Convergence rates for Penalised Least Squares Estimators in PDE-constrained 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 $L_{f}u=\Delta u-2fu$ (Schrödinger equation with attenuation potential $f$) and $L_{f}u=\text{div}(f\nabla u)$ (divergence form equation with conductivity $f$). In both cases, the parameter space \[\mathcal F=\{f\in H^α(\mathcal O)| f > 0\}, ~α>0, \] 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.

40 pages, to appear in SIAM/ASA Journal of Uncertainty Quantification

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