{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T21:55:30Z","timestamp":1780350930821,"version":"3.54.1"},"reference-count":35,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2024]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;abstract&gt;&lt;p&gt;In this research, we constructed a class of nonlinear greedy average block Kaczmarz methods to solve nonlinear problems without computing the Moore-Penrose pseudoinverse of the Jacobian matrix. These kinds of methods adopt the average technique of the Gaussian Kaczmarz method and combine the greedy strategy, which greatly reduces the amount of computation. The local convergence analysis and numerical experiments of the proposed methods are given. The numerical results show the effectiveness of the proposed methods.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2024014","type":"journal-article","created":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T10:42:42Z","timestamp":1711536162000},"page":"305-323","source":"Crossref","is-referenced-by-count":3,"title":["A class of pseudoinverse-free greedy block nonlinear Kaczmarz methods for nonlinear systems of equations"],"prefix":"10.3934","volume":"19","author":[{"given":"Ying","family":"Lv","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li-Li","family":"Xing","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wen-Di","family":"Bao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei-Guo","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhi-Wei","family":"Guo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2024014-1","unstructured":"H. B. An, Z. Z. Bai, Broyden method for nonlinear equation in several variables, <i>Mathematica Numerica Sinica (Chinese Journal)<\/i>, <b>26<\/b> (2004), 385\u2013400."},{"key":"key-10.3934\/nhm.2024014-2","doi-asserted-by":"publisher","unstructured":"H. B. An, Z. Z. Bai, Directional secant method for nonlinear equations, <i>J. Comput. Appl. Math.<\/i>, <b>175<\/b> (2005), 291\u2013304. https:\/\/doi.org\/10.1016\/j.cam.2004.05.013","DOI":"10.1016\/j.cam.2004.05.013"},{"key":"key-10.3934\/nhm.2024014-3","doi-asserted-by":"publisher","unstructured":"Z. Z. Bai, L. Wang, On convergence rates of Kaczmarz-type methods with different selection rules of working rows, <i>Appl. Numer. Math.<\/i>, <b>186<\/b> (2023), 289\u2013319. https:\/\/doi.org\/10.1016\/j.apnum.2023.01.013","DOI":"10.1016\/j.apnum.2023.01.013"},{"key":"key-10.3934\/nhm.2024014-4","doi-asserted-by":"publisher","unstructured":"Z. Z. Bai, W. T. Wu, On greedy randomized Kaczmarz method for solving large sparse linear systems, <i>SIAM J. Sci. Comput.<\/i>, <b>40<\/b> (2018), A592\u2013A606. https:\/\/doi.org\/10.1137\/17M1137747","DOI":"10.1137\/17M1137747"},{"key":"key-10.3934\/nhm.2024014-5","doi-asserted-by":"publisher","unstructured":"J. Q. Chen, Z. D. Huang, On the error estimate of the randomized double block Kaczmarz method, <i>Appl. Math. Comput.<\/i>, <b>370<\/b> (2020), 124907. https:\/\/doi.org\/10.1016\/j.amc.2019.124907","DOI":"10.1016\/j.amc.2019.124907"},{"key":"key-10.3934\/nhm.2024014-6","doi-asserted-by":"publisher","unstructured":"Q. P. Chen, W. R. Hao, A homotopy training algorithm for fully connected neural networks, <i>P. Roy. Soc. A-Math. Phy.