{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T02:08:08Z","timestamp":1776305288866,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,7,23]],"date-time":"2024-07-23T00:00:00Z","timestamp":1721692800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,23]],"date-time":"2024-07-23T00:00:00Z","timestamp":1721692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003945","name":"Link\u00f6ping University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100003945","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider a stationary iteration for solving a linear system of arbitrary order. The method includes, e.g. Kaczmarz iteration, the Landweber iteration and the SOR (Gauss-Seidel) iteration. A study of the behavior of the iterates, both theoretically and experimentally, is performed. In particular we compare the behavior with and without noise in the data. The results give insight into the interplay between noise free and noisy iterates. For comparision we also included a Krylov type method CGLS in the experiments. As expected CGLS works well for noise free data but also tends to amplify the noise faster than the other methods, thus making it more critical when to stop the iterations.<\/jats:p>","DOI":"10.1007\/s11075-024-01890-0","type":"journal-article","created":{"date-parts":[[2024,7,23]],"date-time":"2024-07-23T07:01:47Z","timestamp":1721718107000},"page":"603-625","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Noise propagation in linear stationary iterations"],"prefix":"10.1007","volume":"99","author":[{"given":"Tommy","family":"Elfving","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,23]]},"reference":[{"key":"1890_CR1","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1177\/016173468400600107","volume":"6","author":"AH Andersen","year":"1984","unstructured":"Andersen, A.H., Kak, A.C.: Simultaneous algebraic reconstruction technique (sart): A superior implementation of the art algorithm. Ultrason. Imaging 6, 81\u201394 (1984)","journal-title":"Ultrason. Imaging"},{"key":"1890_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511624100","volume-title":"Iterative solution methods","author":"O Axelsson","year":"1994","unstructured":"Axelsson, O.: Iterative solution methods. Cambridge University Press, Cambridge (1994)"},{"key":"1890_CR3","doi-asserted-by":"crossref","unstructured":"Benning, M., Burger, M.: Modern regularization methods for inverse problems. Acta Numerica, pp. 1\u2013111. (2018)","DOI":"10.1017\/S0962492918000016"},{"key":"1890_CR4","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. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996)"},{"issue":"4","key":"1890_CR5","doi-asserted-by":"publisher","first-page":"444","DOI":"10.1137\/1023097","volume":"23","author":"Y Censor","year":"1981","unstructured":"Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23(4), 444\u2013466 (1981)","journal-title":"SIAM Rev."},{"issue":"6","key":"1890_CR6","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/26\/6\/065008","volume":"26","author":"Y Censor","year":"2010","unstructured":"Censor, Y., Davidi, R., Herman, G.T.: Perturbation resilience and superiorization of iterative algorithms. Inverse Problems 26(6), 065008 (2010)","journal-title":"Inverse Problems"},{"issue":"1","key":"1890_CR7","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/BF01396307","volume":"41","author":"Y Censor","year":"1983","unstructured":"Censor, Y., Eggermont, P.P.B., Gordon, D.: Strong underrelaxation in Kaczmarz\u2019s method for inconsistent systems. Numer. Math. 41(1), 83\u201392 (1983)","journal-title":"Numer. Math."},{"issue":"1","key":"1890_CR8","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1137\/050639399","volume":"30","author":"Y Censor","year":"2007","unstructured":"Censor, Y., Elfving, T., Herman, G.T., Nikazad, T.: On diagonally relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30(1), 473\u2013504 (2007)","journal-title":"SIAM J. Sci. Comput."},{"key":"1890_CR9","doi-asserted-by":"publisher","first-page":"1050","DOI":"10.1109\/42.959302","volume":"20","author":"Y Censor","year":"2001","unstructured":"Censor, Y., Gordon, D., Gordon, R.: Bicav: An inherently parallel algorithm for sparse systems with pixel-dependent weighting. IEEE Trans. Med. Imaging 20, 1050\u20131060 (2001)","journal-title":"IEEE Trans. Med. Imaging"},{"key":"1890_CR10","first-page":"326","volume-title":"Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari","author":"G Cimmino","year":"1938","unstructured":"Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, pp. 326\u2013333. La Ricerca Scientifica, XVI(II) (1938)"},{"key":"1890_CR11","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/0024-3795(81)90139-7","volume":"40","author":"PPB Eggermont","year":"1981","unstructured":"Eggermont, P.P.B., Herman, G.T., Lent, A.: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra Appl. 40, 37\u201367 (1981)","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"1890_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01396365","volume":"35","author":"T Elfving","year":"1980","unstructured":"Elfving, T.: Block-iterative methods for consistent and inconsistent linear equations. Numer. Math. 35(1), 1\u201312 (1980)","journal-title":"Numer. Math."