Approximation with Evolutionary Optimized Tensor Product Bernstein Polynomials

This paper introduces a novel approximation technique that uses Tensor Product Bernstein Polynomials (TPBPs) as base functions. An Evolution Strategy (ES) is proposed for finding suitable control points for TPBPs so that the resulting approximation functions fit given finite samplings of data points very well. Also for more difficult test data sets taken from functions containing discontinuities, the approach is numerically robust. Moreover, due to the ES it is flexible concerning the used error measurement. One major advantage of the method over many other approximation techniques is the possibility to manually adapt the control points of a TPBP in a very intuitive way to achieve specific changes. Another important aspect of the new technique is its good generalization ability: Samples other than those used in the optimization process for determining control points are also mapped to their counterparts with only small errors. The new approach has been compared to the well known Least Square Method (LSM) for doing polynomial regression using various sample sets. Especially for structurally more complex polynomials (as TPBPs of higher degrees), the generalization capabilities of the LSM are rather poor. Although the ES does not usually find the globally optimal approximation with the smallest error, the found solutions are preferable because they generalize very well.