{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T19:43:07Z","timestamp":1775936587780,"version":"3.50.1"},"reference-count":26,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,1,1]]},"abstract":"Abstract<\/jats:title>\n \n In 2010, Qinian Jin considered a regularized Levenberg\u2013Marquardt\nmethod in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation\n \n \n \n \n \n F<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n =<\/m:mo>\n y<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {F(x)=y}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n \n \n F<\/m:mi>\n :<\/m:mo>\n \n \n D<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n F<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2282<\/m:mo>\n X<\/m:mi>\n \u2192<\/m:mo>\n Y<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {F:D(F)\\subset X\\rightarrow Y}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a nonlinear operator between Hilbert spaces\n X<\/jats:italic>\n and\n Y<\/jats:italic>\n and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg\u2013Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration\n \n \n \n \n {<\/m:mo>\n \n x<\/m:mi>\n n<\/m:mi>\n \u03b4<\/m:mi>\n <\/m:msubsup>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\{x_{n}^{\\delta}\\}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is defined as\n <\/jats:p>\n \n \n \n \n \n \n \n x<\/m:mi>\n \n n<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n \u03b4<\/m:mi>\n <\/m:msubsup>\n =<\/m:mo>\n \n \n x<\/m:mi>\n n<\/m:mi>\n \u03b4<\/m:mi>\n <\/m:msubsup>\n -<\/m:mo>\n \n \n \n (<\/m:mo>\n \n \n \n \u03b1<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n I<\/m:mi>\n <\/m:mrow>\n +<\/m:mo>\n \n \n F<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n \n (<\/m:mo>\n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n *<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n F<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n F<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n \n (<\/m:mo>\n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n *<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n F<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n x<\/m:mi>\n n<\/m:mi>\n \u03b4<\/m:mi>\n <\/m:msubsup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n -<\/m:mo>\n \n y<\/m:mi>\n \u03b4<\/m:mi>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n .<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n x^{\\delta}_{n+1}=x^{\\delta}_{n}-(\\alpha_{n}I+F^{\\prime}(x_{0})^{*}F^{\\prime}(x%\n_{0}))^{-1}F^{\\prime}(x_{0})^{*}(F(x^{\\delta}_{n})-y^{\\delta}).<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n <\/jats:p>\n \n Here\n \n \n \n \n {<\/m:mo>\n \n \u03b1<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\{\\alpha_{n}\\}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a decreasing sequence of nonnegative numbers which converges to zero,\n \n \n \n \n \n F<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {F^{\\prime}(x_{0})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n denotes the Fr\u00e9chet derivative of\n F<\/jats:italic>\n at an initial guess\n \n \n \n \n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n \u2208<\/m:mo>\n \n D<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n F<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {x_{0}\\in D(F)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for the exact solution\n \n \n \n \n x<\/m:mi>\n \u2020<\/m:mo>\n <\/m:msup>\n <\/m:math>\n \n {x^{\\dagger}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n \n \n (<\/m:mo>\n \n \n F<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n *<\/m:mo>\n <\/m:msup>\n <\/m:math>\n \n {(F^{\\prime}(x_{0}))^{*}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n denote the adjoint of\n \n \n \n \n \n F<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {F^{\\prime}(x_{0})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In our proposed method, we need to calculate Fr\u00e9chet derivative of\n F<\/jats:italic>\n only at an initial guess\n \n \n \n \n x<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n {x_{0}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Hence, it is more economic to use in numerical computations than the Levenberg\u2013Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method.\n <\/jats:p>","DOI":"10.1515\/cmam-2022-0006","type":"journal-article","created":{"date-parts":[[2022,7,21]],"date-time":"2022-07-21T13:01:25Z","timestamp":1658408485000},"page":"251-276","source":"Crossref","is-referenced-by-count":2,"title":["Simplified Levenberg\u2013Marquardt Method in Hilbert Spaces"],"prefix":"10.1515","volume":"23","author":[{"given":"Pallavi","family":"Mahale","sequence":"first","affiliation":[{"name":"Department of Mathematics , Visvesvaraya National Institute of Technology Nagpur , Maharashtra - 440010 , India"}]},{"given":"Farheen M.","family":"Shaikh","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Visvesvaraya National Institute of Technology Nagpur , Maharashtra - 440010 , India"}]}],"member":"374","published-online":{"date-parts":[[2022,7,22]]},"reference":[{"key":"2026041118490624699_j_cmam-2022-0006_ref_001","unstructured":"A. B. Bakushinski\u012d,\nThe problems of the convergence of the iteratively regularized Gauss\u2013Newton method,\nComput. Math. Math. Phys. 32 (1992), 1353\u20131359."