Numerical differentiation from a viewpoint of regularization theory

Lu, Shuai; Pereverzev, Sergei

doi:10.1090/S0025-5718-06-01857-6

Numerical differentiation from a viewpoint of regularization theory
HTML articles powered by AMS MathViewer

by Shuai Lu and Sergei V. Pereverzev;

Math. Comp. 75 (2006), 1853-1870

DOI: https://doi.org/10.1090/S0025-5718-06-01857-6

Published electronically: May 15, 2006

PDF | Request permission

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

Bob Anderssen, Frank de Hoog, and Markus Hegland, A stable finite difference ansatz for higher order differentiation of non-exact data, Bull. Austral. Math. Soc. 58 (1998), no. 2, 223–232. MR 1642035, DOI 10.1017/S0004972700032196
J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization, Inverse Problems 16 (2000), no. 4, L31–L38. MR 1776470, DOI 10.1088/0266-5611/16/4/101
Peter Deuflhard, Heinz W. Engl, and Otmar Scherzer, A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse Problems 14 (1998), no. 5, 1081–1106. MR 1654603, DOI 10.1088/0266-5611/14/5/002
Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1408680, DOI 10.1007/978-94-009-1740-8
Alexander Goldenshluger and Sergei V. Pereverzev, Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations, Probab. Theory Related Fields 118 (2000), no. 2, 169–186. MR 1790080, DOI 10.1007/s440-000-8013-3
C. W. Groetsch, Differentiation of approximately specified functions, Amer. Math. Monthly 98 (1991), no. 9, 847–850. MR 1133003, DOI 10.2307/2324275
Martin Hanke and Otmar Scherzer, Inverse problems light: numerical differentiation, Amer. Math. Monthly 108 (2001), no. 6, 512–521. MR 1840657, DOI 10.2307/2695705
Helmut Harbrecht, Sergei Pereverzev, and Reinhold Schneider, Self-regularization by projection for noisy pseudodifferential equations of negative order, Numer. Math. 95 (2003), no. 1, 123–143. MR 1993941, DOI 10.1007/s00211-002-0417-x
Markus Hegland, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization, Appl. Anal. 59 (1995), no. 1-4, 207–223. MR 1378036, DOI 10.1080/00036819508840400
Patricia K. Lamm, A survey of regularization methods for first-kind Volterra equations, Surveys on solution methods for inverse problems, Springer, Vienna, 2000, pp. 53–82. MR 1766739
O. V. Lepskiĭ, A problem of adaptive estimation in Gaussian white noise, Teor. Veroyatnost. i Primenen. 35 (1990), no. 3, 459–470 (Russian); English transl., Theory Probab. Appl. 35 (1990), no. 3, 454–466 (1991). MR 1091202, DOI 10.1137/1135065
Fengshan Liu and M. Zuhair Nashed, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators, Partial differential equations and applications, Lecture Notes in Pure and Appl. Math., vol. 177, Dekker, New York, 1996, pp. 353–361. MR 1371608
Peter Mathé and Sergei V. Pereverzev, Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods, SIAM J. Numer. Anal. 38 (2001), no. 6, 1999–2021. MR 1856240, DOI 10.1137/S003614299936175X
Peter Mathé and Sergei V. Pereverzev, Moduli of continuity for operator valued functions, Numer. Funct. Anal. Optim. 23 (2002), no. 5-6, 623–631. MR 1923828, DOI 10.1081/NFA-120014755
Peter Mathé and Sergei V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems 19 (2003), no. 3, 789–803. MR 1984890, DOI 10.1088/0266-5611/19/3/319
Peter Mathé and Sergei V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems 19 (2003), no. 6, 1263–1277. MR 2036530, DOI 10.1088/0266-5611/19/6/003
P.Mathe, S.V.Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Math. Comp. (Accepted).
M. T. Nair, E. Schock, and U. Tautenhahn, Morozov’s discrepancy principle under general source conditions, Z. Anal. Anwendungen 22 (2003), no. 1, 199–214. MR 1962084, DOI 10.4171/ZAA/1140
Arnold Neumaier, Solving ill-conditioned and singular linear systems: a tutorial on regularization, SIAM Rev. 40 (1998), no. 3, 636–666. MR 1642811, DOI 10.1137/S0036144597321909
Sergei Pereverzev and Eberhard Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. MR 2192331, DOI 10.1137/S0036142903433819
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
Alexander G. Ramm and Alexandra B. Smirnova, On stable numerical differentiation, Math. Comp. 70 (2001), no. 235, 1131–1153. MR 1826578, DOI 10.1090/S0025-5718-01-01307-2
Ulrich Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim. 19 (1998), no. 3-4, 377–398. MR 1624930, DOI 10.1080/01630569808816834
U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems 18 (2002), no. 1, 191–207. MR 1893590, DOI 10.1088/0266-5611/18/1/313
Alexandre Tsybakov, On the best rate of adaptive estimation in some inverse problems, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 9, 835–840 (English, with English and French summaries). MR 1769957, DOI 10.1016/S0764-4442(00)00278-0
Y. B. Wang, X. Z. Jia, and J. Cheng, A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Problems 18 (2002), no. 6, 1461–1476. MR 1955897, DOI 10.1088/0266-5611/18/6/301