# 7.7 Solving Design for Manufacturability Problems

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 multi-dimensional interval and let be the probability density of the variable . Often we have

i.e., each conforms to a normal distribution. (The type of distribution does not matter for our discussion, but in practice the variables usually conform to a normal distribution.)

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 general case the probability densities depend on parameters and we have to choose these parameters such that

becomes maximal.

In the case where all variables conform to a normal distribution we have the probability densities

and have to maximize

Hence design for manufacturability problems can be rigorously stated as the problem of maximizing . This is illustrated in Figure 7.5.

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 2003-05-08