{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T11:59:31Z","timestamp":1770897571242,"version":"3.50.1"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Natural Science Foundation Project of CQ CSTC","award":["cstc2021jcyj-msxmX0034"],"award-info":[{"award-number":["cstc2021jcyj-msxmX0034"]}]},{"name":"Intelligent MR Diagnosis of the Liver by Linking Model and Data-driven Processes"},{"name":"Science and Technology Research Program of Chongqing Municipal Educational Commission","award":["KJZD-M202200501"],"award-info":[{"award-number":["KJZD-M202200501"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2025,4]]},"DOI":"10.1007\/s10444-025-10228-w","type":"journal-article","created":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T07:25:21Z","timestamp":1743751521000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Low-rank exponential integrators for stiff differential Riccati equations"],"prefix":"10.1007","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9976-6732","authenticated-orcid":false,"given":"Hao","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alfio","family":"Borz\u00ec","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,2]]},"reference":[{"key":"10228_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8081-7","volume-title":"Matrix Riccati Equations in Control and Systems Theory","author":"H Abou-Kandil","year":"2003","unstructured":"Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkh\u00e4user, Basel (2003)"},{"key":"10228_CR2","unstructured":"Alekseev, V.M.: An estimate for the perturbations of the solution of ordinary differential equations. Westnik Moskov Univ. Ser. I Math. Mekh. 2, 28\u201336 (1961)"},{"key":"10228_CR3","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1137\/100788860","volume":"33","author":"AH Al-Mohy","year":"2011","unstructured":"Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488\u2013511 (2011)","journal-title":"SIAM J. Sci. Comput."},{"key":"10228_CR4","volume":"410","author":"M Behr","year":"2021","unstructured":"Behr, M., Benner, P., Heiland, J.: Galerkin trial spaces and Davison-Maki methods for the numerical solution of differential Riccati equations. Appl. Math. Comput. 410, 126401 (2021)","journal-title":"Appl. Math. Comput."},{"key":"10228_CR5","doi-asserted-by":"crossref","unstructured":"Benner, P., Saak, J.: A semi-discretized heat transfer model for optimal cooling of steel profiles. In: Benner, P., Mehrmann, V., Sorensen, D. (eds.) Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng., volume 45, Springer-Verlag, Berlin\/Heidelberg, Germany, pp. 353\u2013356. (2005)","DOI":"10.1007\/3-540-27909-1_19"},{"key":"10228_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974829","volume-title":"Model Reduction and Approximation: Theory and Algorithms","author":"P Benner","year":"2017","unstructured":"Benner, P., Cohen, A., Ohlberger, M., Willcox, K.: Model Reduction and Approximation: Theory and Algorithms. SIAM, Philidelphia (2017)"},{"key":"10228_CR7","doi-asserted-by":"publisher","first-page":"2950","DOI":"10.1109\/TAC.2013.2258495","volume":"58","author":"P Benner","year":"2013","unstructured":"Benner, P., Mena, H.: Rosenbrock methods for solving differential Riccati equation. IEEE Trans. Automat. Control 58, 2950\u20132957 (2013)","journal-title":"IEEE Trans. Automat. Control"},{"key":"10228_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/jnma-2016-1039","volume":"26","author":"P Benner","year":"2018","unstructured":"Benner, P., Mena, H.: Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations. J. Numer. Math. 26, 1\u201320 (2018)","journal-title":"J. Numer. Math."},{"key":"10228_CR9","unstructured":"Benner, P., Lang, N.: Peer methods for the solution of large-scale differential matrix equations. (2018). arXiv preprint arXiv:1807.08524"},{"key":"10228_CR10","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1016\/S0022-0000(67)80025-4","volume":"1","author":"RS Bucy","year":"1967","unstructured":"Bucy, R.S.: Global theory of the Riccati equation. J. Comput. Syst. Sci. 1, 349\u2013361 (1967)","journal-title":"J. Comput. Syst. Sci."