{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T15:03:36Z","timestamp":1778166216137,"version":"3.51.4"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2012,7,8]],"date-time":"2012-07-08T00:00:00Z","timestamp":1341705600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2013,5]]},"DOI":"10.1007\/s11075-012-9612-8","type":"journal-article","created":{"date-parts":[[2012,7,7]],"date-time":"2012-07-07T07:32:27Z","timestamp":1341646347000},"page":"65-87","source":"Crossref","is-referenced-by-count":171,"title":["Old and new parameter choice rules for discrete ill-posed problems"],"prefix":"10.1007","volume":"63","author":[{"given":"Lothar","family":"Reichel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Rodriguez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2012,7,8]]},"reference":[{"issue":"4","key":"9612_CR1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0041-5553(84)90253-2","volume":"24","author":"AB Bakushinskii","year":"1984","unstructured":"Bakushinskii, A.B.: Remarks on choosing a regularization parameter using quasi-optimality and ratio criterion. USSR Comp. Math. Math. Phys. 24(4), 181\u2013182 (1984)","journal-title":"USSR Comp. Math. Math. Phys."},{"key":"9612_CR2","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1515\/JIIP.2009.002","volume":"17","author":"F Bauer","year":"2009","unstructured":"Bauer, F., Kindermann, S.: Recent results on the quasi-optimality principle. J. Inverse Ill-Posed Probl. 17, 5\u201318 (2009)","journal-title":"J. Inverse Ill-Posed Probl."},{"key":"9612_CR3","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1016\/j.matcom.2011.01.016","volume":"81","author":"F Bauer","year":"2011","unstructured":"Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problem. Math. Comput. Simul. 81, 1795\u20131841 (2011)","journal-title":"Math. Comput. Simul."},{"key":"9612_CR4","doi-asserted-by":"crossref","unstructured":"Bauer, F., Rei\u00df, M.: Regularization independent of the noise-level: an analysis of quasi-optimality. Inverse Probl. 24, 055009 (16 pp) (2008)","DOI":"10.1088\/0266-5611\/24\/5\/055009"},{"key":"9612_CR5","doi-asserted-by":"crossref","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)"},{"key":"9612_CR6","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s11075-008-9163-1","volume":"49","author":"C Brezinski","year":"2008","unstructured":"Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with applications to regularization. Numer. Algorithms 49, 85\u2013104 (2008)","journal-title":"Numer. Algorithms"},{"key":"9612_CR7","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s11075-008-9243-2","volume":"51","author":"C Brezinski","year":"2009","unstructured":"Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms 51, 61\u201376 (2009)","journal-title":"Numer. Algorithms"},{"key":"9612_CR8","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1023\/A:1022383005969","volume":"39","author":"D Calvetti","year":"1999","unstructured":"Calvetti, D., Golub, G.H., Reichel, L.: Estimation of the L-curve via Lanczos bidiagonalization. BIT 39, 603\u2013619 (1999)","journal-title":"BIT"},{"key":"9612_CR9","first-page":"20","volume":"14","author":"D Calvetti","year":"2002","unstructured":"Calvetti, D., Hansen, P.C., Reichel, L.: L-curve curvature bounds via Lanczos bidiagonalization. Electron. Trans. Numer. Anal. 14, 20\u201335 (2002)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9612_CR10","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1023\/A:1021918118380","volume":"42","author":"D Calvetti","year":"2002","unstructured":"Calvetti, D., Lewis, B., Reichel, L.: GMRES, L-curves, and discrete ill-posed problems. BIT 42, 44\u201365 (2002)","journal-title":"BIT"},{"key":"9612_CR11","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1023\/A:1026083619097","volume":"43","author":"D Calvetti","year":"2003","unstructured":"Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT 43, 263\u2013283 (2003)","journal-title":"BIT"},{"key":"9612_CR12","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/S0168-9274(01)00179-9","volume":"43","author":"JL Castellanos","year":"2002","unstructured":"Castellanos, J.L., G\u00f3mez, S., Guerra, V.: The triangle method for finding the corner of the L-curve. Appl. Numer. Math. 43, 359\u2013373 (2002)","journal-title":"Appl. Numer. Math."},{"key":"9612_CR13","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/s10543-010-0266-4","volume":"50","author":"M Donatelli","year":"2010","unstructured":"Donatelli, M.: Fast transforms for high order boundary conditions in deconvolution problems. BIT 50, 559\u2013576 (2010)","journal-title":"BIT"},{"key":"9612_CR14","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/BF01934412","volume":"22","author":"L Eld\u00e9n","year":"1982","unstructured":"Eld\u00e9n, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT 22, 487\u2013501 (1982)","journal-title":"BIT"},{"key":"9612_CR15","doi-asserted-by":"crossref","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":"9612_CR16","first-page":"105","volume-title":"Numerical Analysis 1993","author":"GH Golub","year":"1994","unstructured":"Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Griffiths, D.F., Watson, G.