{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T10:30:30Z","timestamp":1766485830698,"version":"3.40.5"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we study the simplified generalized Gauss\u2013Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form <jats:inline-formula id=\"j_cmam-2017-0045_ineq_9999_w2aab3b7e1034b1b6b1aab1c14b1b1Aa\">\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 stretchy=\"false\">(<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mo stretchy=\"false\">)<\/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:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0045_eq_0267.png\"\/>\n                        <jats:tex-math>{F(x)=y}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> where <jats:inline-formula id=\"j_cmam-2017-0045_ineq_9998_w2aab3b7e1034b1b6b1aab1c14b1b3Aa\">\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 stretchy=\"false\">(<\/m:mo>\n                                       <m:mi>F<\/m:mi>\n                                       <m:mo stretchy=\"false\">)<\/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>Y<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0045_eq_0269.png\"\/>\n                        <jats:tex-math>{F:D(F)\\subseteq X\\to Y}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is a nonlinear operator between Hilbert spaces <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic>. Under suitable nonlinearly conditions on <jats:italic>F<\/jats:italic>, we obtain an order optimal error estimate under the Morozov type stopping rule.<\/jats:p>","DOI":"10.1515\/cmam-2017-0045","type":"journal-article","created":{"date-parts":[[2017,11,3]],"date-time":"2017-11-03T19:15:03Z","timestamp":1509736503000},"page":"687-702","source":"Crossref","is-referenced-by-count":8,"title":["Simplified Generalized Gauss\u2013Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales"],"prefix":"10.1515","volume":"18","author":[{"given":"Pallavi","family":"Mahale","sequence":"first","affiliation":[{"name":"Department of Mathematics , Visvesvaraya National Institute of Technology Nagpur , Nagpur , Maharashtra 440010 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pradeep Kumar","family":"Dadsena","sequence":"additional","affiliation":[{"name":"Government Engineering College Jagdalpur , Bastar, Chattishgarah 494001 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,10,27]]},"reference":[{"key":"2025051309550992439_j_cmam-2017-0045_ref_001_w2aab3b7e1034b1b6b1ab2ab1Aa","unstructured":"A. B.  Bakushinski\u012d,\nOn a convergence problem of the iterative-regularized Gauss\u2013Newton method,\nComput. Math. Math. Phys. 32 (1992), no. 9, 1503\u20131509."},{"key":"2025051309550992439_j_cmam-2017-0045_ref_002_w2aab3b7e1034b1b6b1ab2ab2Aa","unstructured":"A. B.  Bakushinski\u012d,\nIterative methods without saturation for solving degenerate nonlinear operator equations,\nDokl. Akad. Nauk 344 (1995), no. 1, 7\u20138."},{"key":"2025051309550992439_j_cmam-2017-0045_ref_003_w2aab3b7e1034b1b6b1ab2ab3Aa","unstructured":"H.  Egger,\nPreconditioning Iterative Regularization Methods in Hilbert scales,\nDissertation, Johannes Kepler Universit\u00e4t, Linz, 2005."},{"key":"2025051309550992439_j_cmam-2017-0045_ref_004_w2aab3b7e1034b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"H.  Egger,\nSemi-iterative regularization in Hilbert scales,\nSIAM J. Numer. Anal. 44 (2006), no. 1, 66\u201381.","DOI":"10.1137\/040617285"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_005_w2aab3b7e1034b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"H. W.  Engl, M.  Hanke and A.  Neubauer,\nRegularization of Inverse Problems,\nMath. Appl. 375,\nKluwer Academic Publishers, Dordrecht, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_006_w2aab3b7e1034b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"S.  George and M. T.  Nair,\nError bounds and parameter choice strategies for simplified regularization in Hilbert scales,\nIntegral Equations Operator Theory 29 (1997), no. 2, 231\u2013242.","DOI":"10.1007\/BF01191432"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_007_w2aab3b7e1034b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"T.  Hohage,\nLogarithmic convergence rates of the iteratively regularized Gauss\u2013Newton method for an inverse potential and an inverse scattering problem,\nInverse Problems 13 (1997), no. 5, 1279\u20131299.","DOI":"10.1088\/0266-5611\/13\/5\/012"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_008_w2aab3b7e1034b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"Q.  