Abstract
Interpolatory splines are usually useful to reconstruct data that present certain regularity. This paper is devoted to the construction and analysis of a new technique that allows to improve the accuracy of splines near corner singularities in the point values and jump discontinuities in the cell averages. The detection of discontinuities is easily done through the processing of the right hand side of the system of equations of the spline, that contains divided differences. The process of adaption will require some knowledge about the position of the discontinuity and the values of the function and its derivatives at the discontinuity. Using Harten’s multiresolution we can adapt splines to the presence of corner singularities and jump discontinuities. Thanks to the adaption, the smearing of corner singularities does not appear in the reconstruction obtained in the point values and Gibbs phenomenon and diffusion of jump discontinuities is also eliminated. These numerical effects are a consequence of assuming that the data used in the reconstruction comes from the discretization of a continuous function with certain regularity and this might not be true.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbadi, A., Barrera, D., Ibáñez, M.J., Sbibih, D.: A general method for constructing quasi-interpolants from B-splines. J. Comput. Appl. Math. 234(4), 1324–1337 (2010)
Han, X.: Convexity-preserving approximation by univariate cubic splines. J. Comput. Appl. Math. 287, 196–206 (2015)
Barrera, D., Ibáñez, M.J., Sablonnière, P., Sbibih, D.: Near-best univariate spline discrete quasi-interpolants on nonuniform partitions. Constr. Approx. 28(3), 237–251 (2008)
Xu, X.-P., Lang, F.-G.: Quintic B-spline method for function reconstruction from integral values of successive subintervals. Numer. Algor. 66(2), 223–240 (2014)
Baramidze, V.: Smooth bivariate shape-preserving cubic spline approximation. Comput. Aided Geom. Des. 44, 36–55 (2016)
Apprato, D., Gout, C.: A result about scale transformation families in approximation: application to surface fitting from rapidly varying data. Numer. Algor. 23(2), 263–279 (2000)
Bozzini, M., Lenarduzzi, L.: Recovering a function with discontinuities from correlated data. In:: Advanced Topics in Multivariate Approximation. WSP Singapore (1996)
Bozzini, M., Rossini, M.: Approximating surfaces with discontinuities. Math Comput Modell 31(6), 193–213 (2000)
Foley, T.A.: Weighted bicubic spline interpolation to rapidly varying data. ACM Trans. Graph. 6(1), 1–18 (1987)
Gout, C., Komatitsch, D.: Surface fitting of rapidly varying data using rank coding: application to geophysical surfaces. Math. Geol. 32(7), 873–888 (2000)
Gout, C., Le Guyader, C., Romani, L., Saint-Guirons, A.-G.: Approximation of surfaces with fault(s) and/or rapidly varying data, using a segmentation process, d m-splines and the finite element method. Numer. Algor. 48(1), 67–92 (2008)
Gutzmer, T., Iske, A.: Detection of discontinuities in scattered data approximation. Numer. Algor. 16(2), 155–170 (1997)
Parra, M., de Silanes, M., Torrens, J.: Vertical fault detection from scattered data. J Comput Appl Math 73(1), 225–239 (1996)
Salkauskas, K.: c 1 splines for interpolation of rapidly varying data. Rocky Mountain J. Math. 14(1), 239–250 (1984)
Harten, A.: Multiresolution representation of data: a general framework. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)
Harten, A.: Eno schemes with subcell resolution. J. Comput. Phys. 83(1), 148–184 (1989)
Aràndiga, F., Cohen, A., Donat, R., Dyn, N.: Interpolation and approximation of piecewise smooth functions. SIAM J. Numer. Anal. 43(1), 41–57 (2005)
Aràndiga, F., Donat, R.: Nonlinear multiscale decompositions: the approach of A. Harten. Numer. Algor. 23(2-3), 175–216 (2000)
Amat, S., Liandrat, J., Ruiz, J., Trillo, J.: On a compact non-extrapolating scheme for adaptive image interpolation. J. Franklin Inst. 349(5), 1637–1647 (2012)
Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis, 7th edn. Pearson/Addison-Wesley (2004)
Richards, F.: A Gibbs phenomenon for spline functions. J. Approx. Theory. 66(3), 334–351 (1991)
Zhang, Z., Martin, C.F.: Convergence and Gibbs’ phenomenon in cubic spline interpolation of discontinuous functions. J. Comput. Appl. Math. 87(2), 359–371 (1997)
Foster, J., Richards, F.B.: Gibbs-wilbraham splines. Constr. Approx. 11 (1), 37–52 (1995)
de Silanes, M.L., Parra, M., Pasadas, M., Torrens, J.: Spline approximation of discontinuous multivariate functions from scattered data. J. Comput. Appl. Math. 131(1–2), 281–298 (2001)
de Boor, C.: A Practical Guide to Splines, vol. 27. Springer-Verlag, New York (1980)
Acknowledgments
We would like to thank the referees and the editor for their useful suggestions and comments that, with no doubt, have helped to improve the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Sergio Amat and Juan C. Trillo have been supported through the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714 and through the national research project MTM2015-64382-P (MINECO/FEDER).
Juan Ruiz has been supported through the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714, through the national research project MTM2015-64382-P (MINECO/FEDER) and by the Fundación Séneca through the young researchers program Jiménez de la Espada.
Rights and permissions
About this article
Cite this article
Amat, S., Ruiz, J. & Trillo, J.C. On an algorithm to adapt spline approximations to the presence of discontinuities. Numer Algor 80, 903–936 (2019). https://doi.org/10.1007/s11075-018-0511-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-018-0511-5
Keywords
- Splines
- Adaption to discontinuities
- Approximation
- Point-values
- Cell-averages
- Computer aided design (modeling of curves)