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Global search strategies for solving multilinear least-squares problems
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.ORCID iD: 0000-0003-1836-4200
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-9091-4724
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. (Optimization)
2012 (English)In: Sultan Qaboos University Journal for Science, ISSN 1027-524X, Vol. 17, no 1, 12-21 p.Article in journal (Refereed) Published
Abstract [en]

The multilinear least-squares (MLLS) problem is an extension of the linear leastsquares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by results of numerical experiments performed for some problems related to the design of filter networks.

Place, publisher, year, edition, pages
Sultan Qaboos University , 2012. Vol. 17, no 1, 12-21 p.
Keyword [en]
Global optimization; Global search strategies; Multilinear least-squares; Filter
National Category
Computational Mathematics Medical Image Processing
Identifiers
URN: urn:nbn:se:liu:diva-78918OAI: oai:DiVA.org:liu-78918DiVA: diva2:536888
Available from: 2012-08-28 Created: 2012-06-25 Last updated: 2015-09-03Bibliographically approved
In thesis
1. Large-Scale Optimization Methods with Application to Design of Filter Networks
Open this publication in new window or tab >>Large-Scale Optimization Methods with Application to Design of Filter Networks
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Nowadays, large-scale optimization problems are among those most challenging. Any progress in developing methods for large-scale optimization results in solving important applied problems more effectively. Limited memory methods and trust-region methods represent two ecient approaches used for solving unconstrained optimization problems. A straightforward combination of them deteriorates the efficiency of the former approach, especially in the case of large-scale problems. For this reason, the limited memory methods are usually combined with a line search. We develop new limited memory trust-region algorithms for large-scale unconstrained optimization. They are competitive with the traditional limited memory line-search algorithms.

In this thesis, we consider applied optimization problems originating from the design of lter networks. Filter networks represent an ecient tool in medical image processing. It is based on replacing a set of dense multidimensional lters by a network of smaller sparse lters called sub-filters. This allows for improving image processing time, while maintaining image quality and the robustness of image processing.

Design of lter networks is a nontrivial procedure that involves three steps: 1) choosing the network structure, 2) choosing the sparsity pattern of each sub-filter and 3) optimizing the nonzero coecient values. So far, steps 1 and 2 were mainly based on the individual expertise of network designers and their intuition. Given a sparsity pattern, the choice of the coecients at stage 3 is related to solving a weighted nonlinear least-squares problem. Even in the case of sequentially connected lters, the resulting problem is of a multilinear least-squares (MLLS) type, which is a non-convex large-scale optimization problem. This is a very dicult global optimization problem that may have a large number of local minima, and each of them is singular and non-isolated. It is characterized by a large number of decision variables, especially for 3D and 4D lters.

We develop an effective global optimization approach to solving the MLLS problem that reduces signicantly the computational time. Furthermore, we  develop efficient methods for optimizing sparsity of individual sub-filters  in lter networks of a more general structure. This approach offers practitioners a means of nding a proper trade-o between the image processing quality and time. It allows also for improving the network structure, which makes automated some stages of designing lter networks.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. 52 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1561
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-103646 (URN)10.3384/diss.diva-103646 (DOI)978-91-7519-456-1 (ISBN)
Public defence
2014-02-26, Nobel (BL32), B-huset, Campus Valla, Linköping University, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2014-02-03 Created: 2014-01-21 Last updated: 2015-06-02Bibliographically approved

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Andersson, MatsBurdakov, OlegKnutsson, HansZikrin, Spartak

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