In Figure 7.3 and Figure 7.4 the base polynomials of univariate Bernstein polynomials () and their generalization to two dimensions () give an impression how and why approximating functions using these polynomials works.
In the following we discuss two examples illustrating the properties of Bernstein polynomials, namely an analytical function and a twodimensional inverse modeling example.
The example of the function ,


The second, real world example stems from minimizing the leakage current of a novel SRAM storage cell [54]. First, we extracted seven parameters from the drain currents of the select transistor of the storage cell and tried to fit two transfer characteristics (two bulk voltages, two times 27 points). The seven variables were ew, the work function of the gate material, sr, the source resistance, f, a parameter controlling the doping, and four variables pertaining to the ShockleyReadHall model [16, page 71]. In the second step the extracted values were used when minimizing the leakage current.
In the course of the inverse modeling task it was found that two variables, namely the parameter of the gate material (ew) and the parameter controlling the doping (f), have a major influence on the result. For further investigations, these remaining variables were then fixed at the values of the minimum found, and the objective function was evaluated at lattice points with these two most sensitive parameters (cf. Figure 7.2, left). Using these points, two approximations were calculated: the twodimensional Bernstein polynomial (where the variables were scaled to the interval ), and the least squares approximation from the set of all polynomials of degree two or less (cf. Figure 7.2). Again the RSM approximation is misleading.


Clemens Heitzinger 20030508