},{"key":"10228_CR11","doi-asserted-by":"crossref","unstructured":"Carrel, B., Vandereycken, B.: Projected exponential methods for stiff dynamical low-rank approximation problems. (2023). arXiv:2312.00172","DOI":"10.2139\/ssrn.5017212"},{"key":"10228_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113162","volume":"384","author":"H Chen","year":"2021","unstructured":"Chen, H., Wang, Y.: Modified Douglas splitting method for differential matrix equations. J. Comput. Appl. Math. 384, 113162 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"10228_CR13","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1007\/s10915-023-02275-6","volume":"96","author":"H Chen","year":"2023","unstructured":"Chen, H., Borz\u00ec, A.: Positivity preserving exponential integrators for differential Riccati equations. J. Sci. Comput. 96, 50 (2023)","journal-title":"J. Sci. Comput."},{"key":"10228_CR14","doi-asserted-by":"crossref","unstructured":"Damm, T., Dragan, V., Freiling, G.: Coupled Riccati differential equations arising in connection with Nash differential games. Proceedings of the 17th World Congress The International Federation of Automatic Control, Seoul, Korea (2008)","DOI":"10.3182\/20080706-5-KR-1001.00664"},{"key":"10228_CR15","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1137\/0729049","volume":"29","author":"L Dieci","year":"1992","unstructured":"Dieci, L.: Numerical integration of the differential Riccati equation and some related issues. SIAM J. Numer. Anal. 29, 781\u2013815 (1992)","journal-title":"SIAM J. Numer. Anal."},{"key":"10228_CR16","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s002110050030","volume":"67","author":"L Dieci","year":"1994","unstructured":"Dieci, L., Eirola, T.: Positive definiteness in the numerical solution of Riccati differential equations. Numer. Math. 67, 303\u2013313 (1994)","journal-title":"Numer. Math."},{"key":"10228_CR17","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/S0024-3795(01)00534-1","volume":"351\u2013352","author":"G Freiling","year":"2002","unstructured":"Freiling, G.: A survey of nonsymmetric Riccati equations. Linear Algebra Appl. 351\u2013352, 243\u2013270 (2002)","journal-title":"Linear Algebra Appl."},{"key":"10228_CR18","doi-asserted-by":"publisher","first-page":"3128","DOI":"10.1137\/130935501","volume":"52","author":"E Hansen","year":"2014","unstructured":"Hansen, E., Stillfjord, T.: Convergence analysis for splitting of the abstract differential Riccati equation. SIAM J. Numer. Anal. 52, 3128\u20133139 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"10228_CR19","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717778","volume-title":"Functions of Matrices: Theory and Computation","author":"NJ Higham","year":"2008","unstructured":"Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)"},{"key":"10228_CR20","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1017\/S0962492910000036","volume":"19","author":"NJ Higham","year":"2010","unstructured":"Higham, N.J., Al-Mohy, A.H.: Computing matrix functions. Acta Numerica 19, 159\u2013208 (2010)","journal-title":"Acta Numerica"},{"key":"10228_CR21","doi-asserted-by":"publisher","first-page":"1911","DOI":"10.1137\/S0036142995280572","volume":"34","author":"M Hochbruck","year":"1997","unstructured":"Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911\u20131925 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"10228_CR22","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","volume":"19","author":"M Hochbruck","year":"2010","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209\u2013286 (2010)","journal-title":"Acta Numerica"},{"key":"10228_CR23","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in Matrix Analysis","author":"R Horn","year":"1991","unstructured":"Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)"},{"key":"10228_CR24","doi-asserted-by":"publisher","first-page":"1020","DOI":"10.1137\/15M1026791","volume":"54","author":"E Kieri","year":"2016","unstructured":"Kieri, E., Lubich, C., Walach, H.: Discretized dynamical low-rank approximation in the presence of small singular values. SIAM J. Numer. Anal. 54, 1020\u20131038 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"10228_CR25","doi-asserted-by":"publisher","first-page":"A2182","DOI":"10.1137\/19M1264217","volume":"42","author":"G Kirsten","year":"2020","unstructured":"Kirsten, G., Simoncini, V.: Order reduction methods for solving large-scale differential matrix Riccati equations. SIAM J. Sci. Comput. 