{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:54:46Z","timestamp":1740142486531,"version":"3.37.3"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T00:00:00Z","timestamp":1686009600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T00:00:00Z","timestamp":1686009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11701210"],"award-info":[{"award-number":["11701210"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007847","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["20190103029JH","20200201269JC"],"award-info":[{"award-number":["20190103029JH","20200201269JC"]}],"id":[{"id":"10.13039\/100007847","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Education Department Project of Jilin Province","award":["JJKH20211031KJ"],"award-info":[{"award-number":["JJKH20211031KJ"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20210540"],"award-info":[{"award-number":["BK20210540"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"name":"The Natural Science Foundation of the Jiangsu Higher Education Institutions of China","award":["21KJB110015","21KJB110001"],"award-info":[{"award-number":["21KJB110015","21KJB110001"]}]},{"name":"Startup Foundation for Introducing Talent of NJTech","award":["39804131","11901606"],"award-info":[{"award-number":["39804131","11901606"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2023,7]]},"DOI":"10.1007\/s40314-023-02343-5","type":"journal-article","created":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T18:03:21Z","timestamp":1686074601000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A splitting algorithm for constrained optimization problems with parabolic equations"],"prefix":"10.1007","volume":"42","author":[{"given":"Haiming","family":"Song","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0324-3115","authenticated-orcid":false,"given":"Jiachuan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongle","family":"Hao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,6,6]]},"reference":[{"key":"2343_CR1","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s00791-011-0158-4","volume":"14","author":"D Abbeloos","year":"2011","unstructured":"Abbeloos D, Diehl M, Hinze M, Vandewalle S (2011) Nested multigrid methods for time-periodic, parabolic optimal control problems. Comput Vis Sci 14:27\u201338","journal-title":"Comput Vis Sci"},{"key":"2343_CR2","doi-asserted-by":"publisher","first-page":"1484","DOI":"10.1137\/100819333","volume":"50","author":"T Apel","year":"2012","unstructured":"Apel T, Flaig TG (2012) Crank\u2013Nicolson schemes for optimal control problems with evolution equations. SIAM J Numer Anal 50:1484\u20131512","journal-title":"SIAM J Numer Anal"},{"key":"2343_CR3","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1016\/j.cpc.2015.08.025","volume":"197","author":"AT Barker","year":"2015","unstructured":"Barker AT, Stoll M (2015) Domain decomposition in time for PDE-constrained optimization. Comput Phys Commun 197:136\u2013143","journal-title":"Comput Phys Commun"},{"key":"2343_CR4","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1007\/s10589-019-00087-9","volume":"73","author":"E Borgens","year":"2019","unstructured":"Borgens E, Kanzow C (2019) Regularized Jacobi-type ADMM-methods for a class of separable convex optimization problems in Hilbert spaces. Comput Optim Appl 73:755\u2013790","journal-title":"Comput Optim Appl"},{"key":"2343_CR5","doi-asserted-by":"publisher","first-page":"2172","DOI":"10.1137\/070711311","volume":"31","author":"A Borz\u00ed","year":"2009","unstructured":"Borz\u00ed A, von Winckel G (2009) Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J Sci Comput 31:2172\u20132192","journal-title":"SIAM J Sci Comput"},{"key":"2343_CR6","doi-asserted-by":"publisher","first-page":"639","DOI":"10.4208\/nmtma.2017.m1637","volume":"10","author":"MM Butt","year":"2017","unstructured":"Butt MM, Yuan Y (2017) A full multigrid method for distributed control problems constrained by stokes equations. Numer Math Theor Methods Appl 10:639\u2013655","journal-title":"Numer Math Theor Methods Appl"},{"key":"2343_CR7","doi-asserted-by":"publisher","first-page":"A452","DOI":"10.1137\/120895809","volume":"36","author":"T Carraro","year":"2014","unstructured":"Carraro T, Geiger M, Rannacher R (2014) Indirect multiple shooting for nonlinear parabolic optimal control problems with control constraints. SIAM J Sci Comput 36:A452\u2013A481","journal-title":"SIAM J Sci Comput"},{"key":"2343_CR8","doi-asserted-by":"publisher","first-page":"1376","DOI":"10.1007\/s11425-008-0097-9","volume":"51","author":"Y Chen","year":"2008","unstructured":"Chen Y, Huang Y, Yi N (2008) A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations. Sci