<\/i>, <b>475<\/b> (2019), 20190662. https:\/\/doi.org\/10.1098\/rspa.2019.0662","DOI":"10.1098\/rspa.2019.0662"},{"key":"key-10.3934\/nhm.2024014-7","doi-asserted-by":"publisher","unstructured":"K. Du, W. T. Si, X. H. Sun, Randomized extended average block Kaczmarz for solving least squares, <i>SIAM J. Sci. Comput.<\/i>, <b>42<\/b> (2020), A3541\u2013A3559. https:\/\/doi.org\/10.1137\/20M1312629","DOI":"10.1137\/20M1312629"},{"key":"key-10.3934\/nhm.2024014-8","unstructured":"K. Du, X. H. Sun, <i>Pseudoinverse-free randomized block iterative algorithms for consistent and inconsistent linear systems<\/i>, arXiv: 2011.10353 [preprint], (2020), [cited 2024 March 27]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.48550\/arXiv.2011.10353\">https:\/\/doi.org\/10.48550\/arXiv.2011.10353<\/ext-link>"},{"key":"key-10.3934\/nhm.2024014-9","doi-asserted-by":"publisher","unstructured":"K. Du, X. H. Sun, A doubly stochastic block Gauss\u2013Seidel algorithm for solving linear equations, <i>Appl. Math. Comput.<\/i>, <b>408<\/b> (2021), 126373. https:\/\/doi.org\/10.1016\/j.amc.2021.126373","DOI":"10.1016\/j.amc.2021.126373"},{"key":"key-10.3934\/nhm.2024014-10","doi-asserted-by":"publisher","unstructured":"Y. C. Eldar, D. Needell, Acceleration of randomized Kaczmarz method via the Johnson-Lindenstrauss lemma, <i>Numer. Algorithms<\/i>, <b>58<\/b> (2011), 163\u2013177. https:\/\/doi.org\/10.1007\/s11075-011-9451-z","DOI":"10.1007\/s11075-011-9451-z"},{"key":"key-10.3934\/nhm.2024014-11","doi-asserted-by":"publisher","unstructured":"T. Elfving, Block-iterative methods for consistent and inconsistent linear equations, <i>Numer. Math.<\/i>, <b>35<\/b> (1980), 1\u201312. https:\/\/doi.org\/10.1007\/BF01396365","DOI":"10.1007\/BF01396365"},{"key":"key-10.3934\/nhm.2024014-12","doi-asserted-by":"publisher","unstructured":"M. A. Gomes-Ruggiero, J. M. Mart\u00ednez, A. C. Moretti, Comparing algorithms for solving sparse nonlinear systems of equations, <i>SIAM J. Sci. Stat. Comput.<\/i>, <b>13<\/b> (1992), 459\u2013483. https:\/\/doi.org\/10.1137\/0913025","DOI":"10.1137\/0913025"},{"key":"key-10.3934\/nhm.2024014-13","doi-asserted-by":"publisher","unstructured":"R. M. Gower, P. Richt\u00e1rik, Randomized iterative methods for linear systems, <i>SIAM J. Matrix Anal. A.<\/i>, <b>36<\/b> (2015), 1660\u20131690. https:\/\/doi.org\/10.1137\/15M1025487","DOI":"10.1137\/15M1025487"},{"key":"key-10.3934\/nhm.2024014-14","unstructured":"S. Karczmarz, Angenaherte auflosung von systemen linearer glei-chungen, <i>Bull. Int. Acad. Pol. Sic. Let., Cl. Sci. Math. Nat.<\/i>, <b>35<\/b> (1937), 355\u2013357."},{"key":"key-10.3934\/nhm.2024014-15","unstructured":"K. Kawaguchi, Deep learning without poor local minima, In: D. D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, R. Garnett, <i>Advances in neural information processing systems<\/i>, New York: Curran Associates Inc., <b>29<\/b> (2016), 586\u2013594."},{"key":"key-10.3934\/nhm.2024014-16","doi-asserted-by":"publisher","unstructured":"J. Liu, S. J. Wright, An accelerated randomized Kaczmarz algorithm, <i>Math. Comput.<\/i>, <b>85<\/b> (2016), 153\u2013178. http:\/\/dx.doi.org\/10.1090\/mcom\/2971","DOI":"10.1090\/mcom\/2971"},{"key":"key-10.3934\/nhm.2024014-17","doi-asserted-by":"crossref","unstructured":"L. Luk\u0161an, Hybrid methods for large sparse nonlinear least squares, <i>J. Optimiz. Theory App.