Approximation of a Thin Plate Spline Smoother Using Continuous Piecewise Polynomial Functions

A new smoothing method is proposed which can be viewed as a finite element thin plate spline. This approach combines the favourable properties of finite element surface fitting with the ones of thin plate splines. The method is based on first order techniques similar to mixed finite element techniques for the biharmonic equation. The existence of a solution to our smoothing problem is demonstrated and the approximation theory for uniformly spread data is presented, both in the case of exact and noisy data. This convergence analysis seems to be the first for a discrete smoothing spline with data perturbed by white noise. Numerical results verifying our theoretical results and demonstrating our method on a large real life data sets are presented.