{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,12]],"date-time":"2026-04-12T08:35:12Z","timestamp":1775982912975,"version":"3.50.1"},"reference-count":20,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Mahale and Nair [12] considered an iterated\nform of Lavrentiev regularization\nfor obtaining stable approximate solutions for\nill-posed nonlinear equations of the form <jats:inline-formula id=\"j_cmam-2016-0044_ineq_9999_w2aab3b7e2123b1b6b1aab1c13b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mi>F<\/m:mi>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>y<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${F(x)=y}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, where\n<jats:inline-formula id=\"j_cmam-2016-0044_ineq_9998_w2aab3b7e2123b1b6b1aab1c13b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>F<\/m:mi>\n                              <m:mo>:<\/m:mo>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:mi>D<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mi>F<\/m:mi>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>\u2286<\/m:mo>\n                                 <m:mi>X<\/m:mi>\n                                 <m:mo>\u2192<\/m:mo>\n                                 <m:mi>X<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${F:D(F)\\subseteq X\\to X}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is a nonlinear monotone operator and <jats:italic>X<\/jats:italic> is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate\nunder a general source condition. However, the iterations defined\nin [12] require calculation of Fr\u00e9chet derivatives at each iteration. In this paper,\nwe consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Fr\u00e9chet derivative only at the point <jats:inline-formula id=\"j_cmam-2016-0044_ineq_9997_w2aab3b7e2123b1b6b1aab1c13b1c11Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>x<\/m:mi>\n                              <m:mn>0<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>${x_{0}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, i.e., at the initial approximation of the exact solution <jats:inline-formula id=\"j_cmam-2016-0044_ineq_9996_w2aab3b7e2123b1b6b1aab1c13b1c13Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>x<\/m:mi>\n                              <m:mo>\u2020<\/m:mo>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>${x^{\\dagger}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Fr\u00e9chet derivative at the point <jats:inline-formula id=\"j_cmam-2016-0044_ineq_9995_w2aab3b7e2123b1b6b1aab1c13b1c15Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>x<\/m:mi>\n                              <m:mn>0<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>${x_{0}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\ninstead at the unknown exact solution <jats:inline-formula id=\"j_cmam-2016-0044_ineq_9994_w2aab3b7e2123b1b6b1aab1c13b1c17Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>x<\/m:mi>\n                              <m:mo>\u2020<\/m:mo>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>${x^{\\dagger}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> as in [12].<\/jats:p>","DOI":"10.1515\/cmam-2016-0044","type":"journal-article","created":{"date-parts":[[2017,1,11]],"date-time":"2017-01-11T10:00:53Z","timestamp":1484128853000},"page":"269-285","source":"Crossref","is-referenced-by-count":8,"title":["Simplified Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Monotone Operator Equations"],"prefix":"10.1515","volume":"17","author":[{"given":"Pallavi","family":"Mahale","sequence":"first","affiliation":[{"name":"Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur, Maharashtra 440010, India"}]}],"member":"374","published-online":{"date-parts":[[2017,1,11]]},"reference":[{"key":"2023033114222318063_j_cmam-2016-0044_ref_001_w2aab3b7e2123b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Bakushinsky A. and Smirnova A.,\nOn application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems,\nNumer. Funct. Anal. Optim. 26 (2005), 35\u201348.","DOI":"10.1081\/NFA-200051631"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_002_w2aab3b7e2123b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"Bakushinsky A. and Smirnova A.,\nA posteriori stopping rule for regularized fixed point iterations,\nNonlinear Anal. 64 (2006), 1255\u20131261.","DOI":"10.1016\/j.na.2005.06.031"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_003_w2aab3b7e2123b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"Bakushinsky A. and Smirnova A.,\nIterative regularization and generalized discrepancy princiciple for monotone operator equations,\nNumer. Funct. Anal. Optim. 