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.
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.
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.
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.
8. Driscoll, T. A. and B. Fornberg, "A pade-based algorithm for overcoming Gibbs phenomenon," Numerical Algorithms, Vol. 26, 77-92, 2001.
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.
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.
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.
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.
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.
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.
15. Jung, J. and B. D. Shizgal, "Inverse polynomial reconstruction of two-dimensional fourier images," Journal of Science Computing, Vol. 25, 367-399, 2005.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.