Convergence Rates for Penalized Least Squares Estimators in PDE Constrained Regression Problems

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 O, u=g₂ on ∂ O, is observed corrupted by additive Gaussian white noise. Here O is a bounded domain in Rᵈ with smooth boundary ∂ O, and g₁, g₂ are given functions defined on O, ∂ O, respectively. Concrete examples include Lfu=Δ u-2fu (Schrödinger equation with attenuation potential f) and Lfu=div (f∇ u) (divergence form equation with conductivity f). In both cases, the parameter space F=∈ H^α( O)| f > 0\, α>0, where H^α( O) is the usual order α Sobolev space, induces a set of nonlinearly constrained regression functions : f ∈ F\. We study Tikhonov-type penalized 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 “maximum a posteriori” estimator corresponding to some Gaussian process prior. We derive rates of convergence of f and of uf, 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 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.