{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:38:33Z","timestamp":1762349913181,"version":"build-2065373602"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this paper, we consider simplified iterated Lavrentiev regularization in Hilbert scales for obtaining stable approximation solution of an ill-posed nonlinear equation of the form\n                    <jats:inline-formula id=\"j_cmam-2023-0163_ineq_9999\">\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-2023-0163_eq_0211.png\"\/>\n                        <jats:tex-math>{F(x)=y}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula id=\"j_cmam-2023-0163_ineq_9998\">\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 mathvariant=\"script\">\ud835\udc9f<\/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>X<\/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-2023-0163_eq_0215.png\"\/>\n                        <jats:tex-math>{F:\\mathcal{D}(F)\\subseteq X\\to X}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a nonlinear operator on Hilbert space\n                    <jats:italic>X<\/jats:italic>\n                    . We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearly conditions on operator\n                    <jats:italic>F<\/jats:italic>\n                    , we prove convergence of method and obtain a rate of convergence result.\n                  <\/jats:p>","DOI":"10.1515\/cmam-2023-0163","type":"journal-article","created":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T05:58:31Z","timestamp":1746338311000},"page":"907-920","source":"Crossref","is-referenced-by-count":0,"title":["Simplified Iterated Lavrentiev Regularization in Hilbert Scales"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9540-8364","authenticated-orcid":false,"given":"Pallavi","family":"Mahale","sequence":"first","affiliation":[{"name":"Department of Mathematics , 29583 Visvesvaraya National Institute of Technology Nagpur , Maharashtra 440010 , India"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-5236-2458","authenticated-orcid":false,"given":"Ankush","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics , 29583 Visvesvaraya National Institute of Technology Nagpur , Maharashtra 440010 , India"}]}],"member":"374","published-online":{"date-parts":[[2025,4,30]]},"reference":[{"key":"2025110513331578619_j_cmam-2023-0163_ref_001","unstructured":"I. K.  Argyros, S.  George and P.  Jidesh,\nIterative regularization methods for ill-posed operator equations in Hilbert scales,\nNonlinear Stud. 24 (2017), no. 2, 257\u2013271."},{"key":"2025110513331578619_j_cmam-2023-0163_ref_002","unstructured":"A. B.  Bakushinski\u012d,\nOn a convergence problem of the iterative-regularized Gauss\u2013Newton method,\nComput. Math. Phys. 32 (1992), no. 9, 1503\u20131509."},{"key":"2025110513331578619_j_cmam-2023-0163_ref_003","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":"2025110513331578619_j_cmam-2023-0163_ref_004","doi-asserted-by":"crossref","unstructured":"A.  Bakushinski\u012d and A.  Smirnova,\nA study of frozen iteratively regularized Gauss\u2013Newton algorithm for nonlinear ill-posed problems under generalized normal solvability condition,\nJ. Inverse Ill-Posed Probl. 28 (2020), no. 2, 275\u2013286.","DOI":"10.1515\/jiip-2019-0099"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_005","doi-asserted-by":"crossref","unstructured":"A.  Binder, M.  Hanke and O.  Scherzer,\nOn the Landweber iteration for nonlinear ill-posed problems,\nJ. Inverse Ill-Posed Probl. 4 (1996), 381\u2013389.","DOI":"10.1515\/jiip.1996.4.5.381"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_006","unstructured":"H.  Egger,\nPreconditioning iterative regularization methods in Hilbert scales,\nDissertation, Johannes Kepler Universit\u00e4t, Linz, 2005."},{"key":"2025110513331578619_j_cmam-2023-0163_ref_007","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":"2025110513331578619_j_cmam-2023-0163_ref_008","doi-asserted-by":"crossref","unstructured":"H.  Egger and A.  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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":"2025110513331578619_j_cmam-2023-0163_ref_012","doi-asserted-by":"crossref","unstructured":"S.  George, S.  Pareth and M.  Kunhanandan,\nNewton Lavrentiev regularization for ill-posed operator equations in Hilbert scales,\nAppl. Math. Comput. 219 (2013), no. 24, 11191\u201311197.","DOI":"10.1016\/j.amc.2013.05.021"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_013","doi-asserted-by":"crossref","unstructured":"M.  Hanke, A.  Neubauer and O.  Scherzer,\nA convergence analysis of the Landweber iteration for nonlinear ill-posed problems,\nNumer. Math. 72 (1995), no. 1, 21\u201337.","DOI":"10.1007\/s002110050158"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_014","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":"2025110513331578619_j_cmam-2023-0163_ref_015","doi-asserted-by":"crossref","unstructured":"J.  Huang, X.  Luo and R.  Zhang,\nA simplified iteratively regularized projection method for nonlinear ill-posed problems,\nJ. Complexity 72 (2022), Article ID 101664.","DOI":"10.1016\/j.jco.2022.101664"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_016","doi-asserted-by":"crossref","unstructured":"Q.  