On stable numerical differentiation

Ramm, Alexander; Smirnova, Alexandra

doi:10.1090/S0025-5718-01-01307-2

On stable numerical differentiation
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by Alexander G. Ramm and Alexandra B. Smirnova PDF

Math. Comp. 70 (2001), 1131-1153 Request permission

Abstract:

A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with a stepsize $h$ being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed.

References

R. G. Airapetyan, A. G. Ramm, A. B. Smirnova, “Continuous methods for solving nonlinear ill-posed problems”, Operator theory and applications, Fields Institute Communications, 25, Amer. Math. Soc., Providence, RI, 2000, pp. 111-138.
R. S. Anderssen and P. Bloomfield, Numerical differentiation procedures for non-exact data, Numer. Math. 22 (1973/74), 157–182. MR 348976, DOI 10.1007/BF01436965
R. S. Anderssen and F. R. de Hoog, Finite difference methods for the numerical differentiation of nonexact data, Computing 33 (1984), no. 3-4, 259–267 (English, with German summary). MR 773928, DOI 10.1007/BF02242272
Jane Cullum, Numerical differentiation and regularization, SIAM J. Numer. Anal. 8 (1971), 254–265. MR 290567, DOI 10.1137/0708026
T. F. Dolgopolova, Finite-dimensional regularization in numerical differentiation of periodic functions, Ural. Gos. Univ. Mat. Zap. 7 (1969/1970), no. tetrad’ 4, 27–33 (Russian). MR 0288953
T. F. Dolgopolova and V. K. Ivanov, On numerical differentiation, Ž. Vyčisl. Mat i Mat. Fiz. 6 (1966), 570–576 (Russian). MR 199958
Yu. V. Egorov and V. A. Kondrat′ev, On a problem of numerical differentiation, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1989), 80–81 (Russian); English transl., Moscow Univ. Math. Bull. 44 (1989), no. 3, 85–87. MR 1029741
V. K. Ivanov, On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR 145 (1962), 270–272 (Russian). MR 0140944
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
È. V. Kolpakova, Numerical solution of the problem of reconstructing the derivative, Differencial′nye Uravnenija i Vyčisl. Mat. Vyp. 6 (1976), part 1, 137–143, 179 (Russian). MR 0488638
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
T. L. Miller and A. G. Ramm, Estimates of derivatives of random functions. II, J. Math. Anal. Appl. 110 (1985), no. 2, 429–435. MR 805265, DOI 10.1016/0022-247X(85)90305-1
C. K. Pallaghy, U. Luttge, Light-induced and $H^*$-ion fluxes and bioelectric phenomena in mesophyll cells of Atriplex Spongiosa, Zeit. fuer Pflanz. 62 (1970), 417-425.
David L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 84–97. MR 134481, DOI 10.1145/321105.321114
R. Qu, A new approach to numerical differentiation and integration, Math. Comput. Modelling 24 (1996), no. 10, 55–68. MR 1426303, DOI 10.1016/S0895-7177(96)00164-1
A. G. Ramm, Numerical differentiation, Izv. Vysš. Učebn. Zaved. Matematika 1968 (1968), no. 11 (78), 131–134 (Russian). MR 0251904
—, Simplified optimal differentiators, Radiotech.i Electron. 17 (1972), 1325-1328.
A. G. Ramm, On simultaneous approximation of a function and its derivative by interpolation polynomials, Bull. London Math. Soc. 9 (1977), no. 3, 283–288. MR 487163, DOI 10.1112/blms/9.3.283
A. G. Ramm, Stable solutions of some ill-posed problems, Math. Methods Appl. Sci. 3 (1981), no. 3, 336–363. MR 657302, DOI 10.1002/mma.1670030125
A. G. Ramm, Estimates of derivatives of random functions. I, J. Math. Anal. Appl. 102 (1984), no. 1, 244–250. MR 751357, DOI 10.1016/0022-247X(84)90217-8
A. G. Ramm, Random fields estimation theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 48, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. MR 1103995
A. G. Ramm, Inequalities for the derivatives, Math. Inequal. Appl. 3 (2000), no. 1, 129–132. MR 1731921, DOI 10.7153/mia-03-14
A. G. Ramm and A. I. Katsevich, The Radon transform and local tomography, CRC Press, Boca Raton, FL, 1996. MR 1384070
T. J. Rivlin, Optimally stable Lagrangian numerical differentiation, SIAM J. Numer. Anal. 12 (1975), no. 5, 712–725. MR 408203, DOI 10.1137/0712053
Herbert E. Salzer, Some problems in optimally stable Lagrangian differentiation, Math. Comp. 28 (1974), 1105–1115. MR 368391, DOI 10.1090/S0025-5718-1974-0368391-0
Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR 0455365
V. V. Vasin, Regularization of a numerical differentiation problem, Ural. Gos. Univ. Mat. Zap. 7 (1969/1970), no. tetrad’ 2, 29–33 (Russian). MR 0280005
V. V. Vasin, The stable computation of the derivative in the space $C(-\infty ,\,\infty )$, Ž. Vyčisl. Mat i Mat. Fiz. 13 (1973), 1383–1389, 1635 (Russian). MR 347087
V. V. Vasin and A. L. Ageev, Ill-posed problems with a priori information, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1995. MR 1374573, DOI 10.1515/9783110900118