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Recent Developments in the Multivariate Function Decompositions: Enhanced Multivariance Products Representation (EMPR) Under Selective Condition Impositions
Metin Demiralp

Last modified: 2023-06-29


Enhanced Multivariance Products Representation (EMPR) is a High Dimensional Model Representation (HDMR) generalization whose capabilities have been pretty much increased in especially last decade. HDMR finds its roots in Sobol’s studies first and later in Rabitz group studies. The author of this presentation, and his group members, have contributed to the issue in many varieties. On the other hand, the idea of enhanced multivariance product representation has first appeared in Demiralp’s group studies. Today, EMPR has different conceptual extensions useful in the practical applications such that the many true practical applications show the capabilities of EMPR.

In Enhanced Multivariance Products Representation (EMPR), a given multivariate function depending on N separate independent variables is decomposed to a linear combination of less variate functions such that the minimum multivariance component is a constant function while N univariate components, N(N − 1)/2 bivariate functions, N(N − 1)(N − 2)/6 trivariate components and so on follow this very first component. The highest multivariance component depends on N independent variables as the target function does. The total number of the EMPR components is 2N . In HDMR the same philosophy is also taken into consideration; however, therein, the linear combination is provided by using constant coefficients while in EMPR each additive decomposition term’s multivariance is increased to N by using multivariate function factors depending on all independent variables which do not appear in the relevant EMPR component. Hence, in each additive term the multivariance is enhanced to N by using appropriate multivariate function factors.

The determination of EMPR components needs the imposition of certain
conditions. By following the HDMR philosophy EMPR component determi-
nations are based on so-called Sobol’s conditions which state that each non-
constant component’s integral with respect to anyone of its arguments over
the relevant interval of orthogonal geometry vanishes. These conditions need
to be modified in EMPR to involve the relevant support functions. Certain
details of these determinations will be given throughout the presentation.

Until now, in all EMPR applications, all multivariance components are assumed to be existing in the expansion and sufficient number of accompanying
support functions have been considered. Then, all possible Sobol’s-like condi-
tions have been imposed for the determination of entire EMPR-components.
However, starting from some presentations in Minisymposium MS1, we have
begun to use only some of EMPR-components together with the relevant sup-
port functions. Of course, this needs it to reduce the number of the imposi-
tions accordingly even though they are still chosen from the Sobol’s-like con-
ditions. This selective EMPR construction enables us to develop certain ex-
tensions to tridiagonalized EMPR variaties like Tridiagonal Matrix Enhanced
Multivariance Products Representation (TMEMPR), Tridiagonal Kernel En-
hanced Multivariance Products Representation (TKEMPR), Tridiagonal Fol-
mat Enhanced Multivariance Products Representation (TFEMPR), and so
on. Tridiagonalization is related to planar or two-way arrays. This presenta-
tion will focus on the exemplifications on three-way array constructions and
multi-way array based application possibilities will also be briefly discussed.