# Statistical analysis of a non-linear inverse problem for an elliptic PDE

The talk will be about some statistical inverse problems arising in the context of second order elliptic PDEs. Concretely, suppose $f$ is the unknown coefficient function of a second order elliptic partial differential operator $L_f$ on some bounded domain $\mathcal O\subseteq \R^d$, and the unique solution $u_f$ of a corresponding boundary value problem is observed, corrupted by additive Gaussian white noise. Concrete examples include $L_fu=\Delta u-2fu$ (Schr\”odinger equation with attenuation potential $f$) and $L_fu=\text{div} (f\nabla u)$ (divergence form equation with conductivity $f$). I will present some recent results on the convergence rates for Tikhonov-type penalised least squares estimators $\hat f$ for $f$ in these contexts. The penalty functionals are of squared Sobolev-norm type and thus $\hat f$ can also be interpreted as a Bayesian `MAP’-estimator corresponding to some Gaussian process prior. We derive rates of convergence of $\hat f$ and of $u_{\hat f}$, to $f, u_f$, respectively. The rates obtained are minimax-optimal in prediction loss.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.