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Numerical differentiation from a viewpoint of regularization theory
Author(s):
Shuai
Lu;
Sergei
V.
Pereverzev.
Journal:
Math. Comp.
75
(2006),
1853-1870.
MSC (2000):
Primary 65D25;
Secondary 65J20
Posted:
May 15, 2006
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References |
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Additional information
Abstract:
In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.
References:
-
- 1.
- B.Anderssen, F.de Hoog, H.Hegland, A stable finite difference ansatz for higher order differentiation of non-exact data, Bull. Austral. Math. Soc., 22 (1998), 223-232. MR 1642035 (2000d:65037)
- 2.
- J.Cheng, M.Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization, Inverse Problems, 16 (2000), L31-38.MR 1776470 (2001g:65073)
- 3.
- P.Deuflhard, H.W.Engl, O.Scherzer, A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse Problems, 14 (1998), 1081-1106. MR 1654603 (99j:65105)
- 4.
- H.W.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems, Kluwer, Dordrect, 1996. MR 1408680 (97k:65145)
- 5.
- A.Goldenshluger, S.V.Pereverzev, Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations, Probab. Theory Relat. Fields, 118 (2000), 169-186. MR 1790080 (2001h:62055)
- 6.
- C.W.Groetsch, Differentiation of approximately specified functions, Am. Math. Mon., 98 (1991), 847-850. MR 1133003
- 7.
- M.Hanke, O.Scherzer, Inverse Problems light numerical differentiation, Am. Math. Mon., 108 (2001), 512-521. MR 1840657 (2002e:65089)
- 8.
- H.Harbrecht, S.Pereverzev, R.Schneider, Self-regularization by projection for noisy pseudodifferential equations of negative order, Numer. Math., 95 (2003), 123-143. MR 1993941 (2004j:65075)
- 9.
- M.Hegland, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization, Appl. Anal., 59 (1995), 207-223.MR 1378036 (97a:65060)
- 10.
- P.K.Lamm, A survey of regularization methods for first-kind Volterra equations (English summary), Surveys on solution methods for inverse problems, 53-82, Springer, Vienna, 2000. MR 1766739 (2001b:65148)
- 11.
- O.V.Lepskii, A problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl., 36 (1990), 454-466. MR 1091202 (93j:62212)
- 12.
- F.Liu, M.Z.Nashed, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators, Partial Differntial Equations and Application, Dekker, New York (1996), 353-361. MR 1371608 (97m:65112)
- 13.
- P.Mathe, S.V.Pereverzev, Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods, SIAM J. Numer. Analysis, 38 (2001), 1999-2021. MR 1856240 (2002g:62063)
- 14.
- P.Mathe, S.V.Pereverzev, Moduli of continuity for operator valued functions, Numer. Funct. Anal. Optim., 23 (2002), 623-631. MR 1923828 (2003g:47029)
- 15.
- P.Mathe, S.V.Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803. MR 1984890 (2004i:47021)
- 16.
- P.Mathe, S.V.Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 1263-1277.MR 2036530 (2004k:65097)
- 17.
- P.Mathe, S.V.Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Math. Comp. (Accepted).
- 18.
- M.T.Nair, E.Schock, U.Tautenhahn, Morozov's discrepancy principle under general source conditions, Z. Anal. Anwendungen, 22 (2003), 199-214.MR 1962084 (2004a:65069)
- 19.
- A.Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization, SIAM Rev., 40 (1998), 636-666. MR 1642811 (99f:65066)
- 20.
- S.Pereverzev, E.Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Analysis, 43 (2005), 2060-2076. MR 2192331
- 21.
- R.Qu, A new approach to numerical differentiation and integration, Math. Comput. Modelling, 24 (1996), 55-68. MR 1426303 (98b:65024)
- 22.
- A.G.Ramm, A.B.Smirnova, On stable numerical differentiation, Math. Comp., 70 (2001), 1131-1153. MR 1826578 (2002a:65046)
- 23.
- U.Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19 (1998), 377-398. MR 1624930 (99g:65073)
- 24.
- U.Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, 18 (2002), 191-207. MR 1893590 (2002m:47079)
- 25.
- A.Tsybakov, On the best rate of adaptive estimation in some inverse problems, C. R. Acad. Sci. Paris Sèr. I Math., 330 (2000), 835-840. MR 1769957 (2001c:62058)
- 26.
- Y.B.Wang, X.Z.Jia, J.Cheng, A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Problems, 18 (2002), 1461-1476. MR 1955897 (2004b:65086)
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Additional Information:
Shuai
Lu
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, A-4040 Linz, Austria
Email:
shuai.lu@oeaw.ac.at
Sergei
V.
Pereverzev
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Science, Altenbergerstrasse 69, A-4040 Linz, Austria
Email:
sergei.pereverzyev@oeaw.ac.at
DOI:
10.1090/S0025-5718-06-01857-6
PII:
S 0025-5718(06)01857-6
Keywords:
Numerical differentiation,
adaptive regularization,
unknown smoothness,
finite-difference methods,
Tikhonov regularization
Received by editor(s):
November 3, 2004
Received by editor(s) in revised form:
April 19, 2005
Posted:
May 15, 2006
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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