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

@article{Nickl2020ConvergenceRF,
  title={Convergence Rates for Penalized Least Squares Estimators in PDE Constrained Regression Problems},
  author={R. Nickl and S. Geer and Sven Wang},
  journal={SIAM/ASA J. Uncertain. Quantification},
  year={2020},
  volume={8},
  pages={374-413}
}
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… Expand
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