11.4 Summary

In summary, the properties of polynomials of fixed degree arising from least square fits were compared to those of generalized Bernstein polynomials on multidimensional intervals. The properties of the Bernstein polynomials were proven and presented, and it was found that these fulfill the requirements for approximations needed for smoothing Monte Carlo simulation results and translating them from ion implantation ortho-grids to unstructured grids.

The polynomials and the algorithm devised provide the following benefits. First, they converge uniformly when the number of base points goes to infinity. Second, an asymptotic formula gives information about their rate of convergence. Third, total variation is decreased and the approximations do not oscillate more often about any straight line than the original function. This assures suitable smoothing. Fourth, the algorithm works very fast and is easy to implement using the specialized formula given, since the calculation of the actual approximating polynomials is avoided.

Finally, the new algorithm and its RSM counterpart were compared in a real world Monte Carlo ion implantation example, and the new algorithm was found to yield superior results which can immediately be used for further simulations.