Search search
submit Submit
Publication Date
to
Vol. 122
Latest Volume
All Volumes
PIER 180 [2024] PIER 179 [2024] PIER 178 [2023] PIER 177 [2023] PIER 176 [2023] PIER 175 [2022] PIER 174 [2022] PIER 173 [2022] PIER 172 [2021] PIER 171 [2021] PIER 170 [2021] PIER 169 [2020] PIER 168 [2020] PIER 167 [2020] PIER 166 [2019] PIER 165 [2019] PIER 164 [2019] PIER 163 [2018] PIER 162 [2018] PIER 161 [2018] PIER 160 [2017] PIER 159 [2017] PIER 158 [2017] PIER 157 [2016] PIER 156 [2016] PIER 155 [2016] PIER 154 [2015] PIER 153 [2015] PIER 152 [2015] PIER 151 [2015] PIER 150 [2015] PIER 149 [2014] PIER 148 [2014] PIER 147 [2014] PIER 146 [2014] PIER 145 [2014] PIER 144 [2014] PIER 143 [2013] PIER 142 [2013] PIER 141 [2013] PIER 140 [2013] PIER 139 [2013] PIER 138 [2013] PIER 137 [2013] PIER 136 [2013] PIER 135 [2013] PIER 134 [2013] PIER 133 [2013] PIER 132 [2012] PIER 131 [2012] PIER 130 [2012] PIER 129 [2012] PIER 128 [2012] PIER 127 [2012] PIER 126 [2012] PIER 125 [2012] PIER 124 [2012] PIER 123 [2012] PIER 122 [2012] PIER 121 [2011] PIER 120 [2011] PIER 119 [2011] PIER 118 [2011] PIER 117 [2011] PIER 116 [2011] PIER 115 [2011] PIER 114 [2011] PIER 113 [2011] PIER 112 [2011] PIER 111 [2011] PIER 110 [2010] PIER 109 [2010] PIER 108 [2010] PIER 107 [2010] PIER 106 [2010] PIER 105 [2010] PIER 104 [2010] PIER 103 [2010] PIER 102 [2010] PIER 101 [2010] PIER 100 [2010] PIER 99 [2009] PIER 98 [2009] PIER 97 [2009] PIER 96 [2009] PIER 95 [2009] PIER 94 [2009] PIER 93 [2009] PIER 92 [2009] PIER 91 [2009] PIER 90 [2009] PIER 89 [2009] PIER 88 [2008] PIER 87 [2008] PIER 86 [2008] PIER 85 [2008] PIER 84 [2008] PIER 83 [2008] PIER 82 [2008] PIER 81 [2008] PIER 80 [2008] PIER 79 [2008] PIER 78 [2008] PIER 77 [2007] PIER 76 [2007] PIER 75 [2007] PIER 74 [2007] PIER 73 [2007] PIER 72 [2007] PIER 71 [2007] PIER 70 [2007] PIER 69 [2007] PIER 68 [2007] PIER 67 [2007] PIER 66 [2006] PIER 65 [2006] PIER 64 [2006] PIER 63 [2006] PIER 62 [2006] PIER 61 [2006] PIER 60 [2006] PIER 59 [2006] PIER 58 [2006] PIER 57 [2006] PIER 56 [2006] PIER 55 [2005] PIER 54 [2005] PIER 53 [2005] PIER 52 [2005] PIER 51 [2005] PIER 50 [2005] PIER 49 [2004] PIER 48 [2004] PIER 47 [2004] PIER 46 [2004] PIER 45 [2004] PIER 44 [2004] PIER 43 [2003] PIER 42 [2003] PIER 41 [2003] PIER 40 [2003] PIER 39 [2003] PIER 38 [2002] PIER 37 [2002] PIER 36 [2002] PIER 35 [2002] PIER 34 [2001] PIER 33 [2001] PIER 32 [2001] PIER 31 [2001] PIER 30 [2001] PIER 29 [2000] PIER 28 [2000] PIER 27 [2000] PIER 26 [2000] PIER 25 [2000] PIER 24 [1999] PIER 23 [1999] PIER 22 [1999] PIER 21 [1999] PIER 20 [1998] PIER 19 [1998] PIER 18 [1998] PIER 17 [1997] PIER 16 [1997] PIER 15 [1997] PIER 14 [1996] PIER 13 [1996] PIER 12 [1996] PIER 11 [1995] PIER 10 [1995] PIER 09 [1994] PIER 08 [1994] PIER 07 [1993] PIER 06 [1992] PIER 05 [1991] PIER 04 [1991] PIER 03 [1990] PIER 02 [1990] PIER 01 [1989]
All Journals
PIER PIER B PIER C PIER M PIER Letters
2011-11-16
A Fast Inverse Polynomial Reconstruction Method Based on Conformal Fourier Transformation
By
Zhe Liu,

