Computational complexity investigations for high-dimensional model representation algorithms used in multivariate interpolation problems

Abstract

Abstract:In multivariate interpolation problems, increase in both the number of independent variables of the sought function and the number of nodes appearing in the data set cause computational and mathematical difficulties. It may be a better way to deal with less variate partitioned data sets instead of an N -dimensional data set in a multivariate interpolation problem. New algorithms such as High Dimensional Model Representation (HDMR), Generalized HDMR, Factorized HDMR, Hybrid HDMR are developed or rearranged for these types of problems. Up to now, the efficiency of the methods in mathematical sense were discussed in several papers. In this work, the efficiency of these methods in computational sense will be discussed. This investigation will be done by using several numerical implementations.

Constancy maximization based weight optimization in high dimensional model representation

“…There are twelve curves, each corresponds to a different n value. Figures 7,8,9, 10 are plotted for the functions (1 + x) 2 e x , sin(x), √ 1 + x and ln(1 + x 2 ) respectively. When these figures are examined carefully it is seen that the optimized weight There are two tables in this section.…”

“…Amongst these an interval arithmetic based work which mainly construct lower and upper bounds to the function whose given values have uncertainties via an interpolatory HDMR [7], and, another work of the same researchers [8] about computational complexity investigations for HDMR based applications of multivariate interpolation problems can be mentioned. A factorized form of HDMR is developed and named Factorized HDMR [9,10] by the same group for the problems having dominantly multiplicative analytical structures.…”