\u00a0A. (eds.) Numerical Analysis 1993, pp.\u00a0105\u2013156. Longman, Essex (1994)"},{"key":"9612_CR17","doi-asserted-by":"crossref","DOI":"10.1515\/9781400833887","volume-title":"Matrices, Moments and Quadrature with Applications","author":"GH Golub","year":"2010","unstructured":"Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton (2010)"},{"key":"9612_CR18","first-page":"3","volume-title":"Workshop on Scientific Computing","author":"GH Golub","year":"1997","unstructured":"Golub, G.H., von Matt, U.: Tikhonov regularization for large scale problems. In: Golub, G.H., Lui, S.H., Luk, F., Plemmons, R. (eds.) Workshop on Scientific Computing, pp. 3\u201326. Springer, New York (1997)"},{"key":"9612_CR19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10618600.1997.10474725","volume":"6","author":"GH Golub","year":"1997","unstructured":"Golub, G.H., von Matt, U.: Generalized cross-validation for large scale problems. J. Comput. Graph. Stat. 6, 1\u201334 (1997)","journal-title":"J. Comput. Graph. Stat."},{"key":"9612_CR20","volume-title":"Conjugate Gradient Type Methods for Ill-Posed Problems","author":"M Hanke","year":"1995","unstructured":"Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Longman, Essex (1995)"},{"key":"9612_CR21","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/BF01731984","volume":"36","author":"M Hanke","year":"1996","unstructured":"Hanke, M.: Limitations of the L-curve method in ill-posed problems. BIT 36, 287\u2013301 (1996)","journal-title":"BIT"},{"key":"9612_CR22","first-page":"253","volume":"3","author":"M Hanke","year":"1993","unstructured":"Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253\u2013315 (1993)","journal-title":"Surv. Math. Ind."},{"key":"9612_CR23","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/BF02219234","volume":"29","author":"PC Hansen","year":"1989","unstructured":"Hansen, P.C.: Regularization, GSVD and truncated GSVD. BIT 29, 491\u2013504 (1989)","journal-title":"BIT"},{"key":"9612_CR24","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1137\/1034115","volume":"34","author":"PC Hansen","year":"1992","unstructured":"Hansen, P.C.: Analysis of the discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561\u2013580 (1992)","journal-title":"SIAM Rev."},{"key":"9612_CR25","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719697","volume-title":"Rank-Deficient and Discrete Ill-Posed Problems","author":"PC Hansen","year":"1998","unstructured":"Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)"},{"key":"9612_CR26","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11075-007-9136-9","volume":"46","author":"PC Hansen","year":"2007","unstructured":"Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189\u2013194 (2007)","journal-title":"Numer. Algorithms"},{"key":"9612_CR27","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.cam.2005.09.026","volume":"198","author":"PC Hansen","year":"2006","unstructured":"Hansen, P.C., Jensen, T.K., Rodriguez, G.: An adaptive pruning algorithm for the discrete L-curve criterion. J. Comput. Appl. Math. 198, 483\u2013492 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"9612_CR28","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.1137\/0914086","volume":"14","author":"PC Hansen","year":"1993","unstructured":"Hansen, P.C., O\u2019Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14, 1487\u20131503 (1993)","journal-title":"SIAM J. Sci. Comput."},{"key":"9612_CR29","doi-asserted-by":"crossref","first-page":"2146","DOI":"10.1016\/j.cam.2011.09.037","volume":"236","author":"U H\u00e4marik","year":"2012","unstructured":"H\u00e4marik, U., Palm, R., Raus, T.: A family of rules for parameter choice in Tikhonov regularization of ill-posed problems with inexact noise level. J. Comput. Appl. Math. 236, 2146\u20132157 (2012)","journal-title":"J. Comput. Appl. Math."},{"key":"9612_CR30","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s10543-009-0239-7","volume":"49","author":"I Hn\u011btynkov\u00e1","year":"2009","unstructured":"Hn\u011btynkov\u00e1, I., Ple\u0161inger, M., Strako\u0161, Z.: The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data. BIT 49, 669\u2013696 (2009)","journal-title":"BIT"},{"key":"9612_CR31","first-page":"233","volume":"38","author":"S Kindermann","year":"2011","unstructured":"Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233\u2013257 (2011)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9612_CR32","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1088\/0266-5611\/22\/5\/021","volume":"22","author":"MA Lukas","year":"2006","unstructured":"Lukas, M.A.: Robust generalized cross-validation for choosing the regularization parameter. Inverse Probl. 