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On stable numerical differentiation
Authors:
Alexander G. Ramm and Alexandra B. Smirnova
Journal:
Math. Comp. 70 (2001), 1131-1153
MSC (2000):
Primary 65D25; Secondary 65D05
Posted:
March 9, 2001
MathSciNet review:
1826578
Full-text PDF Free Access
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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  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.
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R.
S. Anderssen and P.
Bloomfield, Numerical differentiation procedures for non-exact
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(50 #1470)
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S. Anderssen and F.
R. de Hoog, Finite difference methods for the numerical
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(1984), no. 3-4, 259–267 (English, with German summary). MR 773928
(86e:65032), http://dx.doi.org/10.1007/BF02242272
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Jane
Cullum, Numerical differentiation and regularization, SIAM J.
Numer. Anal. 8 (1971), 254–265. MR 0290567
(44 #7747)
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T.
F. Dolgopolova, Finite-dimensional regularization in numerical
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7 (1969/1970), no. tetrad’ 4, 27–33
(Russian). MR
0288953 (44 #6148)
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(25 #4357)
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Ian
Knowles and Robert
Wallace, A variational method for numerical differentiation,
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MR
0488638 (58 #8159)
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L. Miller and A.
G. Ramm, Estimates of derivatives of random functions. II, J.
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9 (1962), 84–97. MR 0134481
(24 #B534)
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R.
Qu, A new approach to numerical differentiation and
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no. 10, 55–68. MR 1426303
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A.
G. Ramm, Numerical differentiation, Izv. Vysš.
Učebn. Zaved. Matematika 1968 (1968), no. 11
(78), 131–134 (Russian). MR 0251904
(40 #5130)
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-, Simplified optimal differentiators, Radiotech.i Electron. 17 (1972), 1325-1328.
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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 0487163
(58 #6823)
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A.
G. Ramm, Stable solutions of some ill-posed problems, Math.
Methods Appl. Sci. 3 (1981), no. 3, 336–363. MR 657302
(83i:65102), http://dx.doi.org/10.1002/mma.1670030125
- 20.
A.
G. Ramm, Estimates of derivatives of random functions. I, J.
Math. Anal. Appl. 102 (1984), no. 1, 244–250.
MR 751357
(85m:93040), http://dx.doi.org/10.1016/0022-247X(84)90217-8
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A.
G. Ramm, Random fields estimation theory, Pitman Monographs
and Surveys in Pure and Applied Mathematics, vol. 48, Longman
Scientific & Technical, Harlow, 1990. MR 1103995
(92g:00012)
- 22.
A.
G. Ramm, Inequalities for the derivatives, Math. Inequal.
Appl. 3 (2000), no. 1, 129–132. MR 1731921
(2000i:65026), http://dx.doi.org/10.7153/mia-03-14
- 23.
A.
G. Ramm and A.
I. Katsevich, The Radon transform and local tomography, CRC
Press, Boca Raton, FL, 1996. MR 1384070
(97g:44009)
- 24.
T.
J. Rivlin, Optimally stable Lagrangian numerical
differentiation, SIAM J. Numer. Anal. 12 (1975),
no. 5, 712–725. MR 0408203
(53 #11968)
- 25.
Herbert
E. Salzer, Some problems in optimally stable
Lagrangian differentiation, Math. Comp. 28 (1974),
1105–1115. MR 0368391
(51 #4632), http://dx.doi.org/10.1090/S0025-5718-1974-0368391-0
- 26.
Andrey
N. Tikhonov and Vasiliy
Y. Arsenin, Solutions of ill-posed problems, V. H. Winston
& Sons, Washington, D.C.: John Wiley & Sons, New York, 1977.
Translated from the Russian; Preface by translation editor Fritz John;
Scripta Series in Mathematics. MR 0455365
(56 #13604)
- 27.
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
(43 #5726)
- 28.
V.
V. Vasin, The stable computation of the derivative in the space
𝐶(-∞,∞), Ž. Vyčisl. Mat. i Mat. Fiz.
13 (1973), 1383–1389, 1635 (Russian). MR 0347087
(49 #11807)
- 29.
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
(97j:65100)
- 1.
- 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. CMP 2000:13
- 2.
- R. S. Anderssen, P. Bloomfield, Numerical differentiation procedures for non-exact data, Numer. Math. 22 (1973/74), 157-182. MR 50:1470
- 3.
- R. S. Anderssen, F. R. de Hoog, Finite difference methods for the numerical differentiation of non-exact data, Computing 33 (1984), 259-267. MR 86e:65032
- 4.
- J. Cullum, Numerical differentiation and regularization, SIAM J. Numer. Analysis. 8 (1971), 254-265. MR 44:7747
- 5.