42, A2182\u2013A2205 (2020)","journal-title":"SIAM J. Sci. Comput."},{"key":"10228_CR26","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1137\/050639703","volume":"29","author":"O Koch","year":"2007","unstructured":"Koch, O., Lubich, C.: Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29, 434\u2013454 (2007)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"10228_CR27","unstructured":"Koskela, A., Mena, H.: Analysis of Krylov subspace approximation to large scale differential Riccati equations. (2018). arXiv preprint arXiv:1705.07507v4"},{"key":"10228_CR28","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1002\/pamm.201410394","volume":"14","author":"N Lang","year":"2014","unstructured":"Lang, N., Mena, H., Saak, J.: An $$LDL^T$$ factorization based ADI algorithm for solving large scale differential matrix equations. Proc. Appl. Math. Mech. 14, 827\u2013828 (2014)","journal-title":"Proc. Appl. Math. Mech."},{"key":"10228_CR29","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.laa.2015.04.006","volume":"480","author":"N Lang","year":"2015","unstructured":"Lang, N., Mena, H., Saak, J.: On the benefits of the $$LDL^T$$ factorization for large-scale differential matrix equation solvers. Linear Algebra Appl. 480, 44\u201371 (2015)","journal-title":"Linear Algebra Appl."},{"key":"10228_CR30","unstructured":"Lang, N.: Numerical methods for large-scale linear time-varying control systems and related differential matrix equations. Dissertation, Technische Universit\u00e4t Chemnitz, Germany (2017)"},{"key":"10228_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113360","volume":"389","author":"D Li","year":"2021","unstructured":"Li, D., Zhang, X., Liu, R.: Exponential integrators for large-scale stiff Riccati differential equations. J. Comput. Appl. Math. 389, 113360 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"10228_CR32","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1137\/S0036144504443389","volume":"46","author":"J-R Li","year":"2004","unstructured":"Li, J.-R., White, J.: Low-rank solution of Lyapunov equations. SIAM Review 46, 693\u2013713 (2004)","journal-title":"SIAM Review"},{"key":"10228_CR33","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s10543-013-0454-0","volume":"54","author":"C Lubich","year":"2014","unstructured":"Lubich, C., Oseledets, I.V.: A projector-splitting integrator for dynamical low-rank approximation. BIT 54, 171\u2013188 (2014)","journal-title":"BIT"},{"key":"10228_CR34","doi-asserted-by":"publisher","first-page":"A2406","DOI":"10.1137\/17M1134500","volume":"40","author":"A M\u00e5lqvist","year":"2018","unstructured":"M\u00e5lqvist, A., Persson, A., Stillfjord, T.: Multiscale differential Riccati equations for linear quadratic regulator problems. SIAM J. Sci. Comput. 40, A2406\u2013A2426 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"10228_CR35","doi-asserted-by":"crossref","unstructured":"Mena, H.: Numerical solution of differential Riccati equations arising in optimal control problems for parabolic partial differential equations. Ph.D. Thesis, Escuela Politecnica Nacional (2007)","DOI":"10.1002\/pamm.200700913"},{"key":"10228_CR36","doi-asserted-by":"publisher","first-page":"602","DOI":"10.1016\/j.cam.2018.01.035","volume":"340","author":"H Mena","year":"2018","unstructured":"Mena, H., Ostermann, A., Pfurtscheller, L., Piazzola, C.: Numerical low-rank approximation of matrix differential equations. J. Comput. Appl. Math. 340, 602\u2013614 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"10228_CR37","doi-asserted-by":"publisher","first-page":"1947","DOI":"10.1137\/18M1177901","volume":"57","author":"A Ostermann","year":"2019","unstructured":"Ostermann, A., Piazzola, C., Walach, H.: Convergence of a low-rank