{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T03:39:14Z","timestamp":1777088354965,"version":"3.51.4"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,5,15]],"date-time":"2007-05-15T00:00:00Z","timestamp":1179187200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>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.<\/p>","DOI":"10.1090\/s0025-5718-06-01857-6","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1853-1870","source":"Crossref","is-referenced-by-count":65,"title":["Numerical differentiation from a viewpoint of regularization theory"],"prefix":"10.1090","volume":"75","author":[{"given":"Shuai","family":"Lu","sequence":"first","affiliation":[]},{"given":"Sergei","family":"Pereverzev","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,15]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1017\/S0004972700032196","article-title":"A stable finite difference ansatz for higher order differentiation of non-exact data","volume":"58","author":"Anderssen, Bob","year":"1998","journal-title":"Bull. Austral. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0004-9727","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"L31--L38","DOI":"10.1088\/0266-5611\/16\/4\/101","article-title":"One new strategy for a priori choice of regularizing parameters in Tikhonov\u2019s regularization","volume":"16","author":"Cheng, J.","year":"2000","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"1081","DOI":"10.1088\/0266-5611\/14\/5\/002","article-title":"A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions","volume":"14","author":"Deuflhard, Peter","year":"1998","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"key":"4","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-1740-8","volume-title":"Regularization of inverse problems","volume":"375","author":"Engl, Heinz W.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0792341570"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s440-000-8013-3","article-title":"Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations","volume":"118","author":"Goldenshluger, Alexander","year":"2000","journal-title":"Probab. Theory Related Fields","ISSN":"https:\/\/id.crossref.org\/issn\/0178-8051","issn-type":"print"},{"issue":"9","key":"6","doi-asserted-by":"publisher","first-page":"847","DOI":"10.2307\/2324275","article-title":"Differentiation of approximately specified functions","volume":"98","author":"Groetsch, C. W.","year":"1991","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"6","key":"7","doi-asserted-by":"publisher","first-page":"512","DOI":"10.2307\/2695705","article-title":"Inverse problems light: numerical differentiation","volume":"108","author":"Hanke, Martin","year":"2001","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s00211-002-0417-x","article-title":"Self-regularization by projection for noisy pseudodifferential equations of negative order","volume":"95","author":"Harbrecht, Helmut","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1-4","key":"9","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1080\/00036819508840400","article-title":"Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization","volume":"59","author":"Hegland, Markus","year":"1995","journal-title":"Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-6811","issn-type":"print"},{"key":"10","isbn-type":"print","first-page":"53","article-title":"A survey of regularization methods for first-kind Volterra equations","author":"Lamm, Patricia K.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3211834702"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1137\/1135065","article-title":"A problem of adaptive estimation in Gaussian white noise","volume":"35","author":"Lepski\u012d, O. V.","year":"1990","journal-title":"Teor. Veroyatnost. i Primenen.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-361X","issn-type":"print"},{"key":"12","isbn-type":"print","first-page":"353","article-title":"Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators","author":"Liu, Fengshan","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0824796985"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1999","DOI":"10.1137\/S003614299936175X","article-title":"Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods","volume":"38","author":"Math\u00e9, Peter","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5-6","key":"14","doi-asserted-by":"publisher","first-page":"623","DOI":"10.1081\/NFA-120014755","article-title":"Moduli of continuity for operator valued functions","volume":"23","author":"Math\u00e9, Peter","year":"2002","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1088\/0266-5611\/19\/3\/319","article-title":"Geometry of linear ill-posed problems in variable Hilbert scales","volume":"19","author":"Math\u00e9, Peter","year":"2003","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"issue":"6","key":"16","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.1088\/0266-5611\/19\/6\/003","article-title":"Discretization strategy for linear ill-posed problems in variable Hilbert scales","volume":"19","author":"Math\u00e9, Peter","year":"2003","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"key":"17","unstructured":"P.Mathe, S.V.Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Math. Comp. (Accepted)."},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"199","DOI":"10.4171\/ZAA\/1140","article-title":"Morozov\u2019s discrepancy principle under general source conditions","volume":"22","author":"Nair, M. T.","year":"2003","journal-title":"Z. Anal. Anwendungen","ISSN":"https:\/\/id.crossref.org\/issn\/0232-2064","issn-type":"print"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"636","DOI":"10.1137\/S0036144597321909","article-title":"Solving ill-conditioned and singular linear systems: a tutorial on regularization","volume":"40","author":"Neumaier, Arnold","year":"1998","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"5","key":"20","doi-asserted-by":"publisher","first-page":"2060","DOI":"10.1137\/S0036142903433819","article-title":"On the adaptive selection of the parameter in regularization of ill-posed problems","volume":"43","author":"Pereverzev, Sergei","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"10","key":"21","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/S0895-7177(96)00164-1","article-title":"A new approach to numerical differentiation and integration","volume":"24","author":"Qu, R.","year":"1996","journal-title":"Math. Comput. Modelling","ISSN":"https:\/\/id.crossref.org\/issn\/0895-7177","issn-type":"print"},{"issue":"235","key":"22","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1090\/S0025-5718-01-01307-2","article-title":"On stable numerical differentiation","volume":"70","author":"Ramm, Alexander G.","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3-4","key":"23","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1080\/01630569808816834","article-title":"Optimality for ill-posed problems under general source conditions","volume":"19","author":"Tautenhahn, Ulrich","year":"1998","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1088\/0266-5611\/18\/1\/313","article-title":"On the method of Lavrentiev regularization for nonlinear ill-posed problems","volume":"18","author":"Tautenhahn, U.","year":"2002","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"issue":"9","key":"25","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1016\/S0764-4442(00)00278-0","article-title":"On the best rate of adaptive estimation in some inverse problems","volume":"330","author":"Tsybakov, Alexandre","year":"2000","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"issue":"6","key":"26","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1088\/0266-5611\/18\/6\/301","article-title":"A numerical differentiation method and its application to reconstruction of discontinuity","volume":"18","author":"Wang, Y. B.","year":"2002","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01857-6\/S0025-5718-06-01857-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01857-6\/S0025-5718-06-01857-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:41:00Z","timestamp":1776782460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01857-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,5,15]]},"references-count":26,"journal-issue":{"issue":"256","published-print":{"date-parts":[[2006,10]]}},"alternative-id":["S0025-5718-06-01857-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01857-6","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2006,5,15]]}}}