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Exact Minimizers in Real Interpolation: Characterization and Appliations
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-8188-7672
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.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1650
Mathematics
##### Identifiers
ISBN: 978-91-7519-102-7 (print)OAI: oai:DiVA.org:liu-118357DiVA, id: diva2:814591
##### Public defence
2015-06-12, Nobel (BL 32), B-huset, Campus Valla, Linköping, 09:00 (English)
##### Supervisors
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2015-05-27Bibliographically approved
##### List of papers
1. Characterization of optimal decompositions in real interpolation
Open this publication in new window or tab >>Characterization of optimal decompositions in real interpolation
2014 (English)In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 185, p. 1-11Article in journal (Refereed) Published
##### Abstract [en]

We use duality in convex analysis to characterize optimal decompositions for the $L$-functional. We also describe a geometry of optimal decompositions for the $\small L_{p,1}$-functional for the couple $\small \left( \ell^{p},X \right)$ on $\small \mathbb{R}^{n}$.

Elsevier, 2014
##### Keywords
Optimal decompositions, Real interpolation, Duality
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-107565 (URN)10.1016/j.jat.2014.05.015 (DOI)000340691300001 ()
##### Note

Available from: 2014-06-16 Created: 2014-06-16 Last updated: 2017-12-05
2. Subdifferentiability of Infimal Convolution on Banach Couples
Open this publication in new window or tab >>Subdifferentiability of Infimal Convolution on Banach Couples
2015 (English)In: Functiones et Approximatio Commentarii Mathematici, ISSN 0208-6573, Vol. 52, no 2, p. 311-326Article in journal (Other academic) Published
##### Abstract [en]

We use duality in convex analysis and particularly the famous Attouch-Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.

##### Place, publisher, year, edition, pages
Wydawnictwo Naukowe Uniwersytetu im. Adama Mickiewicza, 2015
##### Keywords
real interpolation, infimal convolution, subdifferentiability
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-118353 (URN)10.7169/facm/2015.52.2.9 (DOI)
##### Note

At the time for thesis presentation publication was in status: Manuscript

Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2016-08-04Bibliographically approved
3. A New Reiterative Algorithm for the Rudin-Osher-Fatemi Denoising Model on the Graph
Open this publication in new window or tab >>A New Reiterative Algorithm for the Rudin-Osher-Fatemi Denoising Model on the Graph
2014 (English)In: Proceedings of The 2nd International Conference on Intelligent Systems and Image Processing 2014, ICISIP2014, 2014, p. 81-88Conference paper, Published paper (Refereed)
##### Abstract [en]

We consider an analogue of the well-known in image processing Rudin-Osher-Fatemi (ROF) denoising model on a general finite directed and connected graph. Then we consider 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 ℓ on the graph by a divergence operator. Based on this result, we propose a new fast reiterative algorithm to find the exact minimizer for the ROF model. Finally we prove convergence of the algorithm and illustrate its performance on some test examples. We would like to note that consideration of an image on the graph is important in the sense that it provides useful insights when instead of rectangular domain we have some manifold, which is a good representation for images arising in applications.

##### Keywords
L–functional, Image processing, Dual BV, Regularization
##### National Category
Mathematics Other Natural Sciences
##### Identifiers
urn:nbn:se:liu:diva-111218 (URN)10.12792/icisip2014.018 (DOI)
##### Conference
The 2nd International Conference on Intelligent Systems and Image Processing 2014, ICISIP2014, September 26 – September 29, 2014, Kitakyushu, Japan
Available from: 2014-10-11 Created: 2014-10-11 Last updated: 2015-05-27Bibliographically approved
4. Exact Minimizers in Real Interpolation: Some additional results
Open this publication in new window or tab >>Exact Minimizers in Real Interpolation: Some additional results
##### 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.

##### Keywords
Regular Banach triple, Optimal decompositions, Complex Banach couple, Real Interpolation, Convex Duality
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-118355 (URN)
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2015-05-27Bibliographically approved
5. Optimal decomposition for infimal convolution on Banach Couples
Open this publication in new window or tab >>Optimal decomposition for infimal convolution on Banach Couples
##### Abstract [en]

Infimal convolution of functions defined on a regular Banach couple is considered. By using a theorem due to Attouch and Brezis, we establish sufficient conditions for this infimal convolution to be subdifferentiable. We also provide a result which gives a dual characterization of optimal decomposition for an infimal convolution in general. We plan tu use these results as tools to study the mathematical properties of exact minimizers for the K–, L–, and E– functionals of the theory of real interpolation. This will be done in a separate study. In this study, we apply our approach to two well–known optimization problems, namely convex and linear programming and provide proofs for duality theorems based on infimal convolution.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 7
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-94152 (URN)LiTH-MAT-R--2013/07--SE (ISRN)
Available from: 2013-06-17 Created: 2013-06-17 Last updated: 2015-05-27

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Niyobuhungiro, Japhet

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