<\/i>, <b>89<\/b> (1996), 575\u2013595.","DOI":"10.1007\/BF02275350"},{"key":"key-10.3934\/nhm.2024014-18","doi-asserted-by":"publisher","unstructured":"A. Ma, D. Needell, A. Ramdas, Convergence properties of the randomized extended Gauss\u2013Seidel and Kaczmarz methods, <i>SIAM J. Matrix Anal. A.<\/i>, <b>36<\/b> (2015), 1590\u20131604. https:\/\/doi.org\/10.1137\/15M1014425","DOI":"10.1137\/15M1014425"},{"key":"key-10.3934\/nhm.2024014-19","doi-asserted-by":"publisher","unstructured":"L. Mirsky, Symmetric gauge functions and unitarily invariant norms, <i>Q. J. Math.<\/i>, <b>11<\/b> (1960), 50\u201359. https:\/\/doi.org\/10.1093\/qmath\/11.1.50","DOI":"10.1093\/qmath\/11.1.50"},{"key":"key-10.3934\/nhm.2024014-20","doi-asserted-by":"crossref","unstructured":"J. J. Mor\u00e9, B. S. Garbow, K. E. Hillstrom, Testing unconstrained optimization software, <i>ACM T. Math. Software<\/i>, <b>7<\/b> (1981), 17\u201341.","DOI":"10.1145\/355934.355936"},{"key":"key-10.3934\/nhm.2024014-21","doi-asserted-by":"publisher","unstructured":"M. S. Morshed, M. S. Islam, M. Noor-E-Alam, Accelerated sampling Kaczmarz Motzkin algorithm for the linear feasibility problem, <i>J. Global Optim.<\/i>, <b>77<\/b> (2020), 361\u2013382. https:\/\/doi.org\/10.1007\/s10898-019-00850-6","DOI":"10.1007\/s10898-019-00850-6"},{"key":"key-10.3934\/nhm.2024014-22","doi-asserted-by":"publisher","unstructured":"I. Necoara, Faster randomized block Kaczmarz algorithms, <i>SIAM J. Matrix Anal. A.<\/i>, <b>40<\/b> (2019), 1425\u20131452. https:\/\/doi.org\/10.1137\/19M1251643","DOI":"10.1137\/19M1251643"},{"key":"key-10.3934\/nhm.2024014-23","doi-asserted-by":"publisher","unstructured":"D. Needell, Randomized Kaczmarz solver for noisy linear systems, <i>BIT<\/i>, <b>50<\/b> (2010), 395\u2013403. https:\/\/doi.org\/10.1007\/s10543-010-0265-5","DOI":"10.1007\/s10543-010-0265-5"},{"key":"key-10.3934\/nhm.2024014-24","doi-asserted-by":"publisher","unstructured":"D. Needell, J. A. Tropp, Paved with good intentions: Analysis of a randomized block Kaczmarz method, <i>Linear Algebra Appl.<\/i>, <b>441<\/b> (2014), 199\u2013221. https:\/\/doi.org\/10.1016\/j.laa.2012.12.022","DOI":"10.1016\/j.laa.2012.12.022"},{"key":"key-10.3934\/nhm.2024014-25","doi-asserted-by":"publisher","unstructured":"D. Needell, R. Zhao, A. Zouzias, Randomized block Kaczmarz method with projection for solving least squares, <i>Linear Algebra Appl<\/i>, <b>484<\/b> (2015), 322\u2013343. https:\/\/doi.org\/10.1016\/j.laa.2015.06.027","DOI":"10.1016\/j.laa.2015.06.027"},{"key":"key-10.3934\/nhm.2024014-26","unstructured":"J. M. Ortega, W. C. Rheinboldt, <i>Iterative Solution of Nonlinear Equations in Several Variables<\/i>, Philadelphia: Society for Industrial and Applied Mathematics, 2000. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1137\/1.9780898719468\">https:\/\/doi.org\/10.1137\/1.9780898719468<\/ext-link>"},{"key":"key-10.3934\/nhm.2024014-27","doi-asserted-by":"publisher","unstructured":"T. Strohmer, R. Vershynin, A randomized Kaczmarz algorithm with exponential convergence, <i>J. Fourier Anal. Appl.