7.1 Introduction

The motivation for solving inverse modeling problems in TCAD and thus optimization problems, where the objective function is computationally expensive, was given in Section 6.2. Automated TCAD optimization is difficult since the evaluation of the objective function is usually very computationally expensive. There are two main approaches: the first is to optimize the given objective function, and the second is to optimize an approximation of the objective function. Both approaches are implemented in the SIESTA (simulation environment for semiconductor technology analysis) framework (cf. Chapter 9). The second approach relies on how good an approximation was chosen, and that it can be evaluated much faster than the original objective function so that conventional optimization algorithms requiring many more evaluations can be applied.

Therefore approximations that preserve the global properties of the function under investigation are needed. The RSM (response surface methodology) [21] approach of using (multivariate) polynomials of degree two can at most preserve the local properties of a given function and is therefore not suited for global optimization tasks. In the following we discuss generalized Bernstein polynomials that provide faithful approximations by converging uniformly to the given function. Apart from being useful for optimization tasks, they can also be used for solving design for manufacturability problems (cf. Section 7.7).

In the mathematical literature primarily the generalization of Bernstein polynomials to multidimensional simplices using barycentric coordinates is considered (e.g. [108,5,35]). Here results for the generalization on multidimensional intervals are proved, since multidimensional intervals, and not simplices, are of prime interest for applications.

Concerning the numerical aspect, an implementation for univariate Bernstein polynomials was presented in [144]. For optimization purposes the multivariate case was implemented using MATHEMATICA.