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 two-dimensional inverse modeling example.
The example of the function
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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 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
),
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.
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Clemens Heitzinger 2003-05-08