{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T17:26:28Z","timestamp":1782321988445,"version":"3.54.5"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,10,3]],"date-time":"2023-10-03T00:00:00Z","timestamp":1696291200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,10,3]],"date-time":"2023-10-03T00:00:00Z","timestamp":1696291200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The solution of nonlinear inverse problems is a challenging task in numerical analysis. In most cases, this kind of problems is solved by iterative procedures that, at each iteration, linearize the problem in a neighborhood of the currently available approximation of the solution. The linearized problem is then solved by a direct or iterative method. Among this class of solution methods, the Gauss\u2013Newton method is one of the most popular ones. We propose an efficient implementation of this method for large-scale problems. Our implementation is based on projecting the nonlinear problem into a sequence of nested subspaces, referred to as Generalized Krylov Subspaces, whose dimension increases with the number of iterations, except for when restarts are carried out. When the computation of the Jacobian matrix is expensive, we combine our iterative method with secant (Broyden) updates to further reduce the computational cost. We show convergence of the proposed solution methods and provide a few numerical examples that illustrate their performance.<\/jats:p>","DOI":"10.1007\/s10915-023-02360-w","type":"journal-article","created":{"date-parts":[[2023,10,3]],"date-time":"2023-10-03T09:02:49Z","timestamp":1696323769000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Efficient Implementation of the Gauss\u2013Newton Method Via Generalized Krylov Subspaces"],"prefix":"10.1007","volume":"97","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6456-4150","authenticated-orcid":false,"given":"A.","family":"Buccini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P. D\u00edaz","family":"de Alba","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F.","family":"Pes","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L.","family":"Reichel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,10,3]]},"reference":[{"key":"2360_CR1","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1553\/etna_vol53s113","volume":"53","author":"G Barbarino","year":"2020","unstructured":"Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case. Electron. Trans. Numer. Anal. 53, 113\u2013216 (2020)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"2360_CR2","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1553\/etna_vol53s28","volume":"53","author":"G Barbarino","year":"2020","unstructured":"Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case. Electron. Trans. Numer. Anal. 53, 28\u2013112 (2020)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"2360_CR3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484","volume-title":"Numerical Methods for Least Squares Problems","author":"\u00c5 Bj\u00f6rck","year":"1996","unstructured":"Bj\u00f6rck, \u00c5.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)"},{"issue":"3","key":"2360_CR4","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1137\/0911026","volume":"11","author":"PN Brown","year":"1990","unstructured":"Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11(3), 450\u2013481 (1990)","journal-title":"SIAM J. Sci. Stat. Comput."},{"issue":"2","key":"2360_CR5","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1137\/0804017","volume":"4","author":"PN Brown","year":"1994","unstructured":"Brown, P.N., Saad, Y.: Convergence theory of nonlinear Newton\u2013Krylov algorithms. SIAM J. Optim. 4(2), 297\u2013330 (1994)","journal-title":"SIAM J. Optim."},{"issue":"92","key":"2360_CR6","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1090\/S0025-5718-1965-0198670-6","volume":"19","author":"CG Broyden","year":"1965","unstructured":"Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19(92), 577\u2013593 (1965)","journal-title":"Math. Comput."},{"issue":"1","key":"2360_CR7","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1007\/s10915-021-01727-1","volume":"90","author":"A Buccini","year":"2022","unstructured":"Buccini, A.: Fast alternating direction multipliers method by generalized Krylov subspaces. J. Sci. Comput. 90(1), 60 (2022)","journal-title":"J. Sci. Comput."},{"key":"2360_CR8","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ac3c54","volume":"38","author":"A Buccini","year":"2022","unstructured":"Buccini, A., D\u00edaz\u00a0de\u00a0Alba, P.: A variational non-linear constrained model for the inversion of FDEM data. Inverse Probl. 38, 014001 (2022)","journal-title":"Inverse Probl."