Particle based sampling and optimization methods for inverse problems

Weissmann, Simon

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URN: urn:nbn:de:bsz:180-madoc-580740
Document Type: Doctoral dissertation
Year of publication: 2020
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Schillings, Claudia
Date of oral examination: 18 November 2020
Publication language: English
Institution: School of Business Informatics and Mathematics > Mathematische Optimierung (Schillings 2017-2022)
License: CC BY 4.0 Creative Commons Attribution 4.0 International (CC BY 4.0)
Subject: 510 Mathematics
Subject headings (SWD): Inverses Problem , Zufallsauswahl , Bayes-Verfahren , Kalman-Filter , Tichonov-Regularisierung
Keywords (English): bayesian inverse problems , ensemble Kalman inversion , optimization methods , sampling methods , regularization methods , gradient flow
Abstract: In this thesis, we present and analyse several ensemble based methods for sampling as well as optimization in inverse problems. Firstly we examine the ensemble Kalman inversion, which has been originally introduced as a sampling method for Bayesian inverse problems, but can also be viewed as derivative free optimization method. Furthermore, we present various transformed methods of the ensemble Kalman inversion, which allow to incorporate box-constraints as well as regularization for the underlying optimization problem. In addition, we also consider a more general class of particle based sampling methods, such as the ensemble Kalman sampler, which is based on an interacting Langevin dynamics, a particle system resulting from an Gaussian approximation, as well as a kernelized Fokker--Planck based particle system. In the last part of this work, we discuss machine learning applications in inverse problems. Here, we consider data-driven regularization, where the regularization parameter will be chosen by solving a bilevel optimization problem. Moreover, we consider an incorporation of neural networks into inverse problems, which will act as a model-informed surrogate for the complex forward model and will be trained with the unknown parameter in a one-shot fashion.

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