Lie-Trotter splitting for stiff matrix differential equations. SIAM J. Numer. Anal. 57, 1947\u20131966 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"10228_CR38","unstructured":"Penzl, T.: LYAPACK Users Guide, Technical Report SFB393\/00-33, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 09107 Chemnitz, Germany (2000). http:\/\/www.tu-chemnitz.de\/sfb393\/sfb00pr.html"},{"key":"10228_CR39","volume-title":"Riccati Differential Equations","author":"WT Reid","year":"1972","unstructured":"Reid, W.T.: Riccati Differential Equations. Academic Press, New York (1972)"},{"key":"10228_CR40","unstructured":"Saak, J.: Efficient solution of large scale algebraic matrix equations in PDE control and model order reduction. Ph. D. thesis, Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany (2009)"},{"key":"10228_CR41","unstructured":"Saak, J., K\u00f6hler, M., Benner, P.: M-M.E.S.S.-2.1-The matrix equations sparse solvers library. (2022). https:\/\/www.mpi-magdeburg.mpg.de\/projects\/mess"},{"key":"10228_CR42","first-page":"1","volume":"29","author":"T Schmelzer","year":"2007","unstructured":"Schmelzer, T., Trefethen, L.N.: Evaluating matrix functions for exponential integrators via Carath\u00e9odory-Fej\u00e9r approximation and contour integrals. Electron. Trans. Numer. Anal. 29, 1\u201318 (2007)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"10228_CR43","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1145\/285861.285868","volume":"24","author":"RB Sidje","year":"1998","unstructured":"Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. (TOMS) 24, 130\u2013156 (1998)","journal-title":"ACM Trans. Math. Softw. (TOMS)"},{"key":"10228_CR44","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 Review 58, 377\u2013441 (2016)","journal-title":"SIAM Review"},{"key":"10228_CR45","volume-title":"Optimal Control and Estimation","author":"RF Stengel","year":"1994","unstructured":"Stengel, R.F.: Optimal Control and Estimation. Dover Publications, New York (1994)"},{"key":"10228_CR46","doi-asserted-by":"publisher","first-page":"2791","DOI":"10.1109\/TAC.2015.2398889","volume":"60","author":"T Stillfjord","year":"2015","unstructured":"Stillfjord, T.: Low-rank second-order splitting of large-scale differential Riccati equations. IEEE Trans. Automat. Control 60, 2791\u20132796 (2015)","journal-title":"IEEE Trans. Automat. Control"},{"key":"10228_CR47","doi-asserted-by":"publisher","first-page":"1129","DOI":"10.1007\/s11075-017-0416-8","volume":"78","author":"T Stillfjord","year":"2018","unstructured":"Stillfjord, T.: Adaptive high-order splitting schemes for large-scale differential Riccati equations. Numer. Algorithms 78, 1129\u20131151 (2018)","journal-title":"Numer. Algorithms"},{"key":"10228_CR48","doi-asserted-by":"publisher","first-page":"3598","DOI":"10.1137\/18M1178815","volume":"56","author":"T Stillfjord","year":"2018","unstructured":"Stillfjord, T.: Singular value decay of operator-valued differential Lyapunov and Riccati equations. SIAM J. Control Optim. 56, 3598\u20133618 (2018)","journal-title":"SIAM J. Control Optim."},{"key":"10228_CR49","doi-asserted-by":"publisher","DOI":"10.1155\/2010\/819064","volume":"2010","author":"J Zhang","year":"2010","unstructured":"Zhang, J., Liu, J.: Matrix bounds for the solution of the continuous algebraic Riccati equation. Math. Probl. Eng. 2010, 819064 (2010)","journal-title":"Math. Probl. Eng."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10228-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-025-10228-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10228-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T09:41:44Z","timestamp":1745574104000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-025-10228-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["10228"],"URL":"https:\/\/doi.org\/10.1007\/s10444-025-10228-w","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4]]},"assertion":[{"value":"11 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 April 2025","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 no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"17"}}