The RSM approach can be summarized as follows. Let be a
continuous function on a multidimensional interval
. On the interval
a
rectangular grid with points
is chosen. The RSM approximation
of
is the multivariate polynomial of degree
or
of a higher, fixed degree, which is determined by a least squares fit
such that
Although it can be argued that the RSM approximation is based on a truncated Taylor series expansion
Furthermore the RSM suffers from the fact that a polynomial of fixed
degree cannot preserve the global properties of the original function:
the set of all polynomials of a certain fixed maximal degree is not
dense in ,
compact. There do not exist any
rigorous statements about approximating or smoothing properties.
Moreover, using more and more points for the least squares fit is not
a remedy and does generally not improve the RSM approximation, while
the computational effort is increased. Hence these evaluations are
wasted. A simple, but important example for this are the functions
which are ubiquitous in TCAD
applications. Other examples are functions containing transitions
from exponential to linear behavior.
Although the RSM approach can be improved by transforming the variables before fitting the polynomials, it has to be known a priori which transformations are useful and should be considered. If this knowledge is available, it can of course be applied to other approximation approaches as well.
Clemens Heitzinger 2003-05-08