Efficient solution techniques for a finite element thin plate spline formulation

We present a new technique for solving the saddle point problem arising from a finite element based thin plate spline formulation. The solver uses the Sherman–Morrison–Woodbury formula to divide the domain into different regions depending on the properties of the data projection matrix. We analyse the conditioning of the resulting system on certain data distributions and use the results to develop effective preconditioners. We show our approach is efficient for a wide range of parameters by testing it on a number of different examples. Numerical results are given in one, two and three dimensions

Error Estimation and Adaptive Refinement of Finite Element Thin Plate Spline

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

Error indicators and adaptive refinement of the discrete thin plate spline smoother

The discrete thin plate spline is a data fitting and smoothing technique for large datasets. Current research only uses uniform grids for this discrete smoother, which may require a fine grid to achieve a certain accuracy. This leads to a large system of equations and high computational costs. Adaptive refinement adapts the precision of the solution to reduce computational costs by refining only in sensitive regions. The error indicator is an essential part of the adaptive refinement as it identifies whether certain regions should be refined. Error indicators are well researched in the finite element method, but they might not work for the discrete smoother as data may be perturbed by noise and not uniformly distributed. Two error indicators are presented: one computes errors by solving an auxiliary problem and the other uses the bounds of the finite element error. Their performances are evaluated and compared with 2D model problems. References H. Chui and A. Rangarajan. A new point matching algorithm for non-rigid registration. Comput. Vis. Image Und., 89 (2–3): 114–141, 2003. doi:10.1016/S1077-3142(03)00009-2. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM T. Math. Software, 15 (4): 326–347, 1989. doi:10.1145/76909.76912. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal., 41(1):208–234, 2003. doi:10.1137/S0036142901383296. G. Sewell. Analysis of a finite element method. Springer-Verlag, 1985. doi:10.1007/978-1-4684-6331-6. R. Sprengel, K. Rohr, and H. S. Stiehl. Thin-plate spline approximation for image registration. In P. IEEE EMBS, volume 3, pages 1190–1191. IEEE, 1996. doi:10.1109/IEMBS.1996.652767. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput., 63(2):374–409, 2015. doi:10.1007/s10915-014-9898-x. G. Wahba. Spline models for observational data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1990. doi:10.1137/1.9781611970128

Adaptive discrete thin plate spline smoother

The discrete thin plate spline smoother fits smooth surfaces to large data sets efficiently. It combines the favourable properties of the finite element surface fitting and thin plate splines. The efficiency of its finite element grid is improved by adaptive refinement, which adapts the precision of the solution. It reduces computational costs by refining only in sensitive regions, which are identified using error indicators. While many error indicators have been developed for the finite element method, they may not work for the discrete smoother. In this article we show three error indicators adapted from the finite element method for the discrete smoother. A numerical experiment is provided to evaluate their performance in producing efficient finite element grids. References F. L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Pat. Anal. Mach. Int. 11.6 (1989), pp. 567–585. doi: 10.1109/34.24792. C. Chen and Y. Li. A robust method of thin plate spline and its application to DEM construction. Comput. Geosci. 48 (2012), pp. 9–16. doi: 10.1016/j.cageo.2012.05.018. L. Fang. Error estimation and adaptive refinement of finite element thin plate spline. PhD thesis. The Australian National University. http://hdl.handle.net/1885/237742. L. Fang. Error indicators and adaptive refinement of the discrete thin plate spline smoother. ANZIAM J. 60 (2018), pp. 33–51. doi: 10.21914/anziamj.v60i0.14061. M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Commun. Stat. Simul. Comput. 19.2 (1990), pp. 433–450. doi: 10.1080/0361091900881286. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Soft. 15.4 (1989), pp. 326–347. doi: 10.1145/76909.76912. R. F. Reiniger and C. K. Ross. A method of interpolation with application to oceanographic data. Deep Sea Res. Oceanographic Abs. 15.2 (1968), pp. 185–193. doi: 10.1016/0011-7471(68)90040-5. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal. 41.1 (2003), pp. 208–234. doi: 10.1137/S0036142901383296. D. Ruprecht and H. Muller. Image warping with scattered data interpolation. IEEE Comput. Graphics Appl. 15.2 (1995), pp. 37–43. doi: 10.1109/38.365004. E. G. Sewell. Analysis of a finite element method. Springer, 2012. doi: 10.1007/978-1-4684-6331-6. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput. 63.2 (2015), pp. 374–409. doi: 10.1007/s10915-014-9898-x. O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24.2 (1987), pp. 337–357. doi: 10.1002/nme.1620240206