2018/09/24 by Richard Nickl, Nickl, Richard, Sara van de Geer +2
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Control Systems and Identification #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1809.08818
openalex publication_date 2018/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 on \mathcal O, u=g2 on ∂ \mathcal O, is observed corrupted by additive Gaussian white noise. Here \mathcal O is a bounded domain in \mathbb Rd with smooth boundary ∂ \mathcal O, and g1, g2 are given functions defined on \mathcal O, ∂ \mathcal O, respectively. Concrete examples include Lfu=Δu-2fu (Schrödinger equation with attenuation potential f) and Lfu=div (f∇ u) (divergence form equation with conductivity f). In both cases, the parameter space \mathcal F=\f∈ Hα(\mathcal O)| f gt; 0\, ~αgt;0, where Hα(\mathcal O) is the usual order α Sobolev space, induces a set of non-linearly constrained regression functions \uf: f ∈ \mathcal F\. We study Tikhonov-type penalised least squares estimators f for f. The penalty functionals are of squared Sobolev-norm type and thus f can also be interpreted as a Bayesian `MAP'-estimator corresponding to some Gaussian process prior. We derive rates of convergence of f and of u 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.