Optimising the degree of data smoothing for locally adaptive finite element bivariate smoothing splines

Abstract

Finite difference and finite element schemes for bivariate thin plate smoothing splines are described. Nested grid SOR iterative methods are known to be able to solve these systems efficiently for large data sets. An iterative Newton procedure for optimising the smoothing parameter to achieve a prescribed residual sum of squares from the data is obtained. It can be added to the SOR iteration with little additional computational cost and is demonstrated on test data to work for a wide range of smoothing parameters. An apparently more accurate version of this procedure, which requires more memory, converges slightly less quickly than the simpler approximation. The simpler method appears to be directly compatible with the SOR iterative method. The Newton method is shown to also work for locally adaptive versions of finite difference smoothing splines. The roughness penalty can be made locally adaptive to respect process-based constraints, such as minimum profile curvature, which depends on the local aspect of the fitted surface. This can be applied to the interpolation of digital elevation models. The weighted residual sum of squares can be made locally adaptive to allow for positional error in data, whether arising from actual data error, or from a finite difference discretisation. This has given rise to an objective method for optimising the grid resolution to the information content of the data.