Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems

We consider PDE constrained nonparametric regression problems in which the parameter $f$ is the unknown coefficient function of a second order elliptic partial differential operator $Lf$, and the unique solution $uf$ of the boundary value problem [Lfu=g1\text{ on } \mathcal O, \quad u=g2 \text{ on }\partial \mathcal O,] is observed corrupted by additive Gaussian white noise. Here $\mathcal O$ is a bounded domain in $\mathbb R^d$ with smooth boundary $\partial \mathcal O$, and $g1, g2$ are given functions defined on $\mathcal O, \partial \mathcal O$, respectively. Concrete examples include $Lfu=\Delta u-2fu$ (Schr\"odinger equation with attenuation potential $f$) and $Lfu=\text{div} (f\nabla u)$ (divergence form equation with conductivity $f$). In both cases, the parameter space [\mathcal F={f\in H^{\alpha(\mathcal} O)| f > 0}, ~\alpha>0, ] where $H^{\alpha(\mathcal} O)$ is the usual order $\alpha$ Sobolev space, induces a set of non-linearly constrained regression functions ${uf: f \in \mathcal F}$. We study Tikhonov-type penalised least squares estimators $\hat f$ for $f$. 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 $f, uf$, respectively. We prove that the rates obtained are minimax-optimal in prediction loss. Our bounds are derived from a general convergence rate result for non-linear inverse problems whose forward map satisfies a modulus of continuity condition, a result of independent interest that is applicable also to linear inverse problems, illustrated in an example with the Radon transform.