Optimal filter and mollifier for piecewise smooth spectral data
Author: Jared Tanner
Journal: Math. Comp. 75 (2006), 767-790
MSC (2000): Primary 41A25, 42A10, 42A16, 42A20, 42C25, 65B10, 65T40
Published electronically: January 23, 2006
MathSciNet review: 2196991
Full-text PDF Free Access
Abstract: We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious Gibbs' oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal joint time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.
- 1. John P. Boyd, A lag-averaged generalization of Euler’s method for accelerating series, Appl. Math. Comput. 72 (1995), no. 2-3, 143–166. MR 1346575, https://doi.org/10.1016/0096-3003(94)00180-C
- 2. J. P. Boyd, The Erfc-Log Filter and the Asymptotics of the Euler and Vandeven Sequence Accelerations, Proceedings of the Third International Conference on Spectral and High Order Methods, (1996) 267-276.
- 3. Oscar P. Bruno, Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics, Topics in computational wave propagation, Lect. Notes Comput. Sci. Eng., vol. 31, Springer, Berlin, 2003, pp. 43–82. MR 2032867, https://doi.org/10.1007/978-3-642-55483-4_2
- 4. Wei Cai, David Gottlieb, and Chi-Wang Shu, On one-sided filters for spectral Fourier approximations of discontinuous functions, SIAM J. Numer. Anal. 29 (1992), no. 4, 905–916. MR 1173176, https://doi.org/10.1137/0729055
- 5. Tobin A. Driscoll and Bengt Fornberg, A Padé-based algorithm for overcoming the Gibbs phenomenon, Numer. Algorithms 26 (2001), no. 1, 77–92. MR 1827318, https://doi.org/10.1023/A:1016648530648
- 6. Knut S. Eckhoff, On a high order numerical method for functions with singularities, Math. Comp. 67 (1998), no. 223, 1063–1087. MR 1459387, https://doi.org/10.1090/S0025-5718-98-00949-1
- 7. Anne Gelb, The resolution of the Gibbs phenomenon for spherical harmonics, Math. Comp. 66 (1997), no. 218, 699–717. MR 1401940, https://doi.org/10.1090/S0025-5718-97-00828-4
- 8. Anne Gelb, A hybrid approach to spectral reconstruction of piecewise smooth functions, J. Sci. Comput. 15 (2000), no. 3, 293–322. MR 1828746, https://doi.org/10.1023/A:1011126400782
- 9. A. Gelb and Zackiewicz, Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method, preprint.
- 10. Anne Gelb and Eitan Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999), no. 1, 101–135. MR 1699594, https://doi.org/10.1006/acha.1999.0262
- 11. Anne Gelb and Eitan Tadmor, Detection of edges in spectral data. II. Nonlinear enhancement, SIAM J. Numer. Anal. 38 (2000), no. 4, 1389–1408. MR 1790039, https://doi.org/10.1137/S0036142999359153
- 12. A. Gelb and J. Tanner, Robust Reprojection Methods for the Resolution of the Gibbs Phenomenon, ACHA, to appear.
- 13. David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64 (1995), no. 211, 1081–1095. MR 1284667, https://doi.org/10.1090/S0025-5718-1995-1284667-0
- 14. David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997), no. 4, 644–668. MR 1491051, https://doi.org/10.1137/S0036144596301390
- 15. David Gottlieb and Eitan Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, Progress and supercomputing in computational fluid dynamics (Jerusalem, 1984) Progr. Sci. Comput., vol. 6, Birkhäuser Boston, Boston, MA, 1985, pp. 357–375. MR 935160
- 16. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
- 17. Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717
- 18. David K. Hoffman and Donald J. Kouri, Hierarchy of local minimum solutions of Heisenberg’s uncertainty principle, Phys. Rev. Lett. 85 (2000), no. 25, 5263–5267. MR 1813690, https://doi.org/10.1103/PhysRevLett.85.5263
- 19. Jae-Hun Jung and Bernie D. Shizgal, Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon, J. Comput. Appl. Math. 172 (2004), no. 1, 131–151. MR 2091135, https://doi.org/10.1016/j.cam.2004.02.003
- 20. J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
- 21. Andrew Majda, James McDonough, and Stanley Osher, The Fourier method for nonsmooth initial data, Math. Comp. 32 (1978), no. 144, 1041–1081. MR 501995, https://doi.org/10.1090/S0025-5718-1978-0501995-4
- 22. E. Tadmor, Spectral Methods for Hyperbolic Problems, from ``Lecture Notes Delivered at Ecole Des Ondes'', January 24-28, 1994. Available at http://www.math.ucla.edu/~tadmor/pub/ spectral-approximations/Tadmor.INRIA-94.pdf
- 23. Eitan Tadmor and Jared Tanner, Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information, Found. Comput. Math. 2 (2002), no. 2, 155–189. MR 1894374
- 24. E. Tadmor and J. Tanner, Adaptive Filters for Piecewise Smooth Spectral Data, IMA J. Numerical Analysis, 25 No. 4 (2005) 535-647.
