Data-based Adaptive Refinement of Finite Element Thin Plate Spline

TL;DR

This work addresses efficient interpolation and smoothing of scattered data by extending the thin plate spline with a mixed finite element formulation (TPSFEM) that scales to large data sets. It develops an iterative data-aware adaptive refinement framework using five error indicators, including regression-based RMSE and four PDE-inspired indicators, and updates the smoothing parameter $\alpha$ via generalized cross-validation. Through numerical experiments on a model peaks function and two bathymetric surveys (Crater Lake and Coastal Region), the authors show that adaptive refinement significantly improves efficiency while maintaining or improving RMSE, with the recovery-based and norm-based indicators often delivering the best performance. The study highlights the importance of incorporating data distribution and the smoothing parameter into refinement decisions and demonstrates practical gains for complex, real-world surfaces under both Dirichlet and Neumann boundaries.

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.

Figures (22)

• Figure 1: Data from a bathymetric survey in a triangular FEM grid. Data points are represented as blue dots.
• Figure 2: $\alpha$ values at each iteration of bounded minimisation with various numbers of data, nodes and noise levels.
• Figure 3: RMSE of $s$ against data sets modelled by $y=e^{-50(x_{1}-0.5)^{2}}e^{-50(x_{2}-0.5)^{2}}$ (a) without noise; and (b) with Gaussian noise. The points represent RMSE in each triangle pair.
• Figure 4: Recovery-based error indicator values for the data set modelled by $y=e^{-50(x_{1}-0.5)^{2}}e^{-50(x_{2}-0.5)^{2}}$ (a) without noise; and (b) with Gaussian noise. The points represent error indicator values in each triangle pair.
• Figure 5: An example local domain. Interior nodes and boundary nodes are represented as filled circles and open circles, respectively. Accuracy is improved by refining edge $N_{1}\text{-}N_{2}$ to introduce a new vertex $N_{5}$ and new edges represented as dashed lines.