{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T17:06:27Z","timestamp":1788368787084,"version":"build-2803163510"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["025717"],"award-info":[{"award-number":["025717"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1216559"],"award-info":[{"award-number":["1216559"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1418377"],"award-info":[{"award-number":["1418377"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub--Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization, parameter for the projected problem by unbiased predictive risk estimation, generalized cross validation, and discrepancy principle techniques is investigated. It is shown that the regularized parameter obtained by the unbiased predictive risk estimator can provide a good estimate which can be used for a full problem that is moderately to severely ill-posed. A similar analysis provides the weight parameter for the weighted generalized cross validation such that the approach is also useful in these cases, and also explains why the generalized cross validation without weighting is not always useful. All results are independent of whether systems are over-or underdetermined. Numerical simulations for standard one-dimensional test problems and two-dimensional data, for both image restoration and tomographic image reconstruction, support the analysis and validate the techniques. The size of the projected problem is found using an extension of a noise revealing function for the projected problem [I. Hn\u011btynkov\u00e1, M. Ple\u0161inger, and Z. Strako\u0161, BIT Numer. Math., 49 (2009), pp. 669--696]. Furthermore, an iteratively reweighted regularization approach for edge preserving regularization is extended for projected systems, providing stabilization of the solutions of the projected systems and reducing dependence on the determination of the size of the projected subspace.<\/jats:p>","DOI":"10.1137\/15m1037925","type":"journal-article","created":{"date-parts":[[2017,3,9]],"date-time":"2017-03-09T11:03:55Z","timestamp":1489057435000},"page":"B221-B243","source":"Crossref","is-referenced-by-count":37,"title":["Hybrid and Iteratively Reweighted Regularization by Unbiased Predictive Risk and Weighted GCV for Projected Systems"],"prefix":"10.1137","volume":"39","author":[{"given":"Rosemary A.","family":"Renaut","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saeed","family":"Vatankhah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vahid E.","family":"Ardestani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,3,9]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/100809787"},{"key":"atypb2","first-page":"149","volume":"28","author":"Chung J.","year":"2008","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/130945363"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/10079687X"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1979.10489751"},{"key":"atypb6","volume-title":"Matrix Computations","author":"Golub G. H.","year":"1996","edition":"3"},{"key":"atypb7","volume-title":"Tomographic X-ray Data of a Walnut, preprint, arXiv:1502.04064","author":"H\u00e4m\u00e4l\u00e4inen K.","year":"2015"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/120876277"},{"key":"atypb9","first-page":"253","volume":"3","author":"Hanke M.","year":"1993","journal-title":"Surv. Math. Ind."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02149761"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719697"},{"key":"atypb12","first-page":"204","volume":"31","author":"Hansen P. C.","year":"2008","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb13","volume-title":"Society for Industrial and Applied Mathematics","author":"Hansen P. C.","year":"2006"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-009-0239-7"},{"key":"atypb15","volume-title":"Some Results on the Regularization of LSQR for Large-Scale Discrete Ill-Posed Problems, preprint, arXiv:1503.01864v3","author":"Huang Y.","year":"2015"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899345960"},{"key":"atypb17","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1007\/BF00966239","volume":"7","author":"Morozov V. A.","year":"1966","journal-title":"Sov. Math. Dokl."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1023\/B:NUMA.0000027762.08431.64"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2003.821103"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0718026"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1145\/355993.356000"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1190\/geo2014-0017.1"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1190\/1.1444596"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2010.10.002"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2009.05.026"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1088\/1742-2132\/11\/4\/045001"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1093\/gji\/ggu397"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/8\/085002"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717570"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2003.819861"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2007.906221"},{"key":"atypb33","volume-title":"Methods in Geochemistry and Geophysics 36","author":"Zhdanov M. S.","year":"2002"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1037925","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:30:10Z","timestamp":1787329810000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1037925"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/15M1037925"],"URL":"https:\/\/doi.org\/10.1137\/15m1037925","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}