Fourier truncation method for high order numerical derivatives. (English) Zbl 1103.65023
Summary: The authors consider a classical ill-posed problem – numerical differentiation with a new method.
They propose the Fourier truncation method to compute high order numerical derivatives. A Hölder-type stability estimate is obtained. A numerical implementation is described. Numerical examples show that the proposed method is effective and stable.
MSC:
| 65D25 | Numerical differentiation |
References:
| [1] | Deans, S. R., The Radon Transform and Some of its Applications. A Wiley-Interscience Publication (1983), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0561.44001 |
| [2] | Cheng, J.; Hon, Y. C.; Wang, Y. B., A numerical method for the discontinuous solutions of Abel integral equations, (Inverse Problems and Spectral Theory. Inverse Problems and Spectral Theory, Contemp. Math., vol. 348 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 233-243 · Zbl 1064.65153 |
| [3] | Gorenflo, R.; Vessella, S., (Abel Integral Equations. Abel Integral Equations, Lecture Notes in Mathematics, vol. 1461 (1991), Springer-Verlag: Springer-Verlag Berlin), Analysis and Applications · Zbl 0717.45002 |
| [4] | Hanke, M.; Scherzer, O., Error analysis of an equation error method for the identification of the diffusion coefficient in a quasi-linear parabolic differential equation, SIAM J. Appl. Math., 59, 1012-1027 (1999), (electronic) · Zbl 0928.35198 |
| [5] | Cullum, J., Numerical differentiation and regularization, SIAM J. Numer. Anal., 8, 254-265 (1971) · Zbl 0224.65005 |
| [6] | Groetsch, C. W., Differentiation of approximately specified functions, Amer. Math. Monthly, 98, 9, 847-850 (1991) · Zbl 0745.26004 |
| [7] | Hanke, M.; Scherzer, O., Inverse problems light: numerical differentiation, Amer. Math. Monthly, 108, 6, 512-521 (2001) · Zbl 1002.65029 |
| [8] | Qu, R., A new approach to numerical differentiation and integration, Math. Comput. Modell., 24, 10, 55-68 (1996) · Zbl 0874.65011 |
| [9] | Ramm, A. G.; Smirnova, A. B., On stable numerical differentiation, Math. Comput., 70, 1131-1153 (2001), (electronic) · Zbl 0973.65015 |
| [10] | Rivlin, T. J., Optimally stable Lagrangian numerical differentiation, SIAM J. Numer. Anal., 12, 712-725 (1975) · Zbl 0322.65010 |
| [11] | Wang, Y. B.; Jia, X. Z.; Cheng, J., A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Problems, 18, 1461-1476 (2002) · Zbl 1041.65024 |
| [12] | Anderssen, R. S.; Hegland, M., For numerical differentiation, dimensionality can be a blessing, Math. Comput., 68, 227, 1121-1141 (1999) · Zbl 0921.65018 |
| [13] | Elden, L.; Berntsson, F.; Reginska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21, 6, 2187-2205 (2000) · Zbl 0959.65107 |
| [14] | Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time Dependent Problems and Difference Methods (1995), Wiley Interscience: Wiley Interscience New York · Zbl 0843.65061 |
| [15] | Van Loan, C. F., Computational Frameworks for the Fast Fourier Transform (1992), SIAM: SIAM Philadelphia · Zbl 0757.65154 |
| [16] | Fu, C.-L., Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation, J. Comput. Appl. Math., 167, 449-463 (2004) · Zbl 1055.65106 |
| [17] | Berntsson, F., A spectral method for solving the sideways heat equation, Inverse problems, 15, 891-906 (1999) · Zbl 0934.35201 |
| [18] | Hào, D. N., A mollification method for ill-posed problems, Numer. Math., 68, 469-506 (1994) · Zbl 0817.65041 |
| [19] | Murio, D. A.; Mejía, C. E.; Zhan, S., Discrete mollification and automatic numerical differentiation, Comput. Math. Appl., 35, 1-16 (1998) · Zbl 0910.65010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
