Statistical (Non)linear Inverse Problems with Unknown Structures: From Deconvolution to Dynamical Systems

This thesis investigates statistical methods for three classes of ill-posed inverse problems. In all models considered, observations arise as noisy versions of a transformed, yet unknown, target quantity. The objective is to develop estimation procedures for this quantity and to quantify their statistical accuracy within a rigorous mathematical framework. In the first part, we study statistical deconvolution problems in a multiplicative measurement error model, where both the density of the signal and that of the measurement errors are unknown. This setting leads to a linear inverse problem with an unknown operator. Based on spectral regularization techniques, we construct estimators and analyze their local and global risk. Furthermore, we develop a data-driven method for selecting the regularization parameter and derive upper bounds for the corresponding oracle-type risk. The second and third parts address regression models in which the regression functions are given by the solutions of ordinary or partial differential equations. The functions are evaluated at deterministic or random design points and observed under additive noise. The goal is to recover unknown parameters of the underlying differential equation. Due to the typically nonlinear nature of the solution operator, these models give rise to ill-posed nonlinear inverse problems. In the second part, we investigate both a penalized least squares approach and a Bayesian framework. In contrast to the existing literature, we analyze both methodologies under mild misspecification of model components such as the solution operator or the noise distribution. We establish consistency of the misspecified penalized least squares estimator and posterior contraction around the true parameter. The abstract results are applied to regression models governed by partial differential equations, with particular emphasis on the Darcy problem and the two-dimensional Navier-Stokes equations, where we derive upper bounds for both prediction and estimation errors. In the third part, we study the minimax optimality of such bounds. Concentrating on autonomous ordinary differential equations, we consider the problem of estimating the underlying vector field. After discussing suitable observation schemes to ensure identifiability, we derive lower bounds for the estimation error and establish the minimax optimality of existing procedures.