22, 1883\u20131902 (2006)","journal-title":"Inverse Probl."},{"key":"9612_CR33","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.cam.2005.05.028","volume":"193","author":"S Morigi","year":"2006","unstructured":"Morigi, S., Reichel, L., Sgallari, F., Zama, F.: Iterative methods for ill-posed problems and semiconvergent sequences. J. Comput. Appl. Math. 193, 157\u2013167 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"9612_CR34","doi-asserted-by":"crossref","first-page":"3974","DOI":"10.1016\/j.laa.2010.08.033","volume":"436","author":"A Neuman","year":"2012","unstructured":"Neuman, A., Reichel, L., Sadok, H.: Implementations of range restricted iterative methods for linear discrete ill-posed problems. Linear Algebra Appl. 436, 3974\u20133990 (2012)","journal-title":"Linear Algebra Appl."},{"key":"9612_CR35","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s11075-011-9491-4","volume":"59","author":"A Neuman","year":"2012","unstructured":"Neuman, A., Reichel, L., Sadok, H.: Algorithms for range restricted iterative methods for linear discrete ill-posed problems. Numer. Algorithms 59, 325\u2013331 (2012)","journal-title":"Numer. Algorithms"},{"key":"9612_CR36","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1145\/355984.355989","volume":"8","author":"CC Paige","year":"1982","unstructured":"Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. Trans. Math. Software 8, 43\u201371 (1982)","journal-title":"Trans. Math. Software"},{"key":"9612_CR37","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1145\/321105.321114","volume":"9","author":"DL Phillips","year":"1962","unstructured":"Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9, 84\u201397 (1962)","journal-title":"J. ACM"},{"key":"9612_CR38","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s11075-008-9244-1","volume":"51","author":"L Reichel","year":"2009","unstructured":"Reichel, L., Rodriguez, G., Seatzu, S.: Error estimates for large-scale ill-posed problems. Numer. Algorithms 51, 341\u2013361 (2009)","journal-title":"Numer. Algorithms"},{"key":"9612_CR39","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/j.cam.2007.01.025","volume":"219","author":"L Reichel","year":"2008","unstructured":"Reichel, L., Sadok, H.: A new L-curve for ill-posed problems. J. Comput. Appl. Math. 219, 493\u2013508 (2008)","journal-title":"J. Comput. Appl. Math."},{"key":"9612_CR40","first-page":"63","volume":"33","author":"L Reichel","year":"2009","unstructured":"Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. 33, 63\u201383 (2009)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9612_CR41","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1137\/S1064827593252672","volume":"17","author":"T Regi\u0144ska","year":"1996","unstructured":"Regi\u0144ska, T.: A regularization parameter in discrete ill-posed problems. SIAM J. Sci. Comput. 17, 740\u2013749 (1996)","journal-title":"SIAM J. Sci. Comput."},{"key":"9612_CR42","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0022-247X(72)90259-4","volume":"37","author":"CB Shaw Jr","year":"1972","unstructured":"Shaw, Jr., C.B.: Improvements of the resolution of an instrument by numerical solution of an integral equation. J. Math. Anal. Appl. 37, 83\u2013112 (1972)","journal-title":"J. Math. Anal. Appl."},{"key":"9612_CR43","volume-title":"Solutions of Ill-Posed Problems","author":"A Tikhonov","year":"1977","unstructured":"Tikhonov, A., Arsenin, V.: Solutions of Ill-Posed Problems. Wiley, New York (1977)"},{"key":"9612_CR44","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1088\/0266-5611\/12\/4\/013","volume":"12","author":"CR Vogel","year":"1996","unstructured":"Vogel, C.R.: Non-convergence of the L-curve regularization parameter selection method. Inverse Probl. 12, 535\u2013547 (1996)","journal-title":"Inverse Probl."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-012-9612-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-012-9612-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-012-9612-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,26]],"date-time":"2024-04-26T15:01:56Z","timestamp":1714143716000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-012-9612-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,7,8]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["9612"],"URL":"https:\/\/doi.org\/10.1007\/s11075-012-9612-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,7,8]]}}}