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PLENARY LECTURE
Residual Norm Minimizing Exponential Matrix Factorization
for Solving Ode Sets
Professor Metin Demiralp
Informatics Institute
Istanbul Technical University
ITU Bilisim Enstitusu Ayazaga Yerleskesi
Maslak, 34469, Istanbul, Turkey |
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Abstract: The
fact that the set of first order linear ordinary differential
equations with constant coefficient matrix, has exponential
matrix factor multiplying either a vector or a matrix
depending of the unknown's algebraic nature, vector or matrix,
inspires us to construct an approximation scheme using
exponential matrix factors for non--constant matrix
coefficient cases. There have been certain efforts to this end
especially in the last decade. The weak points of the
constructed algorithms have been strengthened by introducing
new ideas later.
The recent status of this type of works, within the knowledge
of the presenter, is about the norm minimization of the
residual coefficient matrix formed after the consecutive
extraction of certain appropriately parametrized exponential
factors from the solution of the related ODE. In this
approach, first, an exponetial matrix whose argument is a
constant matrix multiplied by a function of the independent
variable of the ODE set under consideration is constructed.
The unknown scalar function is built as being vanished at the
independent variable's value where the initial conditions are
given. Its Taylor series expansion coefficients around this
point are used as the optimization agents to minimize the norm
of the resulting coefficient matrix after this factor's
extraction from the unknown solution. In the second phase
everything is repeated for a new exponential matrix factor
construction. However, this time, the scalar function
multipliying the constant matrix of the argument must be
constructed to provide zero values for the function's itself
and its first derivative at the expansion point of the
relating Taylors series. Then, this iterative procedure is
continued up to some desired number of exponential factors to
get sufficient accuracy.
In this lecture, the scheme outlined above will presented up
to some level of details. Some pitfalls about the structure of
the algorithm and certain dubious points will also be
emphasized. 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. |
