Convergence analysis of critical point regularization with non-convex regularizers

One of the key assumptions in the stability and convergence analysis of variational regularization is the ability of finding global minimizers. However, such an assumption is often not feasible when the regularizer is a black box or non-convex making the search for global minimizers of the involved Tikhonov functional a challenging task. This is in particular the case for the emerging class of learned regularizers defined by neural networks. Instead, standard minimization schemes are applied which typically only guarantee that a critical point is found. To address this issue, in this paper we study stability and convergence properties of critical points of Tikhonov functionals with a possible non-convex regularizer. To this end, we introduce the concept of relative sub-differentiability and study its basic properties. Based on this concept, we develop a convergence analysis assuming relative sub-differentiability of the regularizer. The rationale behind the proposed concept is that critical points of the Tikhonov functional are also relative critical points and that for the latter a convergence theory can be developed. For the case where the noise level tends to zero, we derive a limiting problem representing first-order optimality conditions of a related restricted optimization problem. Besides this, we also give a comparison with classical methods and show that the class of ReLU-networks are appropriate choices for the regularization functional. Finally, we provide numerical simulations that support our theoretical findings and the need for the sort of analysis that we provide in this paper

Convergence analysis of equilibrium methods for inverse problems

Recently, the use of deep equilibrium methods has emerged as a new approach for solving imaging and other ill-posed inverse problems. While learned components may be a key factor in the good performance of these methods in practice, a theoretical justification from a regularization point of view is still lacking. In this paper, we address this issue by providing stability and convergence results for the class of equilibrium methods. In addition, we derive convergence rates and stability estimates in the symmetric Bregman distance. We strengthen our results for regularization operators with contractive residuals. Furthermore, we use the presented analysis to gain insight into the practical behavior of these methods, including a lower bound on the performance of the regularized solutions. In addition, we show that the convergence analysis leads to the design of a new type of loss function which has several advantages over previous ones. Numerical simulations are used to support our findings

Dianthus versicolor FISCH

Das Ziel dieser Arbeit war die phytochemische Untersuchung der mongolischen Arzneipflanze Dianthus versicolor, die in der tradtionellen Medizin gegen Lebererkrankungen und gastrointestinale Beschwerden verwendet wird. Unterschiedliche Konzentrationen eines wässrigen Extraktes führten im Modell der isolierten perfundierten Rattenleber zu einer Steigerung des Gallenflusses, was als Hinweis auf eine Stimulierung der Leber gesehen werden kann. In weiterer Folge wurde das Extrakt aktivitätsgeleitet fraktioniert, wobei eine flavonoidreiche Fraktion eine dem Cynarin vergleichbare choleretische Wirkung zeigte. Darüber hinaus wurde das wässrige Extrakt an verschiedenen glattmuskulären Organpräparaten getestet und wies in höherer Dosierung eine Uterus-kontrahierende Wirkung auf. Eine weiterführende Fraktionierung der Flavonoid-Fraktion mittels CPC, CC und semipräparativer HPLC resultierte in der Isolierung von neun Flavon-C- und O-glykosiden. Die Strukturaufklärung dieser Verbindungen gelang unter Anwendung diverser spektroskopischer und spektrometrischer Verfahren, wie LC-DAD, LC-ESI-MSn, LC-HR-ESI-MS, 1D und 2D-NMR sowie GC-MS nach Hydrolyse. Sieben der isolierten Verbindungen sind neue Strukturen und für D. versicolor noch nicht beschrieben. Für die Analytik der Flavonoide wurden geeignete HPLC-Systeme ausgearbeitet, die die Quantifizierung mit internem oder externem Standard erlauben. Als Alternative zur Quantifizierung mittels HPLC wurde eine UV-spektrophotometrische Methode, basierend auf der Monographie „Passionsblumenkraut“ der Pharmacopoeia Europea, erstellt, welche vergleichbare Ergebnisse liefert. Die im Rahmen dieser Arbeit durchgeführten qualitativen und quantitativen Analysen tragen wesentlich zur Chrarakterisierung der in D. versicolor enthaltenen Flavonoid-Glykoside bei. Darüber hinaus gelang die Isolierung und Strukturaufklärung neuer Substanzen.The aim of this thesis was the phytochemical investigation of the Mongolian medicinal plant Dianthus versicolor accompanied by the testing for its activity on the bile flow. It is used in traditional medicine for various indications, among them for the treatment of liver and gastrointestinal disorders. An aqueous extract, prepared according to the traditional way of intake, was tested in the model of the isolated perfused rat liver in order to examine its influence on the bile flow. Different concentrations led to an increase of the bile flow showing a slight dose dependency. The extract was subsequently fractionated by solid phase extraction and one of the fractions, enriched in flavonoids showed an influence on the bile flow. This effect was comparable to the positive control cynarin, which is known for its cholerectic activity. Furthermore, the extract was examined on isolated organ preparations from the uterus, aorta, heart, arteria pulmonalis and terminal ileum and showed a uterus-constringing activity. Further fractionation of the enriched flavonoid fraction by centrifugal partition chromatography or column chromatography and purification by semipreparative HPLC led to the isolation of nine flavonoid-C- and O-glycosides. Their structures were established on the basis of extensive spectroscopic and spectrometric investigations including LC-DAD, LC-MSn, LC-HR-ESI-MS, 1D and 2D NMR, and by GC-MS analysis after hydrolysis. Seven of the isolated structures are new, and have not been described for D. versicolor so far. For the quantification of the flavonoids different HPLC-DAD methods were established and validated using external or internal standards. As alternative a quite simple UV-spectrophotometric method was developed. It based on a monograph from the European Pharmacopoeia and was slightly modified, showing comparable data to those obtained from HPLC-DAD analysis. The qualitative and quantitative analyses allowed a detailed phytochemical characterization of the flavonoids contained in the aqueous extract of this plant and led to the isolation and structural elucidation of new compounds

Augmented NETT regularization of inverse problems

Abstract: We propose aNETT (augmented NETwork Tikhonov) regularization as a novel data-driven reconstruction framework for solving inverse problems. An encoder-decoder type network defines a regularizer consisting of a penalty term that enforces regularity in the encoder domain, augmented by a penalty that penalizes the distance to the signal manifold. We present a rigorous convergence analysis including stability estimates and convergence rates. For that purpose, we prove the coercivity of the regularizer used without requiring explicit coercivity assumptions for the networks involved. We propose a possible realization together with a network architecture and a modular training strategy. Applications to sparse-view and low-dose CT show that aNETT achieves results comparable to state-of-the-art deep-learning-based reconstruction methods. Unlike learned iterative methods, aNETT does not require repeated application of the forward and adjoint models during training, which enables the use of aNETT for inverse problems with numerically expensive forward models. Furthermore, we show that aNETT trained on coarsely sampled data can leverage an increased sampling rate without the need for retraining