},{"key":"2360_CR9","doi-asserted-by":"crossref","unstructured":"Buccini, A., De\u00a0la Cruz\u00a0Cabrera, O., Koukouvinos, C., Mitrouli, M., Reichel, L.: Variable selection in saturated and supersaturated designs via minimization. Commun. Stat. Simul. Comput. (2021) (in press)","DOI":"10.1080\/03610918.2021.1961151"},{"issue":"2","key":"2360_CR10","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/s11565-022-00430-9","volume":"68","author":"A Buccini","year":"2022","unstructured":"Buccini, A., Pragliola, M., Reichel, L., Sgallari, F.: A comparison of parameter choice rules for $$\\ell ^p$$-$$\\ell ^q$$ minimization. Annnali dell\u2019Universit\u00e0 di Ferrara 68(2), 441\u2013463 (2022)","journal-title":"Annnali dell\u2019Universit\u00e0 di Ferrara"},{"issue":"2","key":"2360_CR11","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1007\/s10444-023-10020-8","volume":"49","author":"A Buccini","year":"2023","unstructured":"Buccini, A., Reichel, L.: Limited memory restarted $$\\ell ^p-\\ell ^q$$ minimization methods using generalized Krylov subspaces. Adv. Comput. Math. 49(2), 26 (2023)","journal-title":"Adv. Comput. Math."},{"key":"2360_CR12","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/S0377-0427(00)00405-2","volume":"123","author":"D Calvetti","year":"2000","unstructured":"Calvetti, D., Reichel, L.: Iterative methods for large continuation problems. J. Comput. Appl. Math. 123, 217\u2013240 (2000)","journal-title":"J. Comput. Appl. Math."},{"key":"2360_CR13","first-page":"19","volume":"15","author":"MT Darvishi","year":"2011","unstructured":"Darvishi, M.T., Shin, B.C.: High-order Newton\u2013Krylov methods to solve systems of nonlinear equations. J. KSIAM 15, 19\u201330 (2011)","journal-title":"J. KSIAM"},{"key":"2360_CR14","first-page":"1","volume":"47","author":"GP Deidda","year":"2017","unstructured":"Deidda, G.P., D\u00edaz\u00a0de Alba, P., Rodriguez, G.: Identifying the magnetic permeability in multi-frequency EM data inversion. Electron. Trans. Numer. Anal. 47, 1\u201317 (2017)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"4","key":"2360_CR15","doi-asserted-by":"publisher","first-page":"1313","DOI":"10.1007\/s11075-019-00843-2","volume":"84","author":"GP Deidda","year":"2020","unstructured":"Deidda, G.P., D\u00edaz de\u00a0Alba, P., Fenu, C., Lovicu, G., Rodriguez, G.: FDEMtools: a MATLAB package for FDEM data inversion. Numer. Algorithms 84(4), 1313\u20131327 (2020)","journal-title":"Numer. Algorithms"},{"issue":"7","key":"2360_CR16","doi-asserted-by":"publisher","first-page":"1772","DOI":"10.3390\/rs15071772","volume":"15","author":"GP Deidda","year":"2023","unstructured":"Deidda, G.P., D\u00edaz de\u00a0Alba, P., Pes, F., Rodriguez, G.: Forward electromagnetic induction modelling in a multilayered half-space: an open-source software tool. Remote Sens. 15(7), 1772 (2023)","journal-title":"Remote Sens."},{"key":"2360_CR17","doi-asserted-by":"crossref","unstructured":"Deidda, G.P., D\u00edaz de\u00a0Alba, P., Rodriguez, G., Vignoli, G.: Smooth and sparse inversion of EMI data from multi-configuration measurements. In: 2018 IEEE 4th International Forum on Research and Technology for Society and Industry (RTSI) (RTSI 2018), pp. 213\u2013218. Palermo, Italy (2018)","DOI":"10.1109\/RTSI.2018.8548416"},{"issue":"7","key":"2360_CR18","doi-asserted-by":"publisher","first-page":"945","DOI":"10.1007\/s11004-020-09855-4","volume":"52","author":"GP Deidda","year":"2020","unstructured":"Deidda, G.P., D\u00edaz de\u00a0Alba, P., Rodriguez, G., Vignoli, G.: Inversion of multiconfiguration complex EMI data with minimum gradient support regularization: a case study. Math. Geosci. 52(7), 945\u2013970 (2020)","journal-title":"Math. Geosci."},{"key":"2360_CR19","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/12\/125014","volume":"30","author":"GP Deidda","year":"2014","unstructured":"Deidda, G.P., Fenu, C., Rodriguez, G.: Regularized solution of a nonlinear problem in electromagnetic sounding. Inverse Probl. 30, 125014 (2014)","journal-title":"Inverse Probl."},{"key":"2360_CR20","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971200","volume-title":"Numerical Methods for Unconstrained Optimization and Nonlinear Equations","author":"JE Dennis Jr","year":"1996","unstructured":"Dennis, J.E., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)"},{"key":"2360_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-1740-8","volume-title":"Regularization of Inverse Problems","author":"HW Engl","year":"1996","unstructured":"Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)"},{"key":"2360_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-53679-8","volume-title":"Generalized Locally Toeplitz Sequences: Theory and Applications","author":"C Garoni","year":"2017","unstructured":"Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 1. Springer, Berlin (2017)"},{"key":"2360_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-02233-4","volume-title":"Generalized Locally Toeplitz Sequences: Theory and Applications","author":"C Garoni","year":"2018","unstructured":"Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 2. Springer, Berlin (2018)"},{"issue":"4","key":"2360_CR24","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1137\/0906055","volume":"6","author":"R Glowinski","year":"1985","unstructured":"Glowinski, R., Keller, H.B., Reinhart, L.: Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems. SIAM J. Sci. Stat. Comput. 