
In this section we discuss how to solve design for manufacturability problems taking advantage of interpolations or good global approximations like those provided by generalized Bernstein polynomials. The problem can be formulated as follows. Let be a real valued continuous function on a multidimensional interval and let be the probability density of the variable . Often we have
Without loss of generality we will consider the case where is to be maximized. Let be a real number. We call a point admissible if and and denote the set of admissible points by
In the case where all variables conform to a normal distribution we have the probability densities
In TCAD problems the evaluation of is computationally very expensive, since it entails simulations of device behavior. Hence the evaluation of is computationally expensive as well and thus the integration and evaluating is prohibitively expensive. If the function is substituted by its approximation on , we substitute the set of admissible points by . Then can be evaluated much faster and hence evaluating becomes feasible and the problem of maximizing can now be attacked using optimization algorithms. Because converges uniformly to (Theorem 7.10), converges to when .
These considerations again underline the importance of the theorems proven in Section 7.4 and conclude this chapter. In the next chapter, another application of this generalization of the Bernstein polynomials is discussed.
Clemens Heitzinger 20030508