{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T22:04:34Z","timestamp":1757455474937,"version":"3.40.5"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,12,5]],"date-time":"2024-12-05T00:00:00Z","timestamp":1733356800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,12,5]],"date-time":"2024-12-05T00:00:00Z","timestamp":1733356800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["20227PCCK","CN00000013 - CUP J33C22001170001"],"award-info":[{"award-number":["20227PCCK","CN00000013 - CUP J33C22001170001"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s10589-024-00629-w","type":"journal-article","created":{"date-parts":[[2024,12,5]],"date-time":"2024-12-05T13:38:10Z","timestamp":1733405890000},"page":"905-932","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Truncated LSQR for matrix least squares problems"],"prefix":"10.1007","volume":"91","author":[{"given":"Lorenzo","family":"Piccinini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0795-5865","authenticated-orcid":false,"given":"Valeria","family":"Simoncini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,5]]},"reference":[{"key":"629_CR1","doi-asserted-by":"publisher","first-page":"656","DOI":"10.1137\/140993867","volume":"36","author":"J Baker","year":"2015","unstructured":"Baker, J., Embree, M., Sabino, J.: Fast singular value decay for Lyapunov solutions with nonnormal coefficients. SIAM J. Matrix Anal. Appl. 36, 656\u2013668 (2015)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"629_CR2","unstructured":"Balabanov, O., Grigori, L.:  Randomized Block Gram\u2013Schmidt Process for Solution of Linear Systems and Eigenvalue Problems, arXiv preprint arXiv:2111.14641, (2021)"},{"key":"629_CR3","doi-asserted-by":"publisher","first-page":"A1450","DOI":"10.1137\/20M138870X","volume":"44","author":"O Balabanov","year":"2022","unstructured":"Balabanov, O., Grigori, L.: Randomized Gram-Schmidt process with application to GMRES. SIAM J. Sci. Comput. 44, A1450\u2013A1474 (2022)","journal-title":"SIAM J. Sci. Comput."},{"key":"629_CR4","doi-asserted-by":"crossref","unstructured":"Bellavia, S., Gondzio, J., Porcelli, M.: A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion J. Sci. Comput. 89, 46 (2021)","DOI":"10.1007\/s10915-021-01654-1"},{"key":"629_CR5","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1553\/etna_vol55s92","volume":"55","author":"AH Bentbib","year":"2022","unstructured":"Bentbib, A.H., Khouia, A., Sadok, H.: The LSQR method for solving tensor least-squares problems. Electron. Trans. Numer. Anal. 55, 92\u2013111 (2022)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"629_CR6","doi-asserted-by":"crossref","unstructured":"Bollh\u00f6fer, M., Eppler, A.\u00a0K.:  A structure preserving FGMRES method for solving large Lyapunov equations, In Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry, M.\u00a0G\u00fcnther, A.\u00a0Bartel, M.\u00a0Brunk, S.\u00a0Schoeps, and M.\u00a0Striebel, eds., vol.\u00a017, 2012, pp.\u00a0131\u2013136","DOI":"10.1007\/978-3-642-25100-9_15"},{"key":"629_CR7","doi-asserted-by":"crossref","unstructured":"Dantas, C.F., Cohen, J.E., Gribonval, R., Learning tensor-structured dictionaries with application to hyperspectral image denoising. In 27th European Signal Processing Conference, EUSIPCO,: A Coru\u00f1a, Spain, September 2\u20136, 2019. IEEE 2019, 1\u20135 (2019)","DOI":"10.23919\/EUSIPCO.2019.8902593"},{"key":"629_CR8","doi-asserted-by":"crossref","unstructured":"Dantas, C.\u00a0F., Da\u00a0Costa, M.\u00a0N., da\u00a0Rocha\u00a0Lopes, R.:  Learning dictionaries as a sum of Kronecker products, IEEE Signal Processing Letters, 24 (2017), pp.