Particle based sampling and optimization methods for inverse problems

Weissmann, Simon

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URL:	https://madoc.bib.uni-mannheim.de/58074
URN:	urn:nbn:de:bsz:180-madoc-580740
Dokumenttyp:	Dissertation
Erscheinungsjahr:	2020
Ort der Veröffentlichung:	Mannheim
Hochschule:	Universität Mannheim
Gutachter:	Schillings, Claudia
Datum der mündl. Prüfung:	18 November 2020
Sprache der Veröffentlichung:	Englisch
Einrichtung:	Fakultät für Wirtschaftsinformatik und Wirtschaftsmathematik > Mathematische Optimierung (Schillings 2017-2022)
Lizenz:	Creative Commons Namensnennung 4.0 International (CC BY 4.0)
Fachgebiet:	510 Mathematik
Normierte Schlagwörter (SWD):	Inverses Problem , Zufallsauswahl , Bayes-Verfahren , Kalman-Filter , Tichonov-Regularisierung
Freie Schlagwörter (Englisch):	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.