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Please use this identifier to cite or link to this item: http://hdl.handle.net/10277/704

Authors: Bianchi, Davide
Internal Tutor: DONATELLI, MARCO
Title: Diffusion equations and inverse problems regularization.
Abstract: The present thesis can be split into two dfferent parts: The first part mainly deals with the porous and fast diffusion equations. Chapter 2 presents these equations in the Euclidean setting highlighting the technical issues that arise when trying to extend results in a Riemannian setting. Chapter 3 is devoted to the construction of exhaustion and cut-o_ functions with controlled gradient and Laplacian, on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, and without any assumptions on the topology or the injectivity radius. The cut-offs are then applied to the study of the fast and porous media diffusion, of Lq-properties of the gradient and of the selfadjointness of Schrödinger-type operators. The second part is concerned with inverse problems regularization applied to image deblurring. In Chapter 5 new variants of the Tikhonov filter method, called fractional and weighted Tikhonov, are presented alongside their saturation properties and converse results on their convergence rates. New iterated fractional Tikhonov regularization methods are then introduced. In Chapter 6 the modified linearized Bregman algorithm is investigated. It is showed that the standard approach based on the block circulant circulant block preconditioner may provide low quality restored images and different preconditioning strategies are then proposed, which improve the quality of the restoration.
Keywords: Porous medium equation, fast diffusion equation, Tikhonov regularization, Breghan splitting
Issue Date: 2016
Language: eng
Doctoral course: Informatica e matematica del calcolo
Academic cycle: 29
Publisher: Università degli Studi dell'Insubria
Citation: Bianchi, D.Diffusion equations and inverse problems regularization. (Doctoral Thesis, Università degli Studi dell'Insubria, 2016).

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