Abstract: The method of finite elements is quite frequently used in the solution of boundary value problems modelled as either ordinary differential equations (ODEs) or partial differential equations (PDEs). It is based on basically weak derivative and Sobolev space concepts. An appropriate bilinear form is constructed from the given equations and accompanying boundary conditions by using these concepts. Then the unknown function is approximately expressed in terms of appropriately defined spline functions over certain convenient subregions of the problem geometry. The linear combination constructed towards this goal contains certain undetermined constants which appear in an algebraic equation, that is linear for the linear ODEs or PDEs, together with linear boundary impositions like Dirichlet or Neumann conditions. The important issue in this approach is the construction of the spline basis set, which is realized in such a way that the resulting algebraic equations possess rather simple structure to be solved for the unknowns. Finite elements can be used for almost anything either derived from ODEs or PDEs, or directly from a function in certain ways although the most desired cases are the differential equations.
Recent developments, especially in last decade, made it possible to decompose a multivariate function or its image under certain appropriate operators to some components which are ordered in ascending multivariance. This approach which was originally proposed by Sobol has been extended to more general representations after the studies by Rabitz group in Princeton and Demiralp group in ?Istanbul, even though the number of the scientists concerning with the issues have been increased recently. The method was named High Dimensional Model Representation (HDMR). There are now many different varieties of HDMR in accordance with the certain particularities of the target function, especially after many works in the Group for Science and Methods of Computing (GfSaMoC, supervised and conducted by Demiralp). Beyond those varieties new quite important approaches like Enhanced Multivariance Product Representation (EMPR) which uses support functions to provide more flexibility to quality control in the truncation approximations have also been developed.
Despite HDMR and EMPR are considered for the continuous structures like multivariate functions of more than one independent variables, recent works of GfSaMoC have shown that these methods can be directly used as orthonormal decomposition methods in Multilinear Algebra even though the preliminary steps to this end were taken by Sobol, Rabitz and some other authors.
Some studies have been realized in GfSaMoC to understand what happens if the HDMR or EMPR geometry is taken to zero limit in the volume. What we have seen was that the constancy measurer of HDMR becomes 1 at the zero volume limit. In other words, the constant component of HDMR was becoming overwhelmingly dominant in that limit, or more precisely, HDMR was becoming composed of just a single constant component. This limiting behaviour was bringing the opportunity of approximating the function under HDMR by just constant component or at most univariate terms when the geometric volume of HDMR diminishes. This urged us to divide the HDMR geometry to certain subgeometries such that the function under consideration can be expressed by at most univariate terms in HDMR for each subregion. The result was a piecewise function whose discontinuities can be smoothened by taking some higher HDMR components or by using an optimisation technique to choose best subregioning through suppressing the function value jumps at the borders of each subregion.
What we have mentioned above can be accordingly modified for the EMPR approach also. Some related theorems about the zero volume properties of HDMR and EMPR together with certain illustrative implementations will be presented during the speech.

Brief Biography of the Speaker:
Metin Demiralp was born in Turkey on 4 May 1948. His education from elementary school to university was entirely 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 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 months long postdoctoral visit to the same group in 1979–1980. Metin Demiralp has more than 90 papers in well known and prestigious scientific journals, and, more than 170 contributions 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 is one of the principal members of Turkish Academy of Sciences since 1994. He is also a member of European Mathematical Society and the chief–editor of WSEAS Transactions on Computers currently. 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: Fluctuation Free 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.