},{"issue":"1","key":"1890_CR13","first-page":"219","volume":"2","author":"T Elfving","year":"2018","unstructured":"Elfving, T.: Row and column based iterations. J. Appl. Anal. Optimization. 2(1), 219\u2013236 (2018)","journal-title":"J. Appl. Anal. Optimization."},{"issue":"1","key":"1890_CR14","doi-asserted-by":"publisher","first-page":"A573","DOI":"10.1137\/17M1133828","volume":"40","author":"T Elfving","year":"2018","unstructured":"Elfving, T., Hansen, P.C.: Unmatched projector\/backprojector pairs: perturbation and convergence analysis. SIAM J. Sci. Comput. 40(1), A573\u2013A591 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"1890_CR15","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/5\/055007","volume":"30","author":"T Elfving","year":"2014","unstructured":"Elfving, T., Hansen, P.C., Nikazad, T.: Semi-convergence properties of Kaczmarz\u2019s method. Inverse Problems 30(5), 055007 (2014)","journal-title":"Inverse Problems"},{"issue":"3","key":"1890_CR16","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1007\/s11075-016-0176-x","volume":"74","author":"T Elfving","year":"2017","unstructured":"Elfving, T., Hansen, P.C., Nikazad, T.: Convergence analysis for column-action methods in image reconstruction. Numer. Algorithms 74(3), 905\u2013924 (2017)","journal-title":"Numer. Algorithms"},{"issue":"4","key":"1890_CR17","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1088\/0266-5611\/23\/4\/004","volume":"23","author":"T Elfving","year":"2007","unstructured":"Elfving, T., Nikazad, T.: Stopping rules for Landweber-type interation. Inverse Problems 23(4), 1417\u20131432 (2007)","journal-title":"Inverse Problems"},{"issue":"11","key":"1890_CR18","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/11\/115011","volume":"25","author":"T Elfving","year":"2009","unstructured":"Elfving, T., Nikazad, T.: Properties of a class of block-iterative methods. Inverse Problems 25(11), 115011 (2009)","journal-title":"Inverse Problems"},{"key":"1890_CR19","first-page":"321","volume":"37","author":"T Elfving","year":"2010","unstructured":"Elfving, T., Nikazad, T., Hansen, P.C.: Semi-convergence and relaxation parameters for a class of SIRT algorithms. Electron. Trans. Numer. Anal. 37, 321\u2013336 (2010)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"1890_CR20","volume-title":"Regularization of Inverse problems","author":"HW Engl","year":"2000","unstructured":"Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse problems. Kluwer, Dordrecht (2000)"},{"key":"1890_CR21","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1016\/0022-5193(70)90109-8","volume":"29","author":"R Gordon","year":"1970","unstructured":"Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction technique (art) for three-dimensional electron microscopy and x-ray photography. J. Theo. Biol. 29, 471\u2013481 (1970)","journal-title":"J. Theo. Biol."},{"key":"1890_CR22","unstructured":"Greenbaum, A.: Iterative methods for solving linear systems, vol. 17 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1997)"},{"key":"1890_CR23","doi-asserted-by":"publisher","first-page":"518","DOI":"10.1137\/1008107","volume":"8","author":"TNE Greville","year":"1966","unstructured":"Greville, T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518\u2013521 (1966)","journal-title":"SIAM Rev."},{"issue":"3","key":"1890_CR24","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/BF01385727","volume":"60","author":"M Hanke","year":"1991","unstructured":"Hanke, M.: Accelerated Landweber iterations for the solution of ill-posed equations. Numer. Math. 60(3), 341\u2013373 (1991)","journal-title":"Numer. Math."},{"key":"1890_CR25","doi-asserted-by":"publisher","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":"1890_CR26","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718836","volume-title":"Discrete Inverse Problems: Insight and Algorithms","author":"PC Hansen","year":"2010","unstructured":"Hansen, P.C.: Discrete Inverse Problems: Insight and Algorithms. SIAM, Philadelphia (2010)"},{"issue":"1","key":"1890_CR27","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s11075-017-0430-x","volume":"79","author":"PC Hansen","year":"2018","unstructured":"Hansen, P.C., J\u00f8rgensen, J.S.: AIR Tools II: algebraic iterative reconstruction methods, improved implementation. Numer. Algorithms 79(1), 107\u2013137 (2018)","journal-title":"Numer. Algorithms"},{"key":"1890_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-723-7","volume-title":"Fundamentals of computerized tomography","author":"GT Herman","year":"2009","unstructured":"Herman, G.T.: Fundamentals of computerized tomography, 2nd edn. Advances in Pattern