On the Transformational HDMR Method

摘要

High Dimensional Model Representation Method (HDMR) is a divide and con quer technique originally suggested by I.M. Sobol later developed by H. Rabitz and M. Demiralp. Various requirements urged these authors as well as others to develop different versions of HDMR. Some of these like Cut HDMR, Mul ticut HDMR, Random Sampling HDMR etc. were suggested for tackling with large numbers of data processing. Such HDMR versions however, will be out of the scope of this talk. An alternative route was dealing with functions of multiplicative nature rather than of additive nature. To this end Demiralp's group developed HDMR versions like Factorized HDMR, Hybrid HDMR and Logarithmic HDMR. The last of these, is based on the idea of transforming a function of multiplicative nature to one of additive nature simply by taking its logarithm. Applying the standard (i.e. Plain) HDMR to this newly obtained function, HDMR components are obtained. This is followed by exponentia tion of HDMR components. It was the fundamental idea behind Logarithmic HDMR which led to the possibility of using alternative transformations and inverse transformations. As a result the more general idea of Transformational HDMR was suggested again by the Demiralp group. Transformational HDMR has been used to develop new approximations of various functions. To this end Rational as well as Conical and Mobius Transformations were used. The major issue to be discussed will be these recently developed approximation techniques which are believed to give good alternatives, to say the least, to approximation methods like Pade or Hermite-Pade.