Optimal knots allocation in the cubic and bicubic spline interpolation problems

Abstract

Interpolation, together with approximation, are two major and ubiquitous prob- lems in Mathematics, but also in almost every scienti c eld. Another inter- esting question is the optimal knots placement when interpolating or approxi- mating certain functions using splines. In this work, a powerful methodology is presented for optimal knots placement when interpolating a curve, or a surface, using cubic or bicubic splines, respectively. For this, a Multi-Objective-Genetic Algorithm (MOGA) has been developed, in a way that ensures avoiding the large number of local minima existing in the problem of random knots place- ment. A new technique is presented to optimize both the number of knots and its optimal placement for cubic or bicubic interpolating splines. The perfor- mance of the proposed methodology has been evaluated using functions of one and two variables, respectively.