On deterministic and statistical consistency for nonlinear inverse problems

Aksel Kaastrup Rasmussen

Research output: Book/ReportPh.D. thesis

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Abstract

This thesis studies consistent reconstruction for nonlinear inverse problems from the perspective of regularization theory and the Bayesian approach. It focuses on three nonlinear inverse problems: the Calderón problem, an inverse problem in photoacoustic tomography, and an inverse Robin problem. These problems have applications across the fields of medical imaging, industrial testing, and, in the case of the latter, even in large-scale ice sheet modeling for sea level predictions. The regularization perspective considers consistency as a property attributed to methods whose reconstructions improve toward the ground truth as the deterministic noise decreases. For a large class of penalized least-squares methods, consistency is guaranteed, while computability is harder to guarantee. In contrast, we offer, in this thesis, a direct, consistent, and computable regularization strategy for the three-dimensional Calderón problem. It is based on truncated frequency information under a prior assumption of a smooth ground truth. Further, we demonstrate the convergence property numerically on synthetic data. The Bayesian perspective has become the preferred perspective when it comes to handling random noise, incorporating prior knowledge, and quantifying uncertainty. Here, consistency enters as a property of a posterior distribution that increasingly concentrates around the ground truth as the random data improves. Guaranteeing consistency is often a first step in reliable estimation and uncertainty quantification for high-dimensional nonlinear inverse problems. We will review the Bayesian perspective, recent consistency results, and offer parallels to the perspective of regularization theory Among new contributions, we address consistent Bayesian reconstruction of piecewise constant parameters: We provide consistent methods for inclusion detection and apply them to a problem in the photoacoustic tomography setting, demonstrating the consistency in a numerical example. Furthermore, we address how to consistently and efficiently reconstruct real analytic parameters with the use of a strong smoothness prior in a Bayesian approach. Here, the inverse Robin problem serves as motivation and a theoretical case study.
Original languageEnglish
PublisherTechnical University of Denmark
Number of pages201
Publication statusPublished - 2023

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  • Computational Uncertainty Quantification for Hybrid Inverse Problems

    Rasmussen, A. K. (PhD Student), Knudsen, K. (Main Supervisor), Tarvainen, T. (Supervisor), Garnier, J. (Examiner) & Kekkonen, H. (Examiner)

    01/09/202011/01/2024

    Project: PhD

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