TUTORIAL III Weight, Geometry and Transformation Dependence of High Dimensional Model Representation (HDMR)

Abstract

High Dimensional Model Representation (HDMR) is one of the very important tools in the approximation of multivariate functions. It has proposed by Sobol, extended by Rabitz via weight and arbitrary orthogonal geometry introduction. Beside many HDMR applications in Rabitz's group, Demiralp's group brought the Hilbert space tools like orthogonality quality measuring functional and transformations to the topic. HDMR is a finite number term including representation. However this number grows awkwardly as the number of independent variables increases. Hence, the tendency of the scientists is to use at most bivariate level truncations as approximants. If the desired quality of approximation is not achieved then not to attempt to use higher variate truncations but to change the structure of the HDMR is preferred. This can be accomplished by using appropriately chosen transformations. Then, not the function itself but its image under the chosen transformation is expanded to HDMR. The dependence of HDMR on weight, geometry and transformation is quite important. Under appropriate choices of these entities the additive nature may become dominant. This dominancy means high quality of univariate or bivariate truncations. As long as there is no limitation on the choices of weight and gemometry, they can be constructed to increase the additivity. Otherwise particular transformations may be needed to get dominancy in additivity. This talk aims at the presentation of certain experimentation results together with the conceptual discussions.