Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

We consider advection–diffusion–reaction problems, where the advective or the reactive term is dominating with respect to the diffusive term. The solutions of these problems are characterized by the so-called layers, which represent localized regions where the gradients of the solutions are rather large or are subjected to abrupt changes. In order to improve the accuracy of the computed solution, it is fundamental to locally increase the number of degrees of freedom by limiting the computational costs. Thus, adaptive refinement, by a posteriori error estimators, is employed. The error estimators are then processed by an anomaly detection algorithm in order to identify those regions of the computational domain that should be marked and, hence, refined. The anomaly detection task is performed in an unsupervised fashion and the proposed strategy is tested on typical benchmarks. The present work shows a numerical study that highlights promising results obtained by bridging together standard techniques, i.e., the error estimators, and approaches typical of machine learning and artificial intelligence, such as the anomaly detection task.

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.