China Ser A 51:1376\u20131390","journal-title":"Sci China Ser A"},{"key":"2343_CR9","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10107-014-0826-5","volume":"155","author":"C Chen","year":"2016","unstructured":"Chen C, He B, Ye Y, Yuan X (2016) The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math Program Ser A 155:57\u201379","journal-title":"Math Program Ser A"},{"key":"2343_CR10","first-page":"36","volume":"40","author":"X Du","year":"2013","unstructured":"Du X, Sarkis M, Schaerer C, Szyld D (2013) Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems. Electron Trans Numer Anal 40:36\u201357","journal-title":"Electron Trans Numer Anal"},{"key":"2343_CR11","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1137\/151004744","volume":"54","author":"W Gong","year":"2016","unstructured":"Gong W, Yan N (2016) Finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold. SIAM J Numer Anal 54:1229\u20131262","journal-title":"SIAM J Numer Anal"},{"key":"2343_CR12","doi-asserted-by":"publisher","first-page":"820","DOI":"10.1007\/s10915-017-0386-y","volume":"72","author":"W Gong","year":"2017","unstructured":"Gong W, Xie H, Yan N (2017) Adaptive multilevel correction method for finite element approximations of elliptic optimal control problems. J Sci Comput 72:820\u2013841","journal-title":"J Sci Comput"},{"key":"2343_CR13","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.11650\/tjm\/7929","volume":"21","author":"H Guan","year":"2017","unstructured":"Guan H, Shi D (2017) A nonconforming finite element method for constrained optimal control problems governed by parabolic equations. Taiwan J Math 21:1193\u20131211","journal-title":"Taiwan J Math"},{"key":"2343_CR14","doi-asserted-by":"publisher","first-page":"1306","DOI":"10.1080\/10556788.2018.1504050","volume":"34","author":"S G\u00fcnther","year":"2019","unstructured":"G\u00fcnther S, Gauger NR, Schroder JB (2019) A non-intrusive parallel-in-time approach for simultaneous optimization with unsteady PDEs. Optim Methods Softw 34:1306\u20131321","journal-title":"Optim Methods Softw"},{"key":"2343_CR15","volume-title":"Perspectives in flow control and optimization","author":"M Gunzburger","year":"1987","unstructured":"Gunzburger M (1987) Perspectives in flow control and optimization. SIAM, Philadelphia"},{"key":"2343_CR16","doi-asserted-by":"publisher","first-page":"2274","DOI":"10.1137\/130922793","volume":"25","author":"B He","year":"2015","unstructured":"He B, Hou L, Yuan X (2015) On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming. SIAM J Optim 25:2274\u20132312","journal-title":"SIAM J Optim"},{"key":"2343_CR17","volume-title":"Optimization with PDE constraints. Mathematical modelling: theory and applications","author":"M Hinze","year":"2009","unstructured":"Hinze M, Pinnau R, Ulbrich M, Ulbrich S (2009) Optimization with PDE constraints. Mathematical modelling: theory and applications. Springer, New York"},{"key":"2343_CR18","doi-asserted-by":"publisher","first-page":"439","DOI":"10.4208\/nmtma.2011.m1017","volume":"4","author":"T Hou","year":"2011","unstructured":"Hou T, Chen Y, Huang Y (2011) A posteriori error estimates of mixed methods for quadratic optimal control problems governed by parabolic equations. Numer Math Theor Methods Appl 4:439\u2013458","journal-title":"Numer Math Theor Methods Appl"},{"key":"2343_CR19","doi-asserted-by":"crossref","unstructured":"Kwok F (2017) On the time-domain decomposition of parabolic optimal control problems. Domain decomposition methods in science and engineering XXIII, 55\u201367. Lect. Notes Comput. Sci. Eng., vol. 116, Springer, Cham","DOI":"10.1007\/978-3-319-52389-7_5"},{"key":"2343_CR20","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1016\/j.cam.2017.06.008","volume":"326","author":"B Li","year":"2017","unstructured":"Li B, Liu J, Xiao M (2017) A new multigrid method for unconstrained parabolic optimal control problems. J Comput Appl Math 326:358\u2013373","journal-title":"J Comput Appl Math"},{"key":"2343_CR21","doi-asserted-by":"crossref","first-page":"2115","DOI":"10.1016\/j.camwa.2017.09.017","volume":"75","author":"J Liu","year":"2018","unstructured":"Liu J, Wang Z (2018) Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems. Comput Math Appl 75:2115\u20132133","journal-title":"Comput Math Appl"},{"key":"2343_CR22","doi-asserted-by":"publisher","first-page":"1032","DOI":"10.1137\/S0036142902397090","volume":"42","author":"W Liu","year":"2004","unstructured":"Liu W, Ma H, Tang T, Yan N (2004) A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations. SIAM J Numer Anal 42:1032\u20131061","journal-title":"SIAM J Numer