Smoothing large data sets using discrete thin plate splines

Traditional thin plate splines use radial basis functions and require the solution of a dense linear system of equations whose size is proportional to the number of data points. Instead of radial basis functions we present a method based on the use of polynomials with local support defined on finite element grids. This method is more efficient when dealing with large data sets as the resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid. Theory is developed for general d-dimensional data sets and model problems are presented in 3D to study the convergence behaviour.