    Dr. Zhe Liu

    School of Electronic Engineering

    University of Electronic Science and Technology of China

    China

    Homepage

Qing Huo Liu,

    Prof. Qing Huo Liu

    Department of Electrical and Computer Engineering

    Duke University

    USA

    Homepage

Chun-Hui Zhu,

    Dr. Chun-Hui Zhu

    Department of Control Science and Engineering

    Harbin Institute of Technology

    China

    Homepage

Jianyu Yang

    Dr. Jianyu Yang

    School of Electronic Engineering

    University of Electronic Science and Technology of China

    China

    Homepage

Progress In Electromagnetics Research, Vol. 122, 119-136, 2012
doi:10.2528/PIER11092008
Abstract
A fast Inverse Polynomial Reconstruction Method (IPRM) is proposed to efficiently eliminate the Gibbs phenomenon in Fourier reconstruction of discontinuous functions. The framework of the fast IPRM is modified by reconstructing the function in discretized elements, then the Conformal Fourier Transform (CFT) and the Chirp Z-Transform (CZT) algorithms are applied to accelerate the evaluation of reconstruction coefficients. The memory cost of the fast IPRM is also significantly reduced, owing to the transformation matrix being discretized in the modified framework. The computation complexity and memory cost of the fast IPRM are O(MN log 2L) and O(MN), respectively, where L is the number of the discretized elements, M is the degree of polynomials for the reconstruction of each element, and N is the number of the Fourier series. Numerical results demonstrate that the fast IPRM method not only inherits the robustness of the Generalized IPRM (G-IPRM) method, but also significantly reduces the computation time and the memory cost. Therefore, the fast IPRM method is useful for the pseudospectral time domain methods and for the volume integral equation of the discontinuous material distributions.
Citation
Zhe Liu, Qing Huo Liu, Chun-Hui Zhu, and Jianyu Yang, "A Fast Inverse Polynomial Reconstruction Method Based on Conformal Fourier Transformation," Progress In Electromagnetics Research, Vol. 122, 119-136, 2012.
doi:10.2528/PIER11092008
References

1. Boyd, J. P., Chebyshev and Fourier Spectral Methods, Dover publications, 2000.

2. Fornberg, B., "A Practical Guide to Pseudospectral Methods," Cambridge University Press, 1996.

3. Liu, Q. H., "The PSTD algorithm: A time-domain method requiring only two cells per wavelength," Microwave and Optical Technology Letters, Vol. 15, 158-165, 1997.
doi:10.1002/(SICI)1098-2760(19970620)15:3<158::AID-MOP11>3.0.CO;2-3

4. Giordano, N. J. and H. Nakanishi, Computational Physics, Prentice Hall Publishing, 1997.

5. Gonzalez, R. C. and R. E. Woods, Digital Image Processing, Addison-Wesley Publishing, 1992.

6. Gottlieb, D. and C. W. Shu, "On the Gibbs phenomenon and its resolution," SIAM Review, Vol. 39, 644-668, 1997.
doi:10.1137/S0036144596301390

7. Gottlieb, D., C. W. Shu, S. Alex, and H. Vandeven, "On the Gibbs phenomenon I: Recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function," Journal of Computational and Applied Mathematics, Vol. 43, 81-98, 1992.
doi:10.1016/0377-0427(92)90260-5

8. Driscoll, T. A. and B. Fornberg, "A pade-based algorithm for overcoming Gibbs phenomenon," Numerical Algorithms, Vol. 26, 77-92, 2001.
doi:10.1023/A:1016648530648

9. Gelb, A. and J. Tanner, "Robust reprojection methods for the resolution of the Gibbs phenomenon," Applied and Computational Harmonic Analysis, Vol. 20, 3-25, 2006.
doi:10.1016/j.acha.2004.12.007

10. Boyd, J. P., "Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations," Journal of Computational Physics, Vol. 204, 253-264, 2005.
doi:10.1016/j.jcp.2004.10.008

11. Min, M., T. Lee, P. F. Fischer, and S. K. Gray, "Fourier spectral simulations and Gegenbauer reconstructions for electromagnetic waves in the presence of a metal nanoparticle ," Journal of Computational Physics, Vol. 213, 730-747, 2006.
doi:10.1016/j.jcp.2005.06.025

12. Shizgal, B. D. and J. Jung, "Towards the resolution of the Gibbs phenomena," Journal of Computational and Applied Mathematics, Vol. 161, 41-65, 2003.
doi:10.1016/S0377-0427(03)00500-4