Recognition. Springer, Dordrecht (2009)","edition":"2"},{"key":"1890_CR29","doi-asserted-by":"crossref","unstructured":"Hn$$\\breve{e}$$tynkov$$\\acute{a}$$, I., Kub$$\\acute{i}$$nov$$\\acute{a}$$, M., Ple$$\\breve{s}$$inger, M.: Noise representation in residuals of LSQR, LSMR, and CRAIG regularization. Linear Algebra Appl. 533, 357\u2013379 (2017)","DOI":"10.1016\/j.laa.2017.07.031"},{"key":"1890_CR30","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix analysis","author":"RA Horn","year":"1985","unstructured":"Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1985)"},{"key":"1890_CR31","first-page":"355","volume":"A35","author":"S Kaczmarz","year":"1937","unstructured":"Kaczmarz, S.: Angen\u00e4herte aufl\u00f6sung von systemen linearer gleichungen. Bulletin de l\u2019Academie Polonaise des Sciences et Lettres A35, 355\u2013357 (1937)","journal-title":"Bulletin de l\u2019Academie Polonaise des Sciences et Lettres"},{"issue":"1","key":"1890_CR32","doi-asserted-by":"publisher","first-page":"149","DOI":"10.3934\/ipi.2014.8.149","volume":"8","author":"S Kindermann","year":"2014","unstructured":"Kindermann, S., Leit\u00e3o, A.: Convergence rates for Kaczmarz-type regularization methods. Inverse Probl. Imaging 8(1), 149\u2013172 (2014)","journal-title":"Inverse Probl. Imaging"},{"key":"1890_CR33","doi-asserted-by":"publisher","first-page":"615","DOI":"10.2307\/2372313","volume":"73","author":"L Landweber","year":"1951","unstructured":"Landweber, L.: An iterative formula for Fredholm integral equations of the first kind. Amer. J. of Math. 73, 615\u2013624 (1951)","journal-title":"Amer. J. of Math."},{"key":"1890_CR34","doi-asserted-by":"publisher","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. Comp. Appl. Math. 193, 157\u2013167 (2006)","journal-title":"J. Comp. Appl. Math."},{"key":"1890_CR35","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1023\/B:BITN.0000014546.51341.53","volume":"43","author":"JG Nagy","year":"2003","unstructured":"Nagy, J.G., Palmer, K.M.: Steepest descent, CG, and iterative regularization of ill-posed problems. BIT 43, 1003\u20131017 (2003)","journal-title":"BIT"},{"key":"1890_CR36","volume-title":"The mathematics of computerized tomography","author":"F Natterer","year":"1986","unstructured":"Natterer, F.: The mathematics of computerized tomography. B.G. Teubner, Stuttgart (1986)"},{"issue":"11","key":"1890_CR37","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1515\/jiip-2015-0082","volume":"25","author":"T Nikazad","year":"2017","unstructured":"Nikazad, T., Abbasi, M., Elfving, T.: Error minimizing relaxation strategies in Landweber and Kaczmarz type iterations. J. Inverse ill-Posed Probl. 25(11), 35\u201356 (2017)","journal-title":"J. Inverse ill-Posed Probl."},{"key":"1890_CR38","doi-asserted-by":"publisher","first-page":"907","DOI":"10.1007\/s11075-016-0122-y","volume":"73","author":"T Nikazad","year":"2016","unstructured":"Nikazad, T., Karimpour, M.: Controlling noise error in block iterative methods. Numer. Algor. 73, 907\u2013925 (2016)","journal-title":"Numer. Algor."},{"key":"1890_CR39","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003","volume-title":"Iterative methods for sparse linear systems","author":"Y Saad","year":"2003","unstructured":"Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition (2003)"},{"key":"1890_CR40","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/0024-3795(94)00114-6","volume":"236","author":"RJ Santos","year":"1996","unstructured":"Santos, R.J.: Equivalence of regularization and truncated iteration for general ill-posed problems. Lin. Alg. Appl. 236, 25\u201333 (1996)","journal-title":"Lin. Alg. Appl."},{"issue":"1","key":"1890_CR41","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1137\/10079015X","volume":"32","author":"X Shi","year":"2011","unstructured":"Shi, X., Wei, Y., Zhang, W.: Convergence of general nonstationary iterative methods for solving singular linear equations. SIAM J. Matrix Anal. Appl. 32(1), 72\u201389 (2011)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1890_CR42","unstructured":"Young, D.M.: Iterative solution of large linear systems. Academic Press (1971)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01890-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01890-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01890-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,10]],"date-time":"2025-05-10T07:51:05Z","timestamp":1746863465000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01890-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,23]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["1890"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01890-0","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,23]]},"assertion":[{"value":"28 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"None.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"Not applicable","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}