Anal"},{"key":"2343_CR23","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1002\/oca.2059","volume":"35","author":"X Luo","year":"2014","unstructured":"Luo X, Chen Y, Huang Y, Hou T (2014) Some error estimates of finite volume element method for parabolic optimal control problems. Optim Control Appl Methods 35:145\u2013165","journal-title":"Optim Control Appl Methods"},{"key":"2343_CR24","doi-asserted-by":"crossref","unstructured":"Meidner D, Vexler B (2008a) A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints. SIAM J Control Optim 47:1150\u20131177","DOI":"10.1137\/070694016"},{"key":"2343_CR25","doi-asserted-by":"crossref","unstructured":"Meidner D, Vexler B (2008b) A priori error estimates for space-time finite element discretization of parabolic optimal control problems. II. Problems with control constraints. SIAM J Control Optim 47:1301\u20131329","DOI":"10.1137\/070694028"},{"key":"2343_CR26","doi-asserted-by":"publisher","first-page":"816","DOI":"10.1002\/nla.814","volume":"19","author":"JW Pearson","year":"2012","unstructured":"Pearson JW, Wathen AJ (2012) A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer Linear Algebra Appl 19:816\u2013829","journal-title":"Numer Linear Algebra Appl"},{"key":"2343_CR27","doi-asserted-by":"publisher","first-page":"1126","DOI":"10.1137\/110847949","volume":"33","author":"JW Pearson","year":"2012","unstructured":"Pearson JW, Stoll M, Wathen AJ (2012) Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems. SIAM J Matrix Anal Appl 33:1126\u20131152","journal-title":"SIAM J Matrix Anal Appl"},{"key":"2343_CR28","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s11075-015-0060-0","volume":"72","author":"MK Riahi","year":"2016","unstructured":"Riahi MK (2016) A new approach to improve ill-conditioned parabolic optimal control problem via time domain decomposition. Numer Algorithms 72:635\u2013666","journal-title":"Numer Algorithms"},{"key":"2343_CR29","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1137\/120877532","volume":"24","author":"A Schiela","year":"2014","unstructured":"Schiela A, Ulbrich S (2014) Operator preconditioning for a class of inequality constrained optimal control problems. SIAM J Optim 24:435\u2013466","journal-title":"SIAM J Optim"},{"key":"2343_CR30","doi-asserted-by":"publisher","first-page":"B1","DOI":"10.1137\/130926365","volume":"37","author":"M Stoll","year":"2015","unstructured":"Stoll M, Breiten T (2015) A low-rank in time approach to PDE-constrained optimization. SIAM J Sci Comput 37:B1\u2013B29","journal-title":"SIAM J Sci Comput"},{"key":"2343_CR31","doi-asserted-by":"crossref","unstructured":"Tr\u00f6ltzsch F (2010) Optimal control of partial differential equations. AMS, Providence","DOI":"10.1090\/gsm\/112"},{"key":"2343_CR32","doi-asserted-by":"publisher","DOI":"10.4208\/aamm.OA-2020-0400","author":"M Wu","year":"2021","unstructured":"Wu M, Ai W, Yuan J, Tian H (2021) A symmetric inertial alternating direction method of multipliers for elliptic equation constrained optimization problem. Adv Appl Math Mech. https:\/\/doi.org\/10.4208\/aamm.OA-2020-0400","journal-title":"Adv Appl Math Mech"},{"key":"2343_CR33","doi-asserted-by":"publisher","first-page":"A2756","DOI":"10.1137\/15M104075X","volume":"38","author":"H Yang","year":"2016","unstructured":"Yang H, Hwang F, Cai X (2016) Nonlinear preconditioning techniques for full-space Lagrange\u2013Newton solution of PDE-constrained optimization problems. SIAM J Sci Comput 38:A2756\u2013A2778","journal-title":"SIAM J Sci Comput"},{"key":"2343_CR34","doi-asserted-by":"publisher","first-page":"727","DOI":"10.4208\/nmtma.OA-2018-0031","volume":"12","author":"C Yang","year":"2019","unstructured":"Yang C, Wang T, Xie X (2019) An interface-unfitted finite element method for elliptic interface optimal control problems. Numer Math Theor Methods Appl 12:727\u2013749","journal-title":"Numer Math Theor Methods Appl"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02343-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02343-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02343-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,14]],"date-time":"2023-12-14T18:55:39Z","timestamp":1702580139000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02343-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,6]]},"references-count":34,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,7]]}},"alternative-id":["2343"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02343-5","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2023,6,6]]},"assertion":[{"value":"26 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 May 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"205"}}