Error Estimation and Adaptive Refinement of Finite Element Thin Plate Spline

Abstract

The thin plate spline smoother is a data fitting and smoothing technique that captures important patterns of potentially noisy data. However, it is computationally expensive for large data sets. The finite element thin plate spline smoother (TPSFEM) combines the thin plate spline smoother and finite element surface fitting to efficiently interpolate large data sets. When the TPSFEM uses uniform finite element grids, it may require a fine grid to achieve the desired accuracy. Adaptive refinement uses error indicators to identify sensitive regions and adapts the precision of the solution dynamically, which reduces the computational cost to achieve the required accuracy. Traditional error indicators were developed for the finite element method to approximate partial differential equations and may not be applicable for the TPSFEM. We examined techniques that may indicate errors for the TPSFEM and adapted four traditional error indicators that use different information to produce efficient adaptive grids. The iterative adaptive refinement process has also been adjusted to handle additional complexities caused by the TPSFEM. The four error indicators presented in this thesis are the auxiliary problem error indicator, recovery-based error indicator, norm-based error indicator and residual-based error indicator. The auxiliary problem error indicator approximates the error by solving auxiliary problems to evaluate approximation quality. The recovery-based error indicator calculates the error by post-processing discontinuous gradients of the TPSFEM. The norm-based error indicator uses an error bound on the interpolation error to indicate large errors. The residual-based error indicator computes interior element residuals and jumps of gradients across elements to estimate the energy norm of the error. Numerical experiments were conducted to evaluate the error indicators' performance on producing efficient adaptive grids, which are measured by the error versus the number of nodes in the grid. A set of one and two-dimensional model problems with various features are chosen to examine the effectiveness of the error indicators. As opposed to the finite element method, error indicators of the TPSFEM may also be affected by noise, data distribution patterns, data sizes and boundary conditions, which are assessed in the experiments. It is found that adaptive grids are significantly more efficient than uniform grids for two-dimensional model problems with difficulties like peaks and singularities. While the TPSFEM may not recover the original solution in the presence of noise or scarce data, error indicators still produce more efficient grids. We also learned that the difference is less obvious when the data has mostly smooth or oscillatory surfaces. Some error indicators that use data may be affected by data distribution patterns and boundary conditions, but the others are robust and produce stable results. Our error indicators also successfully identify sensitive regions for one-dimensional data sets. Lastly, when errors of the TPSFEM cannot be further reduced due to factors like noise, new stopping criteria terminate the iterative process aptly.