13. Jung, J. and B. D. Shizgal, "Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon," Journal of Computational and Applied Mathematics, Vol. 172, 131-151, 2004.
doi:10.1016/j.cam.2004.02.003

14. Jung, J. and B. D. Shizgal, "On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon ," Journal of Computational Physics, Vol. 224, 477-488, 2007.
doi:10.1016/j.jcp.2007.01.018

15. Jung, J. and B. D. Shizgal, "Inverse polynomial reconstruction of two-dimensional fourier images," Journal of Science Computing, Vol. 25, 367-399, 2005.
doi:10.1007/s10915-004-4795-3

16. Hrycak, T. and K. Grochenig, "Pseudospectral fourier reconstruction with the modified inverse polynomial reconstruction method," Journal of Computational Physics, Vol. 229, 933-946, 2010.
doi:10.1016/j.jcp.2009.10.026

17. Jung, J., "A hybrid method for the resolution of the Gibbs phenomenon," Lecture Notes in Computational Science and Engineering, Vol. 76, 219-227, 2011.
doi:10.1007/978-3-642-15337-2_19

18. Paige, C. C. and M. A. Saunders, "LSQR: An algorithm for sparse linear equations and sparse least squares," ACM Transactions on Mathematical Software, Vol. 8, 43-71, 1982.
doi:10.1145/355984.355989

19. Carvalho, S. and L. S. Mendes, "Scattering of EM waves by inhomogeneous dielectrics with the use of the method of moments and 3-D solenoidal basis functions ," Microwave and Optical Technology Letters, Vol. 23, 42-46, 1999.
doi:10.1002/(SICI)1098-2760(19991005)23:1<42::AID-MOP12>3.0.CO;2-N

20. Li, M.-K. and W. C. Chew, "Applying divergence-free condition in solving the volume integral equation," Progress In Electromagnetics Research, Vol. 57, 311-333, 2006.
doi:10.2528/PIER05061303

21. Fan, Z., R.-S. Chen, H. Chen, and D.-Z. Ding, "Weak form nonuniform fast Fourier transform method for solving volume integral equations," Progress In Electromagnetics Research, Vol. 89, 275-289, 2009.
doi:10.2528/PIER08121308

22. Hu, L., L.-W. Li, and T. S. Yeo, "Analysis of scattering by large inhomogeneous bi-anisotropic objects using AIM," Progress In Electromagnetics Research, Vol. 99, 21-36, 2009.
doi:10.2528/PIER09101204

23. Taboada, J. M., M. G. Araujo, J. M. Bertolo, L. Landesa, F. Obelleiro, and J. L. Rodriguez, "MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics," Progress In Electromagnetics Research, Vol. 105, 15-30, 2010.
doi:10.2528/PIER10041603

24. Ergul, O., T. Malas, and L. Gurel, "Solutions of largescale electromagnetics problems using an iterative inner-outer scheme with ordinary and approximate multilevel fast multipole algorithms," Progress In Electromagnetics Research, Vol. 106, 203-223, 2010.
doi:10.2528/PIER10061711

25. Huang, Y., Y. Liu, Q. H. Liu, and J. Zhang, "Improved 3-D GPR detection by NUFFT combined with MPD method," Progress In Electromagnetics Research, Vol. 103, 185-199, 2010.
doi:10.2528/PIER10021005

26. Zhu, X., Z. Zhao, W. Yang, Y. Zhang, Z.-P. Nie, and Q. H. Liu, "Iterative time-reversal mirror method for imaging the buried object beneath rough ground surface," Progress In Electromagnetics Research, Vol. 117, 19-33, 2011.

27. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Government Printing Office Publishing, 1972.

28. Zhu, C. H., Q. H. Liu, Y. Shen, and L. J. Liu, "A high accuracy conformal method for evaluating the discontinuous fourier transform," Progress In Electromagnetics Research, Vol. 109, 425-440, 2010.
doi:10.2528/PIER10082007

29. Rabiner, L. R., R. W. Schafer, and C. M. Rader, "The chirp Z-transform algorithm and its application," IEEE Transaction on Audio Electroacoust, Vol. 17, 86-92, 1969.
doi:10.1109/TAU.1969.1162034

30. Franceschetti, G., R. Lanari, and E. S. Marzouk, "A new two-dimensional squint mode SAR processor," IEEE Transactions on Aerospace and Electronic Systems, Vol. 32, 854-863, 1996.
doi:10.1109/7.489529

Download Full Article (293) View Full Article volume
The EM Academy piers.org jpier.org Who's Who in EM
Copyright © 2022 The Electromagnetics Academy. All Rights Reserved.