},{"key":"2026041118490624699_j_cmam-2022-0006_ref_002","doi-asserted-by":"crossref","unstructured":"B. Blaschke, A. Neubauer and O. Scherzer,\nOn convergence rates for the iteratively regularized Gauss\u2013Newton method,\nIMA J. Numer. Anal. 17 (1997), no. 3, 421\u2013436.","DOI":"10.1093\/imanum\/17.3.421"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_003","doi-asserted-by":"crossref","unstructured":"C. Clason and V. H. Nhu,\nBouligand\u2013Landweber iteration for a non-smooth ill-posed problem,\nNumer. Math. 142 (2019), no. 4, 789\u2013832.","DOI":"10.1007\/s00211-019-01038-6"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_004","doi-asserted-by":"crossref","unstructured":"C. Clason and V. H. Nhu,\nBouligand\u2013Levenberg\u2013Marquardt iteration for a non-smooth ill-posed inverse problem,\nElectron. Trans. Numer. Anal. 51 (2019), 274\u2013314.","DOI":"10.1553\/etna_vol51s274"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_005","unstructured":"H. Egger,\nPreconditioning iterative regularization methods in Hilbert scales,\nDissertation, Johannes Kepler Universit\u00e4t, Linz, 2005."},{"key":"2026041118490624699_j_cmam-2022-0006_ref_006","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke and A. Neubauer,\nRegularization of Inverse Problems,\nMath. Appl. 375,\nKluwer Academic, Dordrecht, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_007","doi-asserted-by":"crossref","unstructured":"M. Hanke,\nA regularizing Levenberg\u2013Marquardt scheme, with applications to inverse groundwater filtration problems,\nInverse Problems 13 (1997), no. 1, 79\u201395.","DOI":"10.1088\/0266-5611\/13\/1\/007"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_008","doi-asserted-by":"crossref","unstructured":"M. Hanke and C. W. Groetsch,\nNonstationary iterated Tikhonov regularization,\nJ. Optim. Theory Appl. 98 (1998), no. 1, 37\u201353.","DOI":"10.1023\/A:1022680629327"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_009","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":"2026041118490624699_j_cmam-2022-0006_ref_010","doi-asserted-by":"crossref","unstructured":"M. Hochbruck and M. H\u00f6nig,\nOn the convergence of a regularizing Levenberg\u2013Marquardt scheme for nonlinear ill-posed problems,\nNumer. Math. 115 (2010), no. 1, 71\u201379.","DOI":"10.1007\/s00211-009-0268-9"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_011","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":"2026041118490624699_j_cmam-2022-0006_ref_012","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":"2026041118490624699_j_cmam-2022-0006_ref_013","doi-asserted-by":"crossref","unstructured":"Q. Jin,\nOn a regularized Levenberg\u2013Marquardt method for solving nonlinear inverse problems,\nNumer. Math. 115 (2010), no. 2, 229\u2013259.","DOI":"10.1007\/s00211-009-0275-x"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_014","doi-asserted-by":"crossref","unstructured":"Q. Jin,\nA general convergence analysis of some Newton-type methods for nonlinear inverse problems,\nSIAM J. Numer. Anal. 49 (2011), no. 2, 549\u2013573.","DOI":"10.1137\/100804231"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_015","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":"2026041118490624699_j_cmam-2022-0006_ref_016","doi-asserted-by":"crossref","unstructured":"Q.-N. Jin,\nOn the iteratively regularized Gauss\u2013Newton method for solving nonlinear ill-posed problems,\nMath. Comp. 69 (2000), no. 232, 1603\u20131623.","DOI":"10.1090\/S0025-5718-00-01199-6"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_017","doi-asserted-by":"crossref","unstructured":"Q.-N. Jin and Z.-Y. Hou,\nOn an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems,\nNumer. Math. 83 (1999), no. 1, 139\u2013159.","DOI":"10.1007\/s002110050442"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_018","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher,\nSome Newton-type methods for the regularization of nonlinear ill-posed problems,\nInverse Problems 13 (1997), no. 3, 729\u2013753.","DOI":"10.1088\/0266-5611\/13\/3\/012"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_019","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":"2026041118490624699_j_cmam-2022-0006_ref_020","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher,\nOn Broyden\u2019s method for the regularization of nonlinear ill-posed problems,\nNumer. Funct. Anal. Optim. 19 (1998), no. 7\u20138, 807\u2013833.","DOI":"10.1080\/01630569808816860"},{"key":"2026041118490624699_j_cmam-2022-0006_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":"2026041118490624699_j_cmam-2022-0006_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":"2026041118490624699_j_cmam-2022-0006_ref_023","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":"2026041118490624699_j_cmam-2022-0006_ref_024","doi-asserted-by":"crossref","unstructured":"A. Rieder,\nOn the regularization of nonlinear ill-posed problems via inexact Newton iterations,\nInverse Problems 15 (1999), no. 1, 309\u2013327.","DOI":"10.1088\/0266-5611\/15\/1\/028"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_025","doi-asserted-by":"crossref","unstructured":"A. Rieder,\nOn convergence rates of inexact Newton regularizations,\nNumer. Math. 88 (2001), no. 2, 347\u2013365.","DOI":"10.1007\/PL00005448"},{"key":"2026041118490624699_j_cmam-2022-0006_ref_026","doi-asserted-by":"crossref","unstructured":"K. Schm\u00fcdgen,\nUnbounded Self-Adjoint Operators on Hilbert Space,\nSpringer, Dordrecht, 2012.","DOI":"10.1007\/978-94-007-4753-1"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0006\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0006\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T18:49:14Z","timestamp":1775933354000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0006\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,22]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,7,22]]},"published-print":{"date-parts":[[2023,1,1]]}},"alternative-id":["10.1515\/cmam-2022-0006"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0006","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,22]]}}}