liu.seSearch for publications in DiVA
Change search
ReferencesLink to record
Permanent link

Direct link
Exact Minimizers in Real Interpolation: Some additional results
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics, School of Pure and Applied Science, College of Science and Technology, University of Rwanda, Rwanda.ORCID iD: 0000-0002-8188-7672
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present some extensions of results presented in our recent papers. First we extend the characterization of optimal decompositions for a Banach couple to optimal decompositions for a Banach triple. Next we show that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that we have proposed for the ROF model with the Split Bregman algorithm. The Split Bregman algorithm can in principle be regarded as a benchmark algorithm for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.

Keyword [en]
Regular Banach triple, Optimal decompositions, Complex Banach couple, Real Interpolation, Convex Duality
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-118355OAI: oai:DiVA.org:liu-118355DiVA: diva2:814579
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2015-05-27Bibliographically approved
In thesis
1. Exact Minimizers in Real Interpolation: Characterization and Appliations
Open this publication in new window or tab >>Exact Minimizers in Real Interpolation: Characterization and Appliations
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The main idea of the thesis is to develop new connections between the theory of real interpolation and applications. Near and exact minimizers for E–, K– and L–functionals of the theory of real interpolation are very important in applications connected to regularization of inverse problems such as image processing. The problem which appears is how to characterize and construct these minimizers. These exact minimizers referred to as optimal decompositions in the thesis, have certain extremal properties that we completely express and characterize in terms of duality. Our characterization generalizes known characterization for a particular Banach couple. The characterization presented in the thesis also makes it possible to understand the geometrical meaning of optimal decomposition for some important particular cases and gives a possibility to construct them. One of the most famous models in image processing is the total variation regularization  published by Rudin, Osher and Fatemi. We propose a new fast algorithm to find the exact minimizer for this model. Optimal decompositions mentioned have some connections to optimization problems which are also pointed out. The thesis is based on results that have been presented in international conferences and have been published in five papers.

In Paper 1, we characterize optimal decomposition for the E–, K– and Lp0,p1 –functional. We also present a geometrical interpretation of optimal decomposition for the Lp,1–functional for the couple (ℓp, X) on Rn. The characterization presented is useful in the sense that it gives insights into the construction of these minimizers.

The characterization mentioned in Paper 1 is based on optimal decomposition for infimal convolution. The operation of infimal convolution is a very important and non–trivial tool in functional analysis and is also very well–known within the context of convex analysis. The L–, K– and E–functionals can be regarded as an infimal convolution of two well–defined functions. Unfortunately tools from convex analysis can not be applied in a straightforward way in this context of couples of spaces. The most important requirement that an infimal convolution would satisfy for a decomposition to be optimal is subdifferentiability. In Paper 2, we have used an approach based on the famous Attouch–Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.

In Paper 3, we apply result from Paper 1 to the well–known Rudin–Osher–Fatemi (ROF) image denoising model on a general finite directed graph. We define the space BV of functions of bounded variation on the graph and show that the unit ball of its dual space can be described as the image of the unit ball of the space `¥ on the graph by a divergence operator. Based on this result, we propose a new fast algorithm to find the exact minimizer for the ROF model. Proof of convergence of the algorithm is presented and its performance on image denoising test examples is illustrated.

In Paper 4, we present some extensions of results presented in Paper 1 and Paper 2. First we extend the results from Banach couples to Banach triples. Then we prove that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that was proposed in Paper 3 with the Split Bregman algorithm which is one of the benchmark algorithms known for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.

In Paper 5, we point out some connections between optimal decompositions mentioned in the thesis and optimization problems. We apply the approach used in Paper 2 to two well–known optimization problems, namely convex and linear programming to investigate connections with standard results in the framework of these problems. It is shown that we can derive proofs for duality theorems for these problems under the assumptions of our approach.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 24 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1650
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-118357 (URN)10.3384/diss.diva-118357 (DOI)978-91-7519-102-7 (print) (ISBN)
Public defence
2015-06-12, Nobel (BL 32), B-huset, Campus Valla, Linköping, 09:00 (English)
Opponent
Supervisors
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2015-05-27Bibliographically approved

Open Access in DiVA

No full text

Search in DiVA

By author/editor
Niyobuhungiro, Japhet
By organisation
Mathematics and Applied MathematicsFaculty of Science & Engineering
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 13 hits
ReferencesLink to record
Permanent link

Direct link