Jin,\nError estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales,\nInverse Problem 16 (1999), 187\u2013197.","DOI":"10.1088\/0266-5611\/16\/1\/315"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_009_w2aab3b7e1034b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Q.  Jin,\nFurther convergence results on the general iteratively regularized Gauss\u2013Newton methods under the discrepancy principle,\nMath. Comp. 82 (2013), no. 283, 1647\u20131665.","DOI":"10.1090\/S0025-5718-2012-02665-2"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_010_w2aab3b7e1034b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"Q.  Jin and U.  Tautenhahn,\nOn the discrepancy principle for some Newton type methods for solving nonlinear inverse problems,\nNumer. Math. 111 (2009), no. 4, 509\u2013558.","DOI":"10.1007\/s00211-008-0198-y"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_011_w2aab3b7e1034b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"B.  Kaltenbacher,\nA posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems,\nNumer. Math. 79 (1998), no. 4, 501\u2013528.","DOI":"10.1007\/s002110050349"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_012_w2aab3b7e1034b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"C.  Kravaris and J. H.  Seinfeld,\nIdentification of parameters in distributed parameter systems by regularization,\nSIAM J. Control Optim. 23 (1985), no. 2, 217\u2013241.","DOI":"10.1137\/0323017"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_013_w2aab3b7e1034b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"S.  Lu, S. V.  Pereverzev, Y.  Shao and U.  Tautenhahn,\nOn the generalized discrepancy principle for Tikhonov regularization in Hilbert scales,\nJ. Integral Equations Appl. 22 (2010), no. 3, 483\u2013517.","DOI":"10.1216\/JIE-2010-22-3-483"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_014_w2aab3b7e1034b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"P.  Mahale and M. T.  Nair,\nA simplified generalized Gauss-Newton method for nonlinear ill-posed problems,\nMath. Comp. 78 (2009), no. 265, 171\u2013184.","DOI":"10.1090\/S0025-5718-08-02149-2"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_015_w2aab3b7e1034b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"M. T.  Nair,\nRole of Hilbert scales in regularization theory,\nSemigroups, Algebras and Operator Theory,\nSpringer Proc. Math. Stat. 142,\nSprinhger, Cham (2015), 159\u2013176.","DOI":"10.1007\/978-81-322-2488-4_13"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_016_w2aab3b7e1034b1b6b1ab2ac16Aa","unstructured":"M. T.  Nair,\nCompact operators and Hilbert scales in ill-posed problems,\nMath. Student 85 (2016), 45\u201361."},{"key":"2025051309550992439_j_cmam-2017-0045_ref_017_w2aab3b7e1034b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"F.  Natterer,\nError bounds for Tikhonov regularization in Hilbert scales,\nAppl. Anal. 18 (1984), no. 1\u20132, 29\u201337.","DOI":"10.1080\/00036818408839508"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_018_w2aab3b7e1034b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"A.  Neubauer,\nTikhonov regularization of nonlinear ill-posed problems in Hilbert scales,\nAppl. Anal. 46 (1992), no. 1\u20132, 59\u201372.","DOI":"10.1080\/00036819208840111"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_019_w2aab3b7e1034b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"U.  Tautenhahn,\nError estimates for regularization methods in Hilbert scales,\nSIAM J. Numer. Anal. 33 (1996), no. 6, 2120\u20132130.","DOI":"10.1137\/S0036142994269411"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_020_w2aab3b7e1034b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"U.  Tautenhahn,\nOn a general regularization scheme for nonlinear ill-posed problems. II. Regularization in Hilbert scales,\nInverse Problems 14 (1998), no. 6, 1607\u20131616.","DOI":"10.1088\/0266-5611\/14\/6\/016"},{"key":"2025051309550992439_j_cmam-2017-0045_ref_021_w2aab3b7e1034b1b6b1ab2ac21Aa","unstructured":"A. N.  Tikhonov and V. Y.  Arsenin,\nSolutions of Ill-Posed Problems,\nV. H. Winston & Sons, Washington, 1977."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/4\/article-p687.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0045\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0045\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T09:55:41Z","timestamp":1747130141000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0045\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,27]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,27]]},"published-print":{"date-parts":[[2018,10,1]]}},"alternative-id":["10.1515\/cmam-2017-0045"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0045","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2017,10,27]]}}}