  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 $L_f$, and the unique solution $u_f$ of the
boundary value problem \[L_fu=g_1\text{ on } \mathcal O, \quad u=g_2 \text{ on
}\partial \mathcal O,\] 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 $g_1, g_2$ are given functions defined on $\mathcal
O, \partial \mathcal O$, respectively. 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 $f$).
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
$\{u_f: f \in \mathcal F\}$.
  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 $f, u_f$, 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.
