{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:17:21Z","timestamp":1760235441178,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,8,22]],"date-time":"2021-08-22T00:00:00Z","timestamp":1629590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42074009"],"award-info":[{"award-number":["42074009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we investigate the ill-conditioned problem of a separable, nonlinear least squares model by using the variable projection method. Based on the truncated singular value decomposition method and the Tikhonov regularization method, we propose an improved Tikhonov regularization method, which neither discards small singular values, nor treats all singular value corrections. By fitting the Mackey\u2013Glass time series in an exponential model, we compare the three regularization methods, and the numerically simulated results indicate that the improved regularization method is more effective at reducing the mean square error of the solution and increasing the accuracy of unknowns.<\/jats:p>","DOI":"10.3390\/axioms10030196","type":"journal-article","created":{"date-parts":[[2021,8,22]],"date-time":"2021-08-22T21:42:12Z","timestamp":1629668532000},"page":"196","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Improved Tikhonov-Regularized Variable Projection Algorithm for Separable Nonlinear Least Squares"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0672-3245","authenticated-orcid":false,"given":"Hua","family":"Guo","sequence":"first","affiliation":[{"name":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"},{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Guolin","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Luyao","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s11263-006-9785-5","article-title":"On the Wiberg algorithm for matrix factorization in the presence of missing components","volume":"72","author":"Okatani","year":"2007","journal-title":"Int. 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