Plenary Lecture

Multivariate Interpolation via Spline Basis Set and High Dimensional Model Representation

Professor Metin Demiralp
Informatics Institute
Istanbul Technical University
ITU Bilisim Enstitusu Ayazaga Yerleskesi
Maslak, 34469, Istanbul, Turkey

Abstract: Recent years brought an increasing tendency to use High Dimensional Model Representation (HDMR) and its certain varieties for the approximation of multivariate functions. It is a divide--and--conquer philosophy and the representation is based on an ordering in ascending multivariance such that the first component of the representation is a constant which is the simplest function structure and the following terms are univariate functions depending on different independent variables, bivariate functions depending on different couples of independent variables and so on.
HDMR was Sobol's idea at the beginning and based on unit hypercubic geometry having origin as one of the corners and unit weight function in the integration which was used in the construction of the representation components. HDMR contains a 2^N number of terms (where N is the level of multivariance, that is, the number of the independent variables) and this number becomes too large when the multivariance tends to increase to even three digit integers. Hence, it is desired to truncate this representation at most at the level of bivariance. Determination of HDMR components produces unique results when the vanishing conditions of Sobol are used. Rabitz extended the Sobol's notion to the hyperprismatic geometry located anywhere and the weight functions differing than 1. Demiralp noticed the relation between the vanishing conditions and the orthogonality amongst the HDMR components.
In this lecture a novel approach based on constancy optimisation in HDMR for the multivariate interpolation is presented. Interpolation is considered on a regular rectangula hyperprismatic grid. Recent findings from certain researches including the PhD studies will also be announced.

Brief Biography of the Speaker:
Metin Demiralp was born in Turkey on 4 May 1948. His education from elementary school to university was all in Turkey. He got his BS, MS, 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 is working on methodology for computational sciences. 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. H. A. Rabitz's group at Princeton University (NJ, USA) at summer and winter semester breaks during the period 1985--2003 after his 14 months long postdoctoral visit to same group in 1979--1980.
Metin Demiralp has roughly 70 papers in well known scientific journals and is the full member of Turkish Academy of Sciences ince 1994. He is also a member of European Mathematical Society and the chief--editor of WSEAS Transactions on Mathematics currently. He has also two important awards of Turkish scientific establishments.