Convergence Rates for Penalized Least Squares Estimators in PDE Constrained Regression Problems
Richard Nickl, Sara van de Geer, Sven Wang
Abstract
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 $L_fu=g_1$ on $\mathcal O, u=g_2$ 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 $L_fu=\Delta u-2fu$ (Schrödinger 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 nonlinearly constrained regression functions $\{u_f: f \in \mathcal F\}$. We study Tikhonov-type penalized 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 “maximum a posteriori” 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 nonlinear 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.