\u00a0559\u2013563","DOI":"10.1109\/LSP.2017.2681159"},{"key":"629_CR9","doi-asserted-by":"publisher","first-page":"2968","DOI":"10.1049\/iet-cta.2019.1400","volume":"14","author":"W Deng","year":"2020","unstructured":"Deng, W., Zeng, X., Hong, Y.: Distributed optimisation approach to least-squares solution of Sylvester equations. IET Control Theor. Appl. 14, 2968\u20132976 (2020)","journal-title":"IET Control Theor. Appl."},{"key":"629_CR10","unstructured":"Eshragh, A., Pietro, O.\u00a0D., Saunders, M.\u00a0A.:  Toeplitz least squares problems, fast algorithms and big data, Tech. Rep., arXiv: 2112.12994, 2021"},{"key":"629_CR11","first-page":"71","volume-title":"Overview of QR Methods for Large Least Squares Problems Involving Kronecker Products","author":"DW Fausett","year":"1995","unstructured":"Fausett, D.W., Hashish, H.: Overview of QR Methods for Large Least Squares Problems Involving Kronecker Products, pp. 71\u201380. De Gruyter, Berlin, Boston (1995)"},{"key":"629_CR12","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1016\/j.camwa.2005.08.009","volume":"50","author":"L Giraud","year":"2005","unstructured":"Giraud, L., Langou, J., Rozloznik, M.: The loss of orthogonality in the Gram-Schmidt orthogonalization process. Comput. Math. Appl. 50, 1069\u20131075 (2005)","journal-title":"Comput. Math. Appl."},{"key":"629_CR13","unstructured":"Gohberg, I., Lancaster, P., Rodman, L.:  Matrices and indefinite scalar products, In Operator Theory: Advances and Applications, vol. 8, Birkh\u00e4user, (1983)"},{"key":"629_CR14","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1137\/0702016","volume":"2","author":"G Golub","year":"1965","unstructured":"Golub, G., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. Soc. Indus. Appl. Math. Ser. B: Numer. Anal. 2, 205\u2013224 (1965)","journal-title":"J. Soc. Indus. Appl. Math. Ser. B: Numer. Anal."},{"key":"629_CR15","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.sysconle.2014.09.006","volume":"73","author":"L Grubisi\u0107","year":"2014","unstructured":"Grubisi\u0107, L., Kressner, D.: On the eigenvalue decay of solutions to operator Lyapunov equations. Syst. Control Lett. 73, 42\u201347 (2014)","journal-title":"Syst. Control Lett."},{"key":"629_CR16","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in Matrix Analysis","author":"RA Horn","year":"1991","unstructured":"Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)"},{"key":"629_CR17","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1016\/0024-3795(92)90031-5","volume":"172","author":"DY Hu","year":"1992","unstructured":"Hu, D.Y., Reichel, L.: Krylov subspace methods for the Sylvester equation. Linear Algebra Appl. 172, 283\u2013313 (1992)","journal-title":"Linear Algebra Appl."},{"key":"629_CR18","doi-asserted-by":"publisher","first-page":"1180","DOI":"10.1137\/130931655","volume":"35","author":"M Ishteva","year":"2014","unstructured":"Ishteva, M., Usevich, K., Markovsky, A.: Factorization approach to structured low-rank approximation with applications. SIAM J. Matrix Anal. Appl. 35, 1180\u20131204 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"629_CR19","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1137\/S0895479898345540","volume":"22","author":"J Kamm","year":"2000","unstructured":"Kamm, J., Nagy, J.: Optimal Kronecker product approximation of block Toeplitz matrices. SIAM J. Matrix Anal. Appl. 22, 155\u2013172 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"629_CR20","doi-asserted-by":"crossref","unstructured":"Kittisopaporn and, A., Chansangiam, P.: Approximated least-squares solutions of a generalized sylvester-transpose matrix equation via gradient-descent iterative algorithm, Adv. Diff. Equ. 2021, 266 (2021)","DOI":"10.1186\/s13662-021-03427-4"},{"key":"629_CR21","doi-asserted-by":"publisher","first-page":"666","DOI":"10.1002\/nla.1919","volume":"21","author":"D Kressner","year":"2014","unstructured":"Kressner, D., Ple\u0161inger, M., Tobler, C.: A preconditioned low-rank CG method for parameter-dependent Lyapunov equations. Num. Lin. Alg. Appl. 21, 666\u2013684 (2014)","journal-title":"Num. Lin. Alg. Appl."