Locally adaptive approaches to finite element bivariate smoothing splines.
by
Michael F. Hutchinson
Centre for Resource and Environmental Studies, Australian National University

Finite difference discretisation, based on a regular two-dimensional grid, has provided the means for efficiently calculating close approximations to bivariate thin plate smoothing splines. When a simple nested grid SOR iteration is used, the number of operations can be made optimal, in the sense that they are proportional to the number of grid points, regardless of grid spacing (Hutchinson 1989, Hancock and Hutchinson 1999). Problems involving millions of data points and grid points can be routinely solved on modest workstations. A particular additional advantage of the finite difference approach is that a variety of locally adaptive constraints can be straightforwardly applied to satisfy known process based conditions. Thus a locally adaptive drainage enforcement algorithm can be added to produce digital elevation models with connected drainage structure. A locally adaptive modification can also be made to the weighting of the residual sum of squares to respect the natural discretisation error that arises with the finite difference discretisation. This has led to a method for optimising the spatial resolution of digital elevation models to the information content of source topographic data A locally adaptive modification to the usual minimum curvature roughness penalty has also been suggested to better respect drainage structure (Hutchinson 1996). This paper extends these locally adaptive strategies to a discretisation based on bilinear finite elements. Bilinear elements can overcome biases in surface shape generated by finite difference discretisations and can better allow for the non-linear constraints due to positional errors in data. Yet they remain computationally efficient in terms of both time and storage. The methods are also readily vectorised.

References:

Hancock, P.A. and Hutchinson, M.F. 1999. Comparison of nested grid and V-cycle multigrid for calculating finite difference smoothing splines. Abstract for this conference.

Hutchinson, M.F. 1989. A new method for gridding elevation and streamline data with automatic removal of spurious pits. Journal of Hydrology 106: 211-232.

Hutchinson, M.F. 1996. A locally adaptive approach to the interpolation of digital elevation models. http://www.ncgia.ucsb.edu/conf/SANTA_FE_CD-ROM/santa_fe.html

Date received: August 17, 1999