- Thesis:
- Master in Mathematics
- Author:
- Maximilian Siebel
- Title:
- Statistical inverse problems: PDE constrained regression models
- Supervisor:
- Jan JOHANNES
- Abstract:
- In this thesis, we are considering a non-parametric and non-linear regression problem, where the corresponding regression function is supposed to be the solution of an elliptic partial differential equation depending on an unknown coefficient function. Based on noisy versions of this solution, we want to recover the unknown coefficient function by defining a Least Squares estimator motivated by the theory of ill-posed linear statistical inverse problems. Afterward, we study the statistical quality of this estimator by deriving concentration inequalities and minimax optimal bounds. This approach is part of a more general model, which will be studied first. For a better understanding, we are further illustrating the statistical behavior of the estimator by numerical experiments.
- Reference:
- R. Nickl, S. A. van de Geer et S. Wang. Convergence rates for penalized least squares estimators in PDE constrained regression problems, SIAM/ASA Journal on Uncertainty Quantification 8(1):374–413, (2020).
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |