{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T13:41:28Z","timestamp":1680270088827},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"Abstract<\/jats:title>\n We give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived.\nWe prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between \n \n \n \n \n T<\/m:mi>\n \u2020<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n T<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n X<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {T^{\\dagger}T(X_{n})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n \n T<\/m:mi>\n *<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n T<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n X<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {T^{*}T(X_{n})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and the principal angles between \n \n \n \n \n X<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n \u2229<\/m:mo>\n \n \n (<\/m:mo>\n \n \n \ud835\udca9<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n T<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2229<\/m:mo>\n \n X<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u22a5<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:math>\n \n {X_{n}\\cap(\\mathcal{N}(T)\\cap X_{n})^{\\perp}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n \n (<\/m:mo>\n \n \n \ud835\udca9<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n T<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n X<\/m:mi>\n n<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2229<\/m:mo>\n \n \ud835\udca9<\/m:mi>\n \u2062<\/m:mo>\n \n \n (<\/m:mo>\n T<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n \u22a5<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {(\\mathcal{N}(T)+X_{n})\\cap\\mathcal{N}(T)^{\\perp}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nAt the end, we consider several specific cases and examples to further illustrate our theorems.<\/jats:p>","DOI":"10.1515\/cmam-2019-0173","type":"journal-article","created":{"date-parts":[[2020,5,26]],"date-time":"2020-05-26T16:32:59Z","timestamp":1590510779000},"page":"783-798","source":"Crossref","is-referenced-by-count":0,"title":["A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems"],"prefix":"10.1515","volume":"20","author":[{"given":"Shukai","family":"Du","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences , University of Delaware , Newark , USA"}]},{"given":"Nailin","family":"Du","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Wuhan University , Wuhan , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2020,4,15]]},"reference":[{"key":"2023033110450936695_j_cmam-2019-0173_ref_001","unstructured":"A. Ben-Israel and T. N. E. Greville,\nGeneralized Inverses: Theory and Applications,\nRobert E. Krieger, Huntington, 1980."},{"key":"2023033110450936695_j_cmam-2019-0173_ref_002","doi-asserted-by":"crossref","unstructured":"G. Bruckner and S. Pereverzev,\nSelf-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems,\nInverse Problems 19 (2003), no. 1, 147\u2013156.","DOI":"10.1088\/0266-5611\/19\/1\/308"},{"key":"2023033110450936695_j_cmam-2019-0173_ref_003","doi-asserted-by":"crossref","unstructured":"D. Chu, L. Lin, R. C. E. Tan and Y. Wei,\nCondition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems,\nNumer. Linear Algebra Appl. 18 (2011), no. 1, 87\u2013103.","DOI":"10.1002\/nla.702"},{"key":"2023033110450936695_j_cmam-2019-0173_ref_004","doi-asserted-by":"crossref","unstructured":"N. Du,\nFinite-dimensional approximation settings for infinite-dimensional Moore\u2013Penrose inverses,\nSIAM J. Numer. Anal. 46 (2008), no. 3, 1454\u20131482.","DOI":"10.1137\/060661120"},{"key":"2023033110450936695_j_cmam-2019-0173_ref_005","doi-asserted-by":"crossref","unstructured":"H. W. Engl,\nOn the convergence of regularization methods for ill-posed linear operator equations,\nImproperly Posed Problems and Their Numerical Treatment (Oberwolfach 1982),\nInternat. Schriftenreihe Numer. Math. 63,\nBirkh\u00e4user, Basel (1983), 81\u201395.","DOI":"10.1007\/978-3-0348-5460-3_6"},{"key":"2023033110450936695_j_cmam-2019-0173_ref_006","unstructured":"H. W. Engl and C. W. Groetsch,\nProjection-regularization methods for linear operator equations of the first kind,\nSpecial Program on Inverse Problems,\nProc. Centre Math. Anal. Austral. Nat. Univ. 17,\nAustralian National University, Canberra (1988), 17\u201331."},{"key":"2023033110450936695_j_cmam-2019-0173_ref_007","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke and A. Neubauer,\nRegularization of Inverse Problems,\nMath. Appl. 375,\nKluwer Academic, Dordrecht, 1996.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"2023033110450936695_j_cmam-2019-0173_ref_008","doi-asserted-by":"crossref","unstructured":"H. W. Engl and A. Neubauer,\nAn improved version of Marti\u2019s method for solving ill-posed linear integral equations,\nMath. Comp. 45 (1985), no. 172, 405\u2013416.","DOI":"10.1090\/S0025-5718-1985-0804932-1"},{"key":"2023033110450936695_j_cmam-2019-0173_ref_009","doi-asserted-by":"crossref","unstructured":"A. Gal\u00e1ntai,\nProjectors and Projection Methods,\nAdv. Math. 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Wiss. 123,\nSpringer, Berlin, 1980."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/4\/article-p783.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0173\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T13:02:16Z","timestamp":1680267736000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0173\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,15]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,8,5]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.1515\/cmam-2019-0173"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2019-0173","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,15]]}}}