},{"key":"629_CR22","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1002\/nla.1973","volume":"22","author":"D Kressner","year":"2015","unstructured":"Kressner, D., Sirkovi\u0107, P.: Truncated low-rank methods for solving general linear matrix equations. Num. Lin. Alg. Appl. 22, 564\u2013583 (2015)","journal-title":"Num. Lin. Alg. Appl."},{"key":"629_CR23","doi-asserted-by":"publisher","first-page":"1688","DOI":"10.1137\/090756843","volume":"31","author":"D Kressner","year":"2010","unstructured":"Kressner, D., Tobler, C.: Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl. 31, 1688\u20131714 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"629_CR24","doi-asserted-by":"publisher","first-page":"1288","DOI":"10.1137\/100799010","volume":"32","author":"D Kressner","year":"2011","unstructured":"Kressner, D., Tobler, C.: Low-rank tensor Krylov subspace methods for parametrized linear systems. SIAM. J. Matrix Anal. Appl. 32, 1288\u20131316 (2011)","journal-title":"SIAM. J. Matrix Anal. Appl."},{"key":"629_CR25","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/5.726791","volume":"86","author":"Y Lecun","year":"1998","unstructured":"Lecun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86, 2278\u20132324 (1998)","journal-title":"Proc. IEEE"},{"key":"629_CR26","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.apnum.2013.04.004","volume":"72","author":"Y Lin","year":"2013","unstructured":"Lin, Y., Simoncini, V.: Minimal residual methods for large scale Lyapunov equations. Appl. Numer. Math. 72, 52\u201371 (2013)","journal-title":"Appl. Numer. Math."},{"key":"629_CR27","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1017\/S0962492920000021","volume":"29","author":"P-G Martinsson","year":"2020","unstructured":"Martinsson, P.-G., Tropp, J.A.: Randomized numerical linear algebra: foundations and algorithms. Acta Numer 29, 403\u2013572 (2020)","journal-title":"Acta Numer"},{"key":"629_CR28","unstructured":"The MathWorks, Inc.,  MATLAB 7, r2020b\u00a0ed.,( 2020)"},{"key":"629_CR29","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1145\/355984.355989","volume":"8","author":"CC Paige","year":"1982","unstructured":"Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. (TOMS) 8, 43\u201371 (1982)","journal-title":"ACM Trans. Math. Softw. (TOMS)"},{"key":"629_CR30","doi-asserted-by":"publisher","first-page":"1383","DOI":"10.1007\/s11075-021-01080-2","volume":"88","author":"D Palitta","year":"2021","unstructured":"Palitta, D., K\u00fcrschner, P.: On the convergence of Krylov methods with low-rank truncations. Numer. Algorith. 88, 1383\u20131417 (2021)","journal-title":"Numer. Algorith."},{"key":"629_CR31","doi-asserted-by":"publisher","first-page":"1820","DOI":"10.1080\/00207160802516875","volume":"87","author":"Z-Y Peng","year":"2010","unstructured":"Peng, Z.-Y.: A matrix LSQR iterative method to solve matrix equation AXB=C. Int\u2019l J. Comput. Math. 87, 1820\u20131830 (2010)","journal-title":"Int\u2019l J. Comput. Math."},{"key":"629_CR32","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S0167-6911(00)00010-4","volume":"40","author":"T Penzl","year":"2000","unstructured":"Penzl, T.: Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Syst. Control Lett. 40, 139\u2013144 (2000)","journal-title":"Syst. Control Lett."},{"key":"629_CR33","unstructured":"Piccinini, L.:  Least Squares Methods for Sylvester-like Linear Matrix Equations, (2023). Master Thesis"},{"key":"629_CR34","unstructured":"Platero, R.