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Abstract: In this paper we present an interpolation method from a surface or a data set by the optimization of a quadratic functional in a bicubic splines functional space. The existence and the uniqueness of the solution of this problem are shown and as well a convergence result of the method is established. The mentioned functional involves some real non negative parameters; the optimal surface is obtained by a suitable optimization of these parameters. Finally, we analyze some numerical and graphic examples in order to prove the efficiency of the presented method. [Copyright &y& Elsevier]

Abstract: We deal with the problem of constructing and/or approximating some curves and surfaces from a noisy data set. The main objective here is to adapt the theory of the discrete smoothing variational splines [A. Kouibia, M. Pasadas, Approximation by discrete variational splines, J. Comput. Appl. Math. 116 (2000) 145–156] to the noisy data case. The problem is formulated and a result of almost sure convergence is established. [Copyright &y& Elsevier]

Abstract: We present an approximation method of surfaces preserving the monotonicity constraints. By minimizing a semi-norm and monotonicity criteria we define the notion of the pseudo-monotone interpolating variational spline in a finite element space. We compute this spline by using a suitable algorithm. Some convergence results are carefully studied. Finally, to show the effectiveness of this method we give some numerical and graphical examples. [Copyright &y& Elsevier]