<\/i>, <b>15<\/b> (2009) 262\u2013278. https:\/\/doi.org\/10.1007\/s00041-008-9030-4","DOI":"10.1007\/s00041-008-9030-4"},{"key":"key-10.3934\/nhm.2024014-28","doi-asserted-by":"publisher","unstructured":"Q. F. Wang, W. G. Li, W. D. Bao, X. Q. Gao, Nonlinear Kaczmarz algorithms and their convergence, <i>J. Comput. Appl. Math.<\/i>, <b>399<\/b> (2022), 113720. https:\/\/doi.org\/10.1016\/j.cam.2021.113720","DOI":"10.1016\/j.cam.2021.113720"},{"key":"key-10.3934\/nhm.2024014-29","doi-asserted-by":"publisher","unstructured":"X. Z. Wang, M. L. Che, C. X. Mo, Y. M. Wei, Solving the system of nonsingular tensor equations via randomized Kaczmarz-like method, <i>J. Comput. Appl. Math.<\/i>, <b>421<\/b> (2023), 114856. https:\/\/doi.org\/10.1016\/j.cam.2022.114856","DOI":"10.1016\/j.cam.2022.114856"},{"key":"key-10.3934\/nhm.2024014-30","doi-asserted-by":"publisher","unstructured":"A. Q. Xiao, J. F. Yin, N. Zheng, On fast greedy block Kaczmarz methods for solving large consistent linear systems, <i>Comput. Appl. Math.<\/i>, <b>42<\/b> (2023), 119. https:\/\/doi.org\/10.1007\/s40314-023-02232-x","DOI":"10.1007\/s40314-023-02232-x"},{"key":"key-10.3934\/nhm.2024014-31","doi-asserted-by":"publisher","unstructured":"R. Yuan, A. Lazaric, R. M. Gower, Sketched Newton-Raphson, <i>SIAM J. Optimiz.<\/i>, <b>32<\/b> (2022), 1555\u20131583. https:\/\/doi.org\/10.1137\/21M139788X","DOI":"10.1137\/21M139788X"},{"key":"key-10.3934\/nhm.2024014-32","doi-asserted-by":"publisher","unstructured":"J. H. Zhang, Y. Q. Wang, J. Zhao, On maximum residual nonlinear Kaczmarz-type algorithms for large nonlinear systems of equations, <i>J. Comput. Appl. Math.<\/i>, <b>425<\/b> (2023), 115065. https:\/\/doi.org\/10.1016\/j.cam.2023.115065","DOI":"10.1016\/j.cam.2023.115065"},{"key":"key-10.3934\/nhm.2024014-33","unstructured":"Y. J. Zhang, H. Y. Li, <i>Greedy capped nonlinear Kaczmarz methods<\/i>, arXiv: 2210.00653 [preprint], (2022), [cited 2024 March 27]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.48550\/arXiv.2210.00653\">https:\/\/doi.org\/10.48550\/arXiv.2210.00653<\/ext-link>"},{"key":"key-10.3934\/nhm.2024014-34","unstructured":"Y. J. Zhang, H. Y. Li, L. Tang, <i>Greedy randomized sampling nonlinear Kaczmarz methods<\/i>, arXiv: 2209.06082 [preprint], (2022), [cited 2024 March 27]. Available from: <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.48550\/arXiv.2209.06082\">https:\/\/doi.org\/10.48550\/arXiv.2209.06082<\/ext-link>"},{"key":"key-10.3934\/nhm.2024014-35","doi-asserted-by":"publisher","unstructured":"A. Zouzias, N. M. Freris, Randomized extended Kaczmarz for solving least squares, <i>SIAM J. Matrix Anal. A.<\/i>, <b>34<\/b> (2013), 773\u2013793. https:\/\/doi.org\/10.1137\/120889897","DOI":"10.1137\/120889897"}],"container-title":["Networks and Heterogeneous Media"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024014?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,1]],"date-time":"2024-04-01T05:12:56Z","timestamp":1711948376000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024014"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024]]}},"URL":"https:\/\/doi.org\/10.3934\/nhm.2024014","relation":{},"ISSN":["1556-1801"],"issn-type":[{"value":"1556-1801","type":"print"}],"subject":[],"published":{"date-parts":[[2024]]}}}