- 25. Hervé Vandeven, Family of spectral filters for discontinuous problems, J. Sci. Comput. 6 (1991), no. 2, 159–192. MR 1140344, https://doi.org/10.1007/BF01062118
- J. P. Boyd, A Lag-Averaged Generalization of Euler's Method for Accelerating Series, Appl. Math. Comput., (1995) 143-166. MR 1346575 (96d:65007)
- J. P. Boyd, The Erfc-Log Filter and the Asymptotics of the Euler and Vandeven Sequence Accelerations, Proceedings of the Third International Conference on Spectral and High Order Methods, (1996) 267-276.
- O.P. Bruno, Fast, High-Order, High-Frequency Integral Methods for Computational Acoustics and Electromagnetics, Topics in Computational Wave Propagation Direct and Inverse Problems Series: Lecture Notes in Computational Science and Engineering, 31 (M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne, eds.), Springer, 2003, 43-83. MR 2032867 (2004k:65260)
- W. Cai, D. Gottlieb, and C.W. Shu, On One-Sided Filters for Spectral Fourier Approximations of Discontinuous Functions, SIAM Journal of Numerical Analysis, 29 (1992) 905-916. MR 1173176 (93e:65021)
- T.A. Driscoll and B. Fornberg, A Padé-Based Algorithm for Overcoming the Gibbs' Phenomenon, Num. Alg. 26 (2001) 77-92. MR 1827318 (2002b:65007)
- K.S. Eckhoff, On a High Order Numerical Method for Functions with Singularities, Math. Comp. 67(223) (1998) 1063-1087. MR 1459387 (98j:65014)
- A. Gelb, The resolution of the Gibbs phenomenon for spherical harmonics, Math. Comp. 66 (1997) 699-717. MR 1401940 (97k:42005)
- A. Gelb, A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions, Journal of Scientific Computing, October 2000. MR 1828746 (2002a:65028)
- A. Gelb and Zackiewicz, Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method, preprint.
- A. Gelb and E. Tadmor, Detection of Edges in Spectral Data, Applied Computational Harmonic Analysis 7 (1999) 101-135. MR 1699594 (2000g:42003)
- A. Gelb and E. Tadmor, Detection of Edges in Spectral Data II. Nonlinear Enhancement, SIAM Journal of Numerical Analysis 38 (2000) 1389-1408.MR 1790039 (2001i:42003)
- A. Gelb and J. Tanner, Robust Reprojection Methods for the Resolution of the Gibbs Phenomenon, ACHA, to appear.
- D. Gottlieb and C.-W. Shu, On The Gibbs Phenomenon IV: recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64 (1995) 1081-1095. MR 1284667 (97b:42004)
- D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution, SIAM Review 39 (1998) 644-668. MR 1491051 (98m:42002)
- D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in ``Progress and Supercomputing in Computational Fluid Dynamics'', Proceedings of 1984 U.S.-Israel Workshop, Progress in Scientific Computing, Vol. 6 (E. M. Murman and S. S. Abarbanel, eds.), Birkhauser, Boston, 1985, 357-375. MR 0935160 (90a:65041)
- I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2000. MR 1773820 (2001c:00002)
- K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001. MR 1843717 (2002h:42001)
- D.K. Hoffman and D.J. Kouri, Hierarchy of Local Minimum Solutions of Heisenberg's Uncertainty Principle, Phy. Rev. Lett. 25 (2002) 5263-5267.MR 1813690 (2001k:81006)
- J.-H. Jung and B.D. Shizgal, Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs' phenomenon, J. Comput. Appl. Math. 172 (2004) 131-151. MR 2091135 (2005d:42002)
- J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, New York, 2003. MR 1937591 (2004h:33001)
- A. Majda, J. McDonough and S. Osher, The Fourier method for nonsmooth initial data, Math. Comput. 30 (1978) 1041-1081. MR 0501995 (80a:65197)
- E. Tadmor, Spectral Methods for Hyperbolic Problems, from ``Lecture Notes Delivered at Ecole Des Ondes'', January 24-28, 1994. Available at http://www.math.ucla.edu/~tadmor/pub/ spectral-approximations/Tadmor.INRIA-94.pdf
- E. Tadmor and J. Tanner, Adaptive Mollifiers - High Resolution Recovery of Piecewise Smooth Data from its Spectral Information, J. Foundations of Comp. Math. 2 (2002) 155-189. MR 1894374 (2003b:42009)
- E. Tadmor and J. Tanner, Adaptive Filters for Piecewise Smooth Spectral Data, IMA J. Numerical Analysis, 25 No. 4 (2005) 535-647.
- H. Vandeven, Family of Spectral Filters for Discontinuous Problems, Journal of Scientific Computings, 6 No. 2 (1991) 159-192. MR 1140344 (92k:65006)
Affiliation: Department of Statistics, Stanford University, Stanford, California 94305-9025
Keywords: Fourier series, filters, time-frequency localization, piecewise smooth, spectral projection
Received by editor(s): May 19, 2004
Received by editor(s) in revised form: January 29, 2005
Published electronically: January 23, 2006
Additional Notes: The author was supported in part by NSF Grants DMS 01-35345 and 04-03041.
Dedicated: This paper is dedicated to Eitan Tadmor for his direction
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.