Convergence rates for Penalised Least Squares Estimators in PDEconstrained regression problems
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 \[L_fu=g_1\text{ on } \mathcal O, \quad u=g_2 \text{ on }\partial \mathcal O,\] 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 $L_fu=\Delta u2fu$ (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 Tikhonovtype penalised least squares estimators $\hat f$ for $f$. The penalty functionals are of squared Sobolevnorm 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 minimaxoptimal 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.08818
 Bibcode:
 2018arXiv180908818N
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis
 EPrint:
 40 pages, to appear in SIAM/ASA Journal of Uncertainty Quantification