Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems
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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 O, u=g_2 on ∂ O, is observed corrupted by additive Gaussian white noise. Here O is a bounded domain in R^d with smooth boundary ∂ O, and g_1, g_2 are given functions defined on O, ∂ O, respectively. Concrete examples include L_fu=Δ u-2fu (Schrödinger equation with attenuation potential f) and L_fu=div (f∇ u) (divergence form equation with conductivity f). In both cases, the parameter space F={f∈ H^α( O)| f > 0}, α>0, where H^α( O) is the usual order α Sobolev space, induces a set of non-linearly constrained regression functions {u_f: f ∈ F}. We study Tikhonov-type penalised least squares estimators f̂ for f. The penalty functionals are of squared Sobolev-norm type and thus f̂ can also be interpreted as a Bayesian `MAP'-estimator corresponding to some Gaussian process prior. We derive rates of convergence of f̂ and of u_f̂, 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 mild 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.
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