Decomposing the Arrays: Most Recent Developments

Prof. Dr. Metin Demiralp
Istanbul Technical University
Informatics Institute
Istanbul, TURKEY
E-mail: metin.demiralp@gmail.com
Web site: http://uzbilim.be.itu.edu.tr/demiralp/homepage
 

Abstract: Even though the word array recalls mostly Cartesian vectors and matrices, last few decays brought many important developments about the definition and utilization of the multiway arrays whose elements are indexed by more-than-two positive integers. Despite the fact that some scientists prefer to use the term “Tensor” we avoid this uti- lization since the tensors do not cover all indexed entities. They are defined mostly for mechanical issues and are expected to satisfy certain specific transformations. In multiway arrays each index can be considered as an independent variable defining the positioning on one of the virtual directions which are assumed to be mutually perpendicular. The indices can vary independently from the other index changes. Hence they can be considered as the position components of the nodes of a rectangular hyperprism. In this sense, vectors are one way arrays while matrices can be considered as two way arrays.
Three way arrays can also be encountered in practise. For example, a color picture can be considered as a three way array whose two ways are for horizontal and vertical positioning of the pixels while the remaining way is used to define the color. By folding the color direction onto the positioning plane it is also possible to use rectangular matrices, that is, two way arrays for picture representations. On the other hand, animations, in unfolded format, can be considered four way arrays.
The number of the ways in array can be increased or decreased by using folding and unfolding operations. However for such a change the possible operation is not unique. Even infinite number of way changing operation can be defined. The important point is that the reversion of the operation should be defined uniquely for a chosen operation. Perhaps the most attractive issue about the arrays is their decomposition to rather simple structures. The most widely used decompositions are spectral decompositions (SD) for square matrices and most generally singular value decompotion (SVD) for rectangular matrices and beyond that for the multiway arrays even though the latter case presents rather complicated structures. These can be considered somehow diagonalization methods.
Recently developed High Dimensional Model Representation (HDMR) and its varieties together with its generalization called Enhanced Multivariate Products Representation (EMPR) and its varieties. We have developed some tridiagonalization techniques for arrays via recursive use of EMPR and related issues. The talk will be designed to report most recent developments in somehow abstract format.

Short biography: Metin Demiralp was born in Türkiye (Turkey) on 4 May 1948. His education from elementary school to university was entirely in Turkey. He got his BS, MS degrees and PhD from the same institution, ˙Istanbul Technical University. He was originally chemical engineer, however, through theoretical chemistry, applied mathematics, and computational science years he was mostly working on methodology for computational sciences and he is continuing to do so. He has a group (Group for Science and Methods of Computing) in Informatics Institute of ˙Istanbul Technical University (he is the founder of this institute). He collaborated with the Prof. Herschel A. Rabitz’s group at Princeton University (NJ, USA) at summer and winter semester breaks during the period 1985-2003 after his 14 month long postdoctoral visit to the same group in 1979-1980. He was also (and still is) in collaboration with a neuroscience group at the Psychology Department in the University of Michigan at Ann Arbour in last three years (with certain publications in journals and proceedings).
Metin Demiralp has more than 100 papers in well known and prestigious scientific journals, and, more than 230 contributions together with various keynote, plenary, and, tutorial talks to the proceedings of various international conferences. He gave many invited talks in various prestigious scientific meetings and academic institutions. He has a good scientific reputation in his country and he was one of the principal members of Turkish Academy of Sciences since 1994. He has resigned on June 2012 because of the governmental decree changing the structure of the academy and putting politicial influence possibility by bringing a member assignation system. Metin Demiralp is also a member of European Mathematical Society. He has also two important awards of turkish scientific establishments.
The important recent foci in research areas of Metin Demiralp can be roughly listed as follows: Probabilistic Evolution Method in Explicit ODE Solutions and in Quantum and Liouville Mechanics, Fluctuation Expansions in Matrix Representations, High Dimensional Model Representations, Space Extension Methods, Data Processing via Multivariate Analytical Tools, Multivariate Numerical Integration via New Efficient Approaches, Matrix Decompositions, Multiway Array Decompositions, Enhanced Multivariate Product Representations, Quantum Optimal Control.