptic PDE] and a ‘noisy version’ the solution u[f] to the boundary value problem corresponding to f is observed. Here, the map from f to u[f] is non-linear and induces a set of functions with a PDE – constraint , and f needs to be modelled with a non-negativity constraint. Under some general conditions on the map from f to u[f], we prove convergence rates for f and the associated plug-in estimator for u[f]. The PDE-constrained regression problem is solved a minimax-optimal way.
Cristina BUTUCEA, Alexandre TSYBAKOV, Julie JOSSE, Eric MOULINES, Mathieu ROSENBAUM