Regularization of Nonlinear Inverse Problems -- From Functional Analysis to Data-Driven Approaches

TL;DR

This work explores regularization techniques for nonlinear inverse problems, emphasizing data-driven methods that incorporate prior experimental data to improve solutions across diverse scientific and industrial applications.

Contribution

It bridges functional analysis and data-driven approaches, advancing regularization methods for nonlinear inverse problems with a focus on integrating experimental data.

Findings

Enhanced regularization techniques for nonlinear inverse problems.

Integration of supervised and unsupervised data improves solution accuracy.

Applicable to medical, geophysical, and industrial imaging.

Abstract

The focus of this book is on the analysis of regularization methods for solving \emph{nonlinear inverse problems}. Specifically, we place a strong emphasis on techniques that incorporate supervised or unsupervised data derived from prior experiments. This approach enables the integration of data-driven insights into the solution of inverse problems governed by physical models. \emph{Inverse problems}, in general, aim to uncover the \emph{inner mechanisms} of an observed system based on indirect or incomplete measurements. This field has far-reaching applications across various disciplines, such as medical or geophysical imaging, as well as, more broadly speaking, industrial processes where identifying hidden parameters is essential.