6(4), 793\u2013832 (1985)","journal-title":"SIAM J. Sci. Stat. Comput."},{"issue":"4","key":"2360_CR25","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1080\/10556789908805730","volume":"10","author":"L Grippo","year":"1999","unstructured":"Grippo, L., Sciandrone, M.: Globally convergent block-coordinate techniques for unconstrained optimization. Optim. Methods Softw. 10(4), 587\u2013637 (1999)","journal-title":"Optim. Methods Softw."},{"issue":"1","key":"2360_CR26","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1088\/0266-5611\/13\/1\/007","volume":"13","author":"M Hanke","year":"1997","unstructured":"Hanke, M.: A regularizing Levenberg\u2013Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl. 13(1), 79 (1997)","journal-title":"Inverse Probl."},{"key":"2360_CR27","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s10543-016-0643-8","volume":"57","author":"G Huang","year":"2017","unstructured":"Huang, G., Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: Majorization-minimization generalized Krylov subspace methods for $$\\ell _p$$-$$\\ell _q$$ optimization applied to image restoration. BIT Numer. Math. 57, 351\u2013378 (2017)","journal-title":"BIT Numer. Math."},{"key":"2360_CR28","doi-asserted-by":"crossref","unstructured":"Kan, Z., Song, N., Peng, H., Chen, B.: Extension of complex step finite difference method to Jacobian-free Newton\u2013Krylov. J. Comput. Appl. Math. 399, Art. 113,732 (2022)","DOI":"10.1016\/j.cam.2021.113732"},{"key":"2360_CR29","volume-title":"Lectures on Numerical Methods in Bifurcation Problems","author":"HB Keller","year":"1987","unstructured":"Keller, H.B.: Lectures on Numerical Methods in Bifurcation Problems. Springer, Berlin (1987)"},{"key":"2360_CR30","doi-asserted-by":"publisher","first-page":"2845","DOI":"10.1016\/j.laa.2011.07.019","volume":"436","author":"J Lampe","year":"2012","unstructured":"Lampe, J., Reichel, L., Voss, H.: Large-scale Tikhonov regularization via reduction by orthogonal projection. Linear Algebra Appl. 436, 2845\u20132865 (2012)","journal-title":"Linear Algebra Appl."},{"key":"2360_CR31","doi-asserted-by":"publisher","first-page":"S30","DOI":"10.1137\/140967982","volume":"37","author":"A Lanza","year":"2015","unstructured":"Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: A generalized Krylov subspace method for $$\\ell _p$$-$$\\ell _q$$ minimization. SIAM J. Sci. Comput. 37, S30\u2013S50 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"2360_CR32","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1553\/etna_vol53s459","volume":"53","author":"F Pes","year":"2020","unstructured":"Pes, F., Rodriguez, G.: The minimal-norm Gauss\u2013Newton method and some of its regularized variants. Electron. Trans. Numer. Anal. 53, 459\u2013480 (2020)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"2360_CR33","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.apnum.2021.09.002","volume":"171","author":"F Pes","year":"2022","unstructured":"Pes, F., Rodriguez, G.: A doubly relaxed minimal-norm Gauss\u2013Newton method for underdetermined nonlinear least-squares problems. Appl. Numer. Math. 171, 233\u2013248 (2022)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"2360_CR34","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1007\/BF01930989","volume":"19","author":"A Ruhe","year":"1979","unstructured":"Ruhe, A.: Accelerated Gauss\u2013Newton algorithms for nonlinear least squares problems. BIT Numer. Math. 19(3), 356\u2013367 (1979)","journal-title":"BIT Numer. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02360-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-023-02360-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02360-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T11:09:16Z","timestamp":1697886556000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-023-02360-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,3]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,11]]}},"alternative-id":["2360"],"URL":"https:\/\/doi.org\/10.1007\/s10915-023-02360-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,3]]},"assertion":[{"value":"26 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 August 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 October 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests. during the current study.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"44"}}