28 (2007), no. 1, 13\u201325.","DOI":"10.1080\/01630560701190315"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_004_w2aab3b7e2123b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"Blaschke B., Neubauer A. and Scherzer O.,\nOn convergence rates for the iteratively regularized Gauss\u2013Newton method,\nIMA J. Numer. Anal. 17 (1997), 421\u2013436.","DOI":"10.1093\/imanum\/17.3.421"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_005_w2aab3b7e2123b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"Deimling K.,\nNonlinear Functional Analysis,\nSpringer, New York, 1985.","DOI":"10.1007\/978-3-662-00547-7"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_006_w2aab3b7e2123b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"Engl H. W., Hanke M. and Neubauer A.,\nRegularization of Inverse Problems,\nKluwer, Dordrecht, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_007_w2aab3b7e2123b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"Engl H. W., Kunisch K. and Neubauer A.,\nConvergence rates for Tikhonov regularization of nonlinear ill-posed problems,\nInverse Problems 5 (1989), 523\u2013540.","DOI":"10.1088\/0266-5611\/5\/4\/007"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_008_w2aab3b7e2123b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"Hohage T.,\nRegularization of exponentially ill-posed problems,\nNumer. Funct. Anal. Optim. 21 (2000), 439\u2013464.","DOI":"10.1080\/01630560008816965"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_009_w2aab3b7e2123b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Jin Q. and Hou Z.,\nOn the choice of the regularization parameter for ordinary and iterated Tikhonov regularization of nonlinear ill-posed problems,\nInverse Problems 13 (1997), 815\u2013827.","DOI":"10.1088\/0266-5611\/13\/3\/016"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_010_w2aab3b7e2123b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"Jin Q. and Hou Z.,\nOn an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems,\nNumer. Math. 83 (1999), 139\u2013159.","DOI":"10.1007\/s002110050442"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_011_w2aab3b7e2123b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"Kaltenbacher B.,\nA posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems,\nNumer. Math. 79 (1998), 501\u2013528.","DOI":"10.1007\/s002110050349"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_012_w2aab3b7e2123b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"Mahale P. and Nair M. T.,\nIterated Lavrentiev regularization for nonlinear ill-posed problems,\nANZIAM J. 51 (2009), 191\u2013217.","DOI":"10.1017\/S1446181109000418"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_013_w2aab3b7e2123b1b6b1ab2ac13Aa","unstructured":"Mahale P. and Nair M. T.,\nLavrentiev regularization of nonlinear ill-posed equations under general source condition,\nJ. Nonlinear Anal. Optim. 4 (2013), no. 2, 193\u2013204."},{"key":"2023033114222318063_j_cmam-2016-0044_ref_014_w2aab3b7e2123b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"Nair M. T. and Tautenhahn U.,\nLavrentiev regularization for linear ill-posed problems under general source conditions,\nJ. Anal. Appl. 23 (2004), 167\u2013185.","DOI":"10.4171\/ZAA\/1192"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_015_w2aab3b7e2123b1b6b1ab2ac15Aa","unstructured":"Santhosh G. and Elmanhdy A. I.,\nAn analysis of Lavrentiev regularization for nonlinear ill-posed problems using an iterative regularization method,\nInt. J. Comput. Appl. Math 3 (2010), 369\u2013381."},{"key":"2023033114222318063_j_cmam-2016-0044_ref_016_w2aab3b7e2123b1b6b1ab2ac16Aa","unstructured":"Santhosh G. and Nair M. T.,\nA derivative-free iterative method for nonlinear ill-posed equations with monotone operators,\nJ. Inverse Ill-Posed Probl. (2016), 10.1515\/jiip-2014-0049."},{"key":"2023033114222318063_j_cmam-2016-0044_ref_017_w2aab3b7e2123b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"Scherzer O., Engl H. W. and Kunisch K.,\nOptimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems,\nSIAM J. Numer. Anal. 30 (1993), 1796\u20131838.","DOI":"10.1137\/0730091"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_018_w2aab3b7e2123b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"Semenova E. V.,\nLavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators,\nComput. Methods Appl. Math. 10 (2010), 444\u2013454.","DOI":"10.2478\/cmam-2010-0026"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_019_w2aab3b7e2123b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"Tautenhahn U.,\nOn the method of Lavrentiev regularization for nonlinear ill-posed problems,\nInverse Problems 18 (2002), 191\u2013207.","DOI":"10.1088\/0266-5611\/18\/1\/313"},{"key":"2023033114222318063_j_cmam-2016-0044_ref_020_w2aab3b7e2123b1b6b1ab2ac20Aa","unstructured":"Tautenhahn U.,\nLavrentiev regularization of nonlinear ill-posed problems,\nVietnam J. Math. 32 (2004), 29\u201341."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/17\/2\/article-p269.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0044\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0044\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T20:40:02Z","timestamp":1680295202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0044\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,11]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,1,10]]},"published-print":{"date-parts":[[2017,4,1]]}},"alternative-id":["10.1515\/cmam-2016-0044"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2016-0044","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,1,11]]}}}