Surface approximation by spline smoothing and generalized cross-validation

A technique is developed to approximate multi-dimensional surfaces based on smoothing splines. The tensor product is used to extend a one-dimensional spline basis to higher dimensions. The method of generalized cross-validation is applied to choose the smoothing parameter which is computed with the aid of the generalized singular value decomposition of the design and penalty matrices. Numerical examples are also presented to illustrate the technique.