Abstract: High Dimensional Model Representation (HDMR) is a function decomposition method basically developed in the last two decades. Although it was first proposed by I. M. Sobol, there has been quite important contributions from H. A. Rabitz and his group. Further developments and various new varieties of HDMR were made in the Group for Science and Methods of Computing in ?Istanbul (Demiralp, Baykara, Tunga et. al.). HDMR is basically for continuous multivariate functions even though very recently linear and multilinear arrays to be decomposed. The roots of those efforts lie in the earlier studies, but not all that comprehensively.
Most prominent versions of HDMR are the Plain, Cut, Multicut, Factorized, Logarithmic, Hybrid ones. There are various successful applications of these on the practical problems coming from Engineering, Science and Physical Chemistry. The general intention is to truncate the HDMR up to and including at most the bivariate terms, although univariate truncation is preferable. The quality of this truncation increases when it dominates in the norm square of function under consideration. This urges the methodologists to develop methods for maximizing this quality via certain appropriately chosen flexibilities. Certain fruitful methods to this end have been developed. Amongst these, Transformational HDMR is the first important step. The most prominent aspect of this approach (THDMR) is the choice of appropriate transformation whose inverse is rather easily obtainable. The Logarithmic HDMR can be thought as the most elementary version of THDMR where additivity and multiplicativity can be interchanged. One of the other approaches within this framework is the affine transformation where the constancy optimization reduces certain rational approximants which can be more efficiently applicable in practice compared to Pad/e approximants. Quite recently conic transformations are brought under focus enabling comparisons with the Hermite–Pad/e approximants by utilizing approximants involving square roots. Another important flexibility to be optimized is the weight function. Univariate and multivariate cases have been investigated and quite non-linear algebraic equations have been obtained. These equations could have been approximated efficiently by using Fluctuation–Free matrix representations, and then, a perturbation expansionhas also been developed to get corrections. What is mentioned above is not the full story of the issue. There have been various interesting developments also, amongst which Enhanced Multivariance Product Representation (EMPR) and its varieties, Generalized HDMR, Discrete and Continuous HDMR and EMPR together with applications to multilinear array decompositions. The presentation will outline all this within the time limitation.

Brief Biography of the Speaker:
N. A. BAYKARA was born in Istanbul,Turkey on 29th July 1948. He received a B.Sc. degree in Chemistry from Bosphorous University in 1972. He obtained his PhD from Salford University, Greater Manchester, Lancashire,U.K. in 1977 with a thesis entitled “Studies in Self Consistent Field Molecular Orbital Theory”, Between the years 1977–1981 and 1985–1990 he worked as a research scientist in the Applied Maths Department of The Scientific Research Council of Turkey. During the years 1981-1985 he did postdoctoral research in the Chemistry Department ofMontreal University, Quebec, Canada. Since 1990 he is employed as a Staff member of Marmara University. He is now an Associate Professor of Applied Mathematics mainly teaching Numerical Analysis courses and is involved in HDMR research and is a member of Group for Science and Methods of Computing in Informatics Institute of Istanbul Technical University. Other research interests for him are “Density Functional Theory” and “Fluctuationlessness Theorem and its Applications” which he is actually involved in. Most recent of his concerns is focused at efficient remainder calculations of Taylor expansion via Fluctuation–Free ?Integration, and Fluctuation–Free Expectation Value Dynamics.