Optimal knots allocation in the cubic and bicubic spline interpolation problems

Interpolation, together with approximation, are two major and ubiquitous problems in Mathematics, but also in almost every scientific field. Another interesting question is the optimal knots placement when interpolating or approximating 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 placement. A new technique is presented to optimize both the number of knots and its optimal placement for cubic or bicubic interpolating splines. The performance of the proposed methodology has been evaluated using functions of one and two variables, respectively.