7.2 The Response Surface Methodology and Its Problems

The RSM approach can be summarized as follows. Let $f$ be a continuous function on a multidimensional interval $I:=\{ (x_1,\ldots,x_m) \mid a_k \le x_k \le b_k \}$. On the interval $I$ a rectangular grid with points $x_k$ is chosen. The RSM approximation $\mathit{rsm}$ of $f$ is the multivariate polynomial of degree $2$ or of a higher, fixed degree, which is determined by a least squares fit such that

$$\sum_k \left( f(x_k)-\mathit{rsm}(x_k) \right)^2$$

is minimal. Computing the coefficients of $\mathit{rsm}$ is a well-known procedure [21,22]. RSM has been used extensively in TCAD applications, e.g. in [24,38,20,23,148,131,146,117,6,60,74,103].

Although it can be argued that the RSM approximation is based on a truncated Taylor series expansion

$$f({\mathbf{r}} + {\mathbf{a}}) = \sum_{k=0}^\infty \Bigl( {1\over... ..._{\mathbf{r'}})^k f({\mathbf{r'}}) \Bigr) \Bigm\vert _{\mathbf{r'}=\mathbf{r}}$$

for a multivariate function $f$, it is important to note that this is a local approximation and quite different from a least squares fit for several points. In the Taylor series expansion convergence occurs when the number of terms and thus the degree of the polynomial increases, whereas in the RSM approach the degree of the approximating polynomial is fixed to an arbitrary low value. Increasing the degree is possible of course, but the choice is still arbitrary and the number of coefficients and thus the number of points required for the least squares fit increases abundantly.

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 $C(X)$, $X \subset {\mathbb{R}}^p$ 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 $e_\lambda: x\mapsto \mathrm{e}^{\lambda x}$ 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.