Plenary Lecture

Exponential Function of a Multivariate Argument within the High Dimensional Model Representation (HDMR) Perspective

Professor N. A. Baykara
Mathematics Department
Marmara University
Istanbul, TURKEY

Abstract: Especially last decade brought important developments in the theory of HighDimensional Model Representation (HDMR) which was first proposed by Sobolin 1993. In its first format, HDMR was a function decomposition in ascend ing multivariance. A given target multivariate function was expressed the sumof a single constant term that is followed by N number of univariate functioncomponent each of which depends on a di erent but single independent variableand bivariate terms group composed of bivariate functions each of which de pends on a di erent couple of independent variables and so on. Sobol proposedvanishing conditions for HDMR components such that each component exceptthe constant one should vanish if it is integrated between 0 and 1 inclusive withrespect to anyone of its arguments. These conditions were su cient to un quelydetermine each HDMR component as long as the target function is integrablein the hypercube whose one corner is located at the origin while its each edgeresides in a di erent axis' positive half.
Herschel Rabitz brought the nonunit weight concept under the product typeassumption (it is product of the univariate functions each of which is a weightfunction depending on a di erent independent variable) and extended the ge ometry from unit hypercube to hyperprisms. Demiralp group revealed manyimportant properties of HDMR and developed various e ective HDMR versionsto increase the truncated HDMR approximants. As state of art developments,factorized, hybrid, transformational HDMR varieties have been arisen while theproduct type hypothesis has also been relaxed. As a quite recently developedand more enthusiastic approach, "Enhanced Multivariate Product Represen tation" has shown up in the stage to more delicately control the truncationapproximant quality. The further studies are also in the air at the present time.This talk focuses on the univariate exponential functions whose argumentis a multivariate function. The argument function has been taken as at mostquadratic form structure even though it may be complicated for more rigorousapplication. What we report here is the application of HDMR in in finite dimen sional space to this type of functions and to show a rather detailed investigationresults together with interpretations.

Brief Biography of the Speaker: N. A. BAYKARA was born in Istanbul,Turkey on 29th July 1948. He received aB.Sc. degree in Chemistry from Bosphorous University in 1972. He obtained his PhDfrom Salford University, Greater Manchester, Lancashire,U.K. in 1977 with a thesis entitled “Studies in Self Consistent Field Molecular Orbital Theory”, Between the years1977–1981 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 hedid postdoctoral research in the Chemistry Department of Montreal University, Quebec,Canada. Since 1990 he is employed as a Staff member of Marmara University. He is now aFull Professor of Applied Mathematics mainly teaching Numerical Analysis courses and isinvolved in HDMR research and is a member of Group for Science and Methods of Computing in Informatics Institute of Istanbul Technical University. Other research interestsof his for him are “Density Functional Theory” and “Fluctuationlessness Theorem and itsApplications” which he is actually involved in. Most recent of his concerns is focused atefficient remainder calculations of Taylor expansion via Fluctuation–Free Ë™Integration, andFluctuation–Free Expectation Value Dynamics.