Jin,\nError estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales,\nInverse Problems 16 (2000), no. 1, 187\u2013197.","DOI":"10.1088\/0266-5611\/16\/1\/315"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_017","doi-asserted-by":"crossref","unstructured":"Q.  Jin,\nOn a class of frozen regularized Gauss\u2013Newton methods for nonlinear inverse problems,\nMath. Comp. 79 (2010), no. 272, 2191\u20132211.","DOI":"10.1090\/S0025-5718-10-02359-8"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_018","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":"2025110513331578619_j_cmam-2023-0163_ref_019","doi-asserted-by":"crossref","unstructured":"J.  Jose and M. P.  Rajan,\nA simplified Landweber iteration for solving nonlinear ill-posed problems,\nInt. J. Appl. Comput. Math. 3 (2017), no. 1, S1001\u2013S1018.","DOI":"10.1007\/s40819-017-0395-4"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_020","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":"2025110513331578619_j_cmam-2023-0163_ref_021","doi-asserted-by":"crossref","unstructured":"B.  Kaltenbacher, A.  Neubauer and O.  Scherzer,\nIterative Regularization Methods for Nonlinear Ill-Posed Problems,\nRadon Ser. Comput. Appl. Math. 6,\nWalter de Gruyter, Berlin, 2008.","DOI":"10.1515\/9783110208276"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_022","doi-asserted-by":"crossref","unstructured":"P.  Mahale,\nSimplified iterated Lavrentiev regularization for nonlinear ill-posed monotone operator equations,\nComput. Methods Appl. Math. 17 (2017), no. 2, 269\u2013285.","DOI":"10.1515\/cmam-2016-0044"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_023","doi-asserted-by":"crossref","unstructured":"P.  Mahale and P. K.  Dadsena,\nSimplified generalized Gauss\u2013Newton method for nonlinear ill-posed operator equations in Hilbert scales,\nComput. Methods Appl. Math. 18 (2018), no. 4, 687\u2013702.","DOI":"10.1515\/cmam-2017-0045"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_024","doi-asserted-by":"crossref","unstructured":"P.  Mahale and M. T.  Nair,\nA simplified generalized Gauss\u2013Newton method for nonlinear ill-posed problems,\nMath. Comp. 78 (2009), no. 265, 171\u2013184.","DOI":"10.1090\/S0025-5718-08-02149-2"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_025","doi-asserted-by":"crossref","unstructured":"P.  Mahale and M. T.  Nair,\nIterated Lavrentiev regularization for nonlinear ill-posed problems,\nANZIAM J. 51 (2009), no. 2, 191\u2013217.","DOI":"10.1017\/S1446181109000418"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_026","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,\nSpringer, New Delhi (2015), 159\u2013176.","DOI":"10.1007\/978-81-322-2488-4_13"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_027","unstructured":"M. T.  Nair,\nCompact operators and Hilbert scales in ill-posed problems,\nMath. Student 85 (2016), no. 1\u20132, 45\u201361."},{"key":"2025110513331578619_j_cmam-2023-0163_ref_028","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":"2025110513331578619_j_cmam-2023-0163_ref_029","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":"2025110513331578619_j_cmam-2023-0163_ref_030","doi-asserted-by":"crossref","unstructured":"A.  Neubauer,\nOn Landweber iteration for nonlinear ill-posed problems in Hilbert scales,\nNumer. Math. 85 (2000), no. 2, 309\u2013328.","DOI":"10.1007\/s002110050487"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_031","doi-asserted-by":"crossref","unstructured":"A.  Neubauer,\nSome generalizations for Landweber iteration for nonlinear ill-posed problems in Hilbert scales,\nJ. Inverse Ill-Posed Probl. 24 (2016), no. 4, 393\u2013406.","DOI":"10.1515\/jiip-2015-0086"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_032","doi-asserted-by":"crossref","unstructured":"D.  Pradeep and M. P.  Rajan,\nA simplified Gauss\u2013Newton iterative scheme with an a posteriori parameter choice rule for solving nonlinear ill-posed problems,\nInt. J. Appl. Comput. Math. 2 (2016), no. 1, 97\u2013112.","DOI":"10.1007\/s40819-015-0050-x"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_033","doi-asserted-by":"crossref","unstructured":"O.  Scherzer,\nConvergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems,\nJ. Math. Anal. Appl. 194 (1995), no. 3, 911\u2013933.","DOI":"10.1006\/jmaa.1995.1335"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_034","doi-asserted-by":"crossref","unstructured":"M. E.  Shobha and S.  George,\nNewton type iteration for Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales,\nJ. Math. 2014 (2014), Article ID 965097.","DOI":"10.1155\/2014\/965097"},{"key":"2025110513331578619_j_cmam-2023-0163_ref_035","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":"2025110513331578619_j_cmam-2023-0163_ref_036","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"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0163\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0163\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:35:43Z","timestamp":1762349743000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0163\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,30]]},"references-count":36,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,6,25]]},"published-print":{"date-parts":[[2025,10,1]]}},"alternative-id":["10.1515\/cmam-2023-0163"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0163","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2025,4,30]]}}}