7.6 Examples

In Figure 7.3 and Figure 7.4 the base polynomials of univariate Bernstein polynomials ($n=5$) and their generalization to two dimensions ($n_1=n_2=3$) 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 two-dimensional inverse modeling example.

The example of the function $f: [0,1]\times[0,1] \to \mathbb{R}$,

$$f(x,y):=(1/2) \mathrm{e}^{-10((x-1/2)^2+(y-1/2)^2)} + \mathrm{e}^{-50((x-1)^2+(y-1)^2)}$$

shows that approximation using generalized Bernstein polynomials resembles the global properties of a given function better than using multivariate polynomials of degree 2 or less, even when using a small number of lattice points. The two approaches are compared in Figure 7.1. In the left hand figure, $f$ is plotted at the $11\cdot11$ lattice points that were used for calculating the two-dimensional Bernstein polynomial $B_{f,10}(x,y)$ and the least squares fit $\mathit{rsm}(x,y)$ of degree 2. $f$ and $B_{f,10}$ have two local maxima on $[0,1]\times [0,1]$, whereas $\mathit{rsm}$ has only one. Their respective values are (up to six digits): $f(0.5,0.5)=0.5$, $f(0.999661,0.999661)=1.00338$; $B_{f,10}(0.500674,0.500674)=0.331634$, $B_{f,10}(1,1)=1.00337$; $\mathit{rsm}(0.696706,0.696706)=0.283076$.

Figure 7.1: Comparison of 11x11 lattice points of f (left), the Bernstein approximation (middle, the variables have been scaled to the interval [0,1]), and the RSM approximation (right) as found by MATHEMATICA's Fit function.

Figure 7.2: Comparison of the computed lattice points (left), the Bernstein approximation (middle), and the RSM approximation (right) as found by MATHEMATICA's Fit 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 Shockley-Read-Hall 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 $11\cdot11$ lattice points with these two most sensitive parameters (cf. Figure 7.2, left). Using these points, two approximations were calculated: the two-dimensional Bernstein polynomial (where the variables were scaled to the interval $[0,1]$), 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.

Figure 7.3: This graph shows the polynomials that occur as summands in a one-dimensional Bernstein polynomial. Each of the summands has exactly one global maximum and the maximums of all summands are equidistant.

Figure 7.4: This graph gives an impression of the polynomials that occur as summands in a two-dimensional Bernstein polynomial. Each of the summands has exactly one global maximum and the maximums of all summands form an equidistant grid. For better visibility only function values in the range [0,0.6] are shown. The function values in the corners are 1.