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Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable.

Curve and surface fitting are important and attractive problems in many applied domains, from CAD techniques to geological prospections. Different methodologies have been developed to find a curve or a surface that best describes some 2D or 3D data, or just to approximate some function of one or several variables. In this paper, a new methodology is presented for optimal knots' placement when approximating functions of one or two variables. When approximating, or fitting, a surface to a given data set inside a rectangle using B-splines, the main idea is to use an appropriate multi-objective genetic algorithm to optimize both the number of random knots and their optimal placement both in the x and y intervals, defining the corresponding rectangle. In any case, we will use cubic B-splines in one variable and a tensor product procedure to construct the corresponding bicubic B-spline basis functions in two variables. The proposed methodology has been tested both for functions of one or two independent variables, in order to evaluate the performance and possible issues of the procedure.