Abstract
Monotonicity-preserving approximation methods are used in numerous applications. These methods are expected to reconstruct a function from a discrete set of data preserving Monotonicity properties (i.e., the reconstructed function has to be monotone wherever the discrete data are). Most Monotonicity-preserving methods sacrifice accuracy in order to obtain Monotonicity. In this work we propose and analyze a Monotonicity-preserving Hermite interpolant that does not fully sacrifice accuracy. A key ingredient in the proposed technique is the approximation of derivatives using ENO (Essentially Non Oscillatory) and WENO (Weighted Essentially Non Oscillatory) interpolatory techniques. Then, an appropriate filtering process ensures the preservation of Monotonicity while maintaining, at the same time, the order of accuracy as high as possible. We perform several numerical experiments which confirm the properties of the proposed algorithms and compare them to other standard interpolatory approximation techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aràndiga, F., Baeza, A., Donat, R.: Discrete multiresolution based on Hermite interpolation: computing derivatives. Commun. Nonlinear Sci. Numer. Simul. 9(2), 263–273 (2004)
Aràndiga, F., Baeza A., Donat, R.: Vector cell-average multiresolution based on Hermite interpolation. Adv. Comput. Math. 28(1), 1–22 (2008)
Aràndiga, F., Belda, A.M. , Mulet, P.: Point-value WENO multiresolution applications to stable image compression. J. Sci. Comput. 43(2), 158–182 (2010)
Aràndiga, F., Donat, R.: Nonlinear multiscale decompositions: the approach of A. Harten. Numer. Algorithms 23, 175–216 (2000)
Belda, A.M.: Técnicas de interpolación WENO y su aplicación al procesamiento de imágenes. PhD thesis, University of Valencia (2010)
Berzins, M.: Adaptive polynomial interpolation on evenly spaced meshes. SIAM Rev. 49(4), 604–627 (2007)
De Boor, C.: A Practical Guide to Splines. Springer, New York (2001)
De Boor, C., B. Swartz, B.: Piecewise monotone interpolation. J. Approx. Theory 21, 411–416 (1977)
Fritsch, F.N., Butland, J.: A method for constructing local monotone piecewise cubic interpolants. SIAM J. Sci. Stat. Comput. 5(2), 300–304 (1984)
Fritsch, F.N., Carlson, R.E.: Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17(2), 238–246 (1980)
Gal, S.G.: Shape-Preserving Approximation by Real and Complex Polynomials. Springer, Birkhäuser (2008)
Harten, A., Engquist, B., Osher S., Chakravarthy, S.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)
Hyman, J.H.: Accurate Monotonicity preserving cubic interpolation. SIAM J. Numer. Anal. 4(4), 645–654 (1983)
Huynh, H.T.: Accurate monotone cubic interpolation. SIAM J. Numer. Anal. 30(1), 57–100 (1993)
Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Kvasov, B.I.: Methods of Shape-Preserving Spline Approximation. World Scientific, Singapore (2000)
Kocić, L.M., Milovanović, G.V.: Shape preserving approximations by polynomials and splines. Comput. Math. Appl. 33(11), 59–97 (1997)
Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 202–212 (1994)
Moler, C.: Numerical Computing with MATLAB. SIAM, Philadelphia (2004)
Steffen, M.: A simple method for monotonic interpolation in one dimension. Astron. Astrophys. 239, 443–450 (1990)
Wolberg, G., Alfy, I.: An energy-minimization framework for monotonic cubic spline interpolation. J. Comput. Appl. Math. 143, 145–188 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: J. Ward.
Rights and permissions
About this article
Cite this article
Aràndiga, F., Baeza, A. & Yáñez, D.F. A new class of non-linear monotone Hermite interpolants. Adv Comput Math 39, 289–309 (2013). https://doi.org/10.1007/s10444-012-9280-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-012-9280-1