Data-based Adaptive Refinement of Finite Element Thin Plate Spline

TL;DR

This paper introduces an adaptive refinement method for the finite element thin plate spline (TPSFEM), improving computational efficiency for large scattered data sets by customizing the grid refinement process.

Contribution

It develops an iterative adaptive refinement process and five error indicators specifically tailored for TPSFEM, incorporating data information not present in PDE-based methods.

Findings

Adaptive refinement improves TPSFEM efficiency.

Error indicators effectively guide refinement process.

Numerical experiments validate the approach.

Abstract

The thin plate spline, as introduced by Duchon, interpolates a smooth surface through scattered data. It is computationally expensive when there are many data points. The finite element thin plate spline (TPSFEM) possesses similar smoothing properties and is efficient for large data sets. Its efficiency is further improved by adaptive refinement that adapts the precision of the finite element grid. Adaptive refinement processes and error indicators developed for partial differential equations may not apply to the TPSFEM as it incorporates information about the scattered data. This additional information results in features not evident in partial differential equations. An iterative adaptive refinement process and five error indicators were adapted for the TPSFEM. We give comprehensive depictions of the process in this article and evaluate the error indicators through a numerical experiment with a model problem and two bathymetric surveys in square and L-shaped domains.