Wavelet-based filters for accurate computation of derivatives
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- by Maurice Hasson;
- Math. Comp. 75 (2006), 259-280
- DOI: https://doi.org/10.1090/S0025-5718-05-01767-9
- Published electronically: June 23, 2005
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Abstract:
Let $f(x)$ be a smooth function whose derivative of a given order must be computed. The signal $f(x)$ is affected by two kinds of perturbation. The perturbation caused by the presence of the machine epsilon $\epsilon _M$ of the computer may be considered to be an extremely high-frequency noise of very small amplitude. The way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient. The second perturbation, caused by the presence of noise, requires first the signal to be treated in some way. It is the purpose of this work to construct a wavelet-based band-pass filter that deals with the two cited perturbations simultaneously. In effect our wavelet acts like a “smoothed difference quotient" whose stepsize is of the same order as that of the usual difference quotient. Moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise.References
- Robert S. Anderssen and Markus Hegland, For numerical differentiation, dimensionality can be a blessing!, Math. Comp. 68 (1999), no. 227, 1121–1141. MR 1620207, DOI 10.1090/S0025-5718-99-01033-9
- Kendall E. Atkinson, An introduction to numerical analysis, 2nd ed., John Wiley & Sons, Inc., New York, 1989. MR 1007135
- Michèle Basseville and Igor V. Nikiforov, Detection of abrupt changes: theory and application, Prentice Hall Information and System Sciences Series, Prentice Hall, Inc., Englewood Cliffs, NJ, 1993. MR 1210954
- Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers. I, Springer-Verlag, New York, 1999. Asymptotic methods and perturbation theory; Reprint of the 1978 original. MR 1721985, DOI 10.1007/978-1-4757-3069-2
- Franca Caliò, Marco Frontini, and Gradimir V. Milovanović, Numerical differentiation of analytic functions using quadratures on the semicircle, Comput. Math. Appl. 22 (1991), no. 10, 99–106. MR 1136179, DOI 10.1016/0898-1221(91)90196-B
- François Chaplais and Sylvain Faure, Wavelets and differentiation, Internal Report, Centre Automatique et Systèmes, École Nationale des Mines de Paris (2000), 1–14.
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- David L. Donoho, Private communication, 2003.
- Dan E. Dudgeon and Russell M. Mersereau, Multidimensional digital signal processing, Prentice-Hall, Englewood Cliffs, N.J., 1984.
- Sylvain Faure, Implémentation sous simulux de la dérivation des signaux à partir des coefficients d’ondelettes, Internal Report, Centre Automatique et Systèmes, École Nationale des Mines de Paris (1999), 1–12.
- Michael S. Floater, Error formulas for divided difference expansions and numerical differentiation, J. Approx. Theory 122 (2003), no. 1, 1–9. MR 1976120, DOI 10.1016/S0021-9045(03)00025-X
- Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
- Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300, DOI 10.1090/cbms/079
- Martin Hanke and Otmar Scherzer, Inverse problems light: numerical differentiation, Amer. Math. Monthly 108 (2001), no. 6, 512–521. MR 1840657, DOI 10.2307/2695705
- Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, and Alexander Tsybakov, Wavelets, approximation, and statistical applications, Lecture Notes in Statistics, vol. 129, Springer-Verlag, New York, 1998. MR 1618204, DOI 10.1007/978-1-4612-2222-4
- Eugenio Hernández and Guido Weiss, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR 1408902, DOI 10.1201/9781420049985
- Peter Hoffman and K. C. Reddy, Numerical differentiation by high order interpolation, SIAM J. Sci. Statist. Comput. 8 (1987), no. 6, 979–987. MR 911068, DOI 10.1137/0908079
- Ian Knowles and Robert Wallace, A variational method for numerical differentiation, Numer. Math. 70 (1995), no. 1, 91–110. MR 1320703, DOI 10.1007/s002110050111
- Alexandru Lupaş and Detlef H. Mache, On the numerical differentiation, Rev. Anal. Numér. Théor. Approx. 26 (1997), no. 1-2, 109–115. MR 1703928
- Malcolm T. McGregor, Numerical differentiation of analytic functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 2 (1991), 11–17. MR 1162961
- Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. MR 1085487
- D. A. Murio, C. E. Mejía, and S. Zhan, Discrete mollification and automatic numerical differentiation, Comput. Math. Appl. 35 (1998), no. 5, 1–16. MR 1612285, DOI 10.1016/S0898-1221(98)00001-7
- Alexander G. Ramm and Alexandra B. Smirnova, On stable numerical differentiation, Math. Comp. 70 (2001), no. 235, 1131–1153. MR 1826578, DOI 10.1090/S0025-5718-01-01307-2
- Harold S. Shapiro, Smoothing and approximation of functions, Van Nostrand Reinhold Mathematical Studies, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. Revised and expanded edition of mimiographed notes (Matscience Report No. 55). MR 412669
- M.-R. Skrzipek, Generalized associated polynomials and their application in numerical differentiation and quadrature, Calcolo 40 (2003), no. 3, 131–147. MR 2025599, DOI 10.1007/s10092-003-0074-1
- Robert S. Strichartz, A guide to distribution theory and Fourier transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994. MR 1276724
- O. L. Vinogradov and V. V. Zhuk, Sharp estimates for errors of numerical differentiation type formulas on trigonometric polynomials, J. Math. Sci. (New York) 105 (2001), no. 5, 2347–2376. Function theory and partial differential equations. MR 1855438, DOI 10.1023/A:1011313229137
- Y. B. Wang, X. Z. Jia, and J. Cheng, A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Problems 18 (2002), no. 6, 1461–1476. MR 1955897, DOI 10.1088/0266-5611/18/6/301
- Yazhen Wang, Jump and sharp cusp detection by wavelets, Biometrika 82 (1995), no. 2, 385–397. MR 1354236, DOI 10.1093/biomet/82.2.385
- J. A. C. Weideman, Numerical integration of periodic functions: a few examples, Amer. Math. Monthly 109 (2002), no. 1, 21–36. MR 1903510, DOI 10.2307/2695765
Bibliographic Information
- Maurice Hasson
- Affiliation: Program in Applied Mathematics, The University of Arizona, Tucson, Arizona 85721-0089
- Email: hasson@math.arizona.edu
- Received by editor(s): July 27, 2004
- Published electronically: June 23, 2005
- Additional Notes: Supported by a VIGRE Postdoctoral Fellowship at the University of Arizona.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 75 (2006), 259-280
- MSC (2000): Primary 41A40, 42A85, 65D25, 65T60
- DOI: https://doi.org/10.1090/S0025-5718-05-01767-9
- MathSciNet review: 2176399