\u00a0E.:  Least Squares Updating for Kronecker Products, (2017). Bachelor Thesis"},{"key":"629_CR35","doi-asserted-by":"publisher","first-page":"13212","DOI":"10.1073\/pnas.0804869105","volume":"105","author":"V Rokhlin","year":"2008","unstructured":"Rokhlin, V., Tygert, M.: A fast randomized algorithm for overdetermined linear least-squares regression. Proc. Natl. Acad. Sci. USA. 105, 13212\u201313217 (2008)","journal-title":"Proc. Natl. Acad. Sci. USA."},{"key":"629_CR36","unstructured":"Sabino, J.:  Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method, PhD thesis, Rice University, (2006)"},{"key":"629_CR37","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1137\/130912839","volume":"58","author":"V Simoncini","year":"2016","unstructured":"Simoncini, V.: Computational methods for linear matrix equations. SIAM Rev. 58, 377\u2013441 (2016)","journal-title":"SIAM Rev."},{"key":"629_CR38","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1137\/22M147880X","volume":"44","author":"V Simoncini","year":"2023","unstructured":"Simoncini, V., Hao, Y.: Analysis of the truncated conjugate gradient method for linear matrix equations. SIAM J. Matrix Anal. Appl. 44, 359\u2013381 (2023)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"629_CR39","doi-asserted-by":"publisher","first-page":"454","DOI":"10.1137\/S1064827502406415","volume":"25","author":"V Simoncini","year":"2003","unstructured":"Simoncini, V., Szyld, D.B.: Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J. Sci. Comput. 25, 454\u2013477 (2003)","journal-title":"SIAM J. Sci. Comput."},{"key":"629_CR40","doi-asserted-by":"publisher","first-page":"1458","DOI":"10.1137\/090759252","volume":"31","author":"GW Stewart","year":"2009","unstructured":"Stewart, G.W.: On the semidefinite B-Arnoldi method. SIAM J. Matrix Anal. Appl. 31, 1458\u20131468 (2009)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"629_CR41","doi-asserted-by":"publisher","first-page":"B1","DOI":"10.1137\/130926365","volume":"37","author":"M Stoll","year":"2015","unstructured":"Stoll, M., Breiten, T.: A low-rank in time approach to PDE-constrained optimization. SIAM J. Sci. Comput. 37, B1\u2013B29 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"629_CR42","unstructured":"Timsit, E., Grigori, L., Balabanov, O.:  Randomized Orthogonal Projection Methods for Krylov Subspace Solvers, arXiv preprint arXiv:2302.07466, (2023)"},{"key":"629_CR43","doi-asserted-by":"publisher","first-page":"1289","DOI":"10.1080\/00207160802275977","volume":"87","author":"M Wang","year":"2010","unstructured":"Wang, M., Wei, M., Feng, Y.: An iterative algorithm for a least squares solution of a matrix equation. Int. J. Comput. Math. 87, 1289\u20131298 (2010)","journal-title":"Int. J. Comput. Math."},{"key":"629_CR44","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/0400000060","volume":"10","author":"DP Woodruff","year":"2014","unstructured":"Woodruff, D.P.: Sketching as a tool for numerical linear algebra. Found. Trends Theor. Comput. Sci. 10, 1\u2013157 (2014)","journal-title":"Found. Trends Theor. Comput. Sci."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-024-00629-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-024-00629-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-024-00629-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,17]],"date-time":"2025-05-17T12:32:10Z","timestamp":1747485130000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-024-00629-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,5]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["629"],"URL":"https:\/\/doi.org\/10.1007\/s10589-024-00629-w","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2024,12,5]]},"assertion":[{"value":"5 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 November 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 December 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}