{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T06:54:58Z","timestamp":1787381698043,"version":"build-2736575974"},"reference-count":56,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/M025179\/1"],"award-info":[{"award-number":["EP\/M025179\/1"]}],"id":[{"id":"10.13039\/501100000266","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>By examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them, we perform numerical experiments to study how they can be used to efficiently solve rank-deficient sparse linear least-squares problems arising from practical applications. The Cholesky factorization of the normal equations breaks down when the least-squares problem is rank-deficient, while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors. To try to resolve these difficulties, we consider a regularization procedure that modifies the diagonal of the unregularized matrix. This leads to matrices that are easier to factorize. We consider both the regularized normal equations and the regularized augmented system. We employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original (unregularized) problem. Furthermore, we look at using limited-memory incomplete Cholesky-based factorizations and how these can offer the potential to solve very large problems.<\/jats:p>","DOI":"10.1137\/16m1065380","type":"journal-article","created":{"date-parts":[[2017,8,24]],"date-time":"2017-08-24T12:50:49Z","timestamp":1503579049000},"page":"C319-C339","source":"Crossref","is-referenced-by-count":6,"title":["On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems"],"prefix":"10.1137","volume":"39","author":[{"given":"Jennifer","family":"Scott","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,8,24]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389335"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7176"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/130931588"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/110846427"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1145\/1391989.1391995"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/140968896"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(97)00171-4"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1145\/2049662.2049670"},{"key":"atypb10","first-page":"1","volume":"38","author":"Davis T. A.","year":"2011","journal-title":"ACM Trans. Math. Software"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1145\/992200.992202"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/11.2.181"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897317661"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/10079687X"},{"key":"atypb15","volume-title":"Research Report SOL 99-1, Department of EESOR","author":"George A.","year":"1999"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/0613022"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479893252623"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(79)90122-8"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-014-9687-3"},{"key":"atypb20","unstructured":"N. I. M. Gould and J. A. Scott,\n                      The State-of-the-Art of Preconditioners for Sparse Linear Least Squares Problems: The Complete Results\n                      , Technical Report RAL-TR-2015-009, Rutherford Appleton Laboratory, Oxfordshire, UK, 2015; available online athttp:\/\/purl.org\/net\/epubs\/work\/23411221."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1145\/3014057"},{"key":"atypb22","volume-title":"Technical report, Department of Computer Science","author":"Greif C.","year":"2015"},{"key":"atypb23","unstructured":"A. Gupta,\n                      WSMP Watson Sparse Matrix Package (Part II: Direct Solution of General Sparse Systems)\n                      , Technical Report RC 21888 (98472), IBM T. J. Watson Research Center, Yorktown Heights, NY, 2000."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1145\/2756548"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/090757216"},{"key":"atypb26","unstructured":"J. D. Hogg and J. A. Scott,\n                      The Effects of Scalings on the Performance of a Sparse Symmetric Indefinite Solver\n                      , Technical Report RAL-TR-2008-007, Rutherford Appleton Laboratory, Oxfordshire, UK, 2008."},{"key":"atypb27","unstructured":"J. D. Hogg and J. A. Scott,\n                      HSL_MA97: A Bit-Compatible Multifrontal Code for Sparse Symmetric Systems\n                      , Technical Report RAL-TR-2011-024, Rutherford Appleton Laboratory, Oxfordshire, UK, 2011; available online athttp:\/\/purl.org\/net\/epubs\/work\/61445."},{"key":"atypb28","unstructured":"J. D. Hogg and J. A. Scott,\n                      A Study of Pivoting Strategies for Tough Sparse Indefinite Systems\n                      , Technical Report RAL-TR-2012-009, Rutherford Appleton Laboratory, Oxfordshire, UK, 2012; available online athttp:\/\/purl.org\/net\/epubs\/work\/63152."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.3390\/a6040702"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1145\/2513109.2513113"},{"key":"atypb31","unstructured":"The HSL Mathematical Software Library\n                      , a collection of Fortran codes for large-scale scientific computation,http:\/\/www.hsl.rl.ac.uk(2016)."},{"key":"atypb32","volume-title":"Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices (Version 4.0), Technical report","author":"Karypis G.","year":"1998"},{"key":"atypb33","first-page":"75","volume":"20","author":"Li N.","year":"2005","journal-title":"Electron. Trans. Numer. 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