- T. F. Dolgopolova, Finite dimensional regularization in the case of numerical differentiation of periodic functions, Ural. Gos. Univ. Mat. Zap. 7(4) (1970), 27-33. MR 44:6148
- 6.
- T. F. Dolgopolova, V. K. Ivanov, Numerical differentiation, Comp. Math and Math. Physics. 6(3) (1966), 570-576. MR 33:8098
- 7.
- Yu. V. Egorov, V. A. Kondrat'ev, On a problem of numerical differentiation, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1989), 80-81. MR 91a:65048
- 8.
- V. K. Ivanov, On linear problems, which are not well-posed, Doklady Akad. Nauk SSSR. 145(2) (1962), 270-272. MR 25:4357
- 9.
- I. Knowles, R. Wallace, A variational method for numerical differentiation, Numer. Math. 70 (1995), 91-110. MR 96h:65031
- 10.
- E. V. Kolpakova, Numerical solution of the problem of reconstructing the derivative, Differencial'nye Uravnenija i Vychisl. Mat. 6 (1976), 137-143. MR 58:8159
- 11.
- C. Lanczos, Applied Analysis, Englewood Cliffs, N.J. Prentice-Hall, 1956. MR 18:823c
- 12.
- T. Miller, A. G. Ramm, Estimates of the derivatives of random functions II, J. Math. Anal. Appl. 110 (1985), 429-435. MR 87f:93076
- 13.
- C. K. Pallaghy, U. Luttge, Light-induced and -ion fluxes and bioelectric phenomena in mesophyll cells of Atriplex Spongiosa, Zeit. fuer Pflanz. 62 (1970), 417-425.
- 14.
- D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Machinery. 9(1) (1962), 84-97. MR 24:B534
- 15.
- R. Qu, A new approach to numerical differentiation and regularization, Math. Comput. Modelling. 24(10) (1996), 55-68. MR 98b:65024
- 16.
- A. G. Ramm, On numerical differentiation, Izv. Vyssh. Uchebn. Zaved. Mat., 11 (1968), 131-135. MR 40:5130
- 17.
- -, Simplified optimal differentiators, Radiotech.i Electron. 17 (1972), 1325-1328.
- 18.
- -, On simultaneous approximation of a function and its derivative by interpolation polynomials, Bull. Lond. Math. Soc. 9 (1977), 283-288. MR 58:6823
- 19.
- -, Stable solutions of some ill-posed problems, Math. Methods Appl. Sci. 3 (1981), 336-363. MR 83i:65102
- 20.
- -, Estimates of the derivatives of random functions, J. Math. Anal. Appl., 102 (1984), 244-250. MR 85m:93040
- 21.
- -, Random fields estimation theory, Longman Scientific and Wiley, New York, 1990. MR 92g:00012
- 22.
- -, Inequalities for the derivatives, Math. Inequal. Appl., 3 (2000), 129-132. MR 2000i:65026
- 23.
- A. G. Ramm, A. I. Katsevich, The Radon transform and local tomography, CRC Press, Boca Raton, Florida, 1996. MR 97g:44009
- 24.
- T. J. Rivlin, Optimally stable Lagrangian numerical differentiation, SIAM J. Numer. Anal. 12(5) (1975), 712-725. MR 53:11968
- 25.
- H. E. Salzer, Some problems in optimally stable Lagrangian differentiation, Math. Comp. 28 (1974), 1105-1115. MR 51:4632
- 26.
- A. N. Tikhonov, V. Y. Arsenin, Solutions of ill-posed problems, John Wiley and Sons, New York, 1977. MR 56:13604
- 27.
- V. V. Vasin, Regularization of a numerical differentiation problem, Ural. Gos. Univ. Mat. Zap. 7(2) (1969), 29-33. MR 43:5726
- 28.
- V. V. Vasin, The stable evaluation of a derivative in
 , Comp. Math and Math. Physics. 13(6) (1973), 1383-1389. MR 49:11807
- 29.
- V. V. Vasin, A. L. Ageev, Ill-posed problems with a priori information, VSP, Utrecht, 1995. MR 97j:65100
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Additional Information
Alexander G. Ramm
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602
Email:
ramm@math.ksu.edu
Alexandra B. Smirnova
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602
Email:
matabs@zeus.cs.gsu.edu
DOI:
http://dx.doi.org/10.1090/S0025-5718-01-01307-2
PII:
S 0025-5718(01)01307-2
Keywords:
Numerical differentiation,
noisy data,
ill-posed problems,
multi-point methods,
regularization
Received by editor(s):
August 5, 1999
Posted:
March 9, 2001
Article copyright:
© Copyright 2001 American Mathematical Society
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