- Thesis:
- Master in Mathematics
- Author:
- Timo Dörzbach
- Title:
- Nonparametric estimation for linear stochastic partial differential equations from local measurements
- Supervisors:
- Maximilian Siebel
- Jan JOHANNES
- Abstract:
- The present thesis introduces from scratch a methodology of parametric and nonparametric estimation for the coefficient and coefficient function, respectively, of the leading second-order differential operator from observations based on a linear stochastic partial differential equation. The procedure relies on time-continuous measurements, which are localized in space. Fixing a finite time horizon, we investigate the asymptotic regime of the observation’s resolution level tending to zero, with a view to rescaling properties of a solution process to the differential equation. Establishing scaling limits of the deterministic as well as the stochastic partial differential equation on growing domains, the underlying heat equation structure is revealed. Furthermore, we construct two minimax-optimal estimators that are robust against lower order perturbations of the differential operator, and achieve the parametric convergence rate even in the nonparametric setup with spatially varying coefficient function. The approach is put into the broader context of statistical inference for stochastic partial differential equations, and the most important recent discoveries based on the novel idea are briefly sketched.
- Reference:
- R. Altmeyer und Markus Reiß. Nonparametric estimation for linear SPDEs from local measurements The Annals of Applied Probability 31(1):1–38, 2021.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |