{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T11:10:03Z","timestamp":1748517003728,"version":"3.41.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T00:00:00Z","timestamp":1745366400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T00:00:00Z","timestamp":1745366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007210","name":"RWTH Aachen University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007210","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.\n<\/jats:p>","DOI":"10.1007\/s00211-025-01465-8","type":"journal-article","created":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T09:56:47Z","timestamp":1745402207000},"page":"781-822","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations"],"prefix":"10.1007","volume":"157","author":[{"given":"Markus","family":"Bachmayr","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henrik","family":"Eisenmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andr\u00e9","family":"Uschmajew","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,23]]},"reference":[{"issue":"2","key":"1465_CR1","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s10543-013-0444-2","volume":"54","author":"A Arnold","year":"2014","unstructured":"Arnold, A., Jahnke, T.: On the approximation of high-dimensional differential equations in the hierarchical Tucker format. BIT 54(2), 305\u2013341 (2014)","journal-title":"BIT"},{"key":"1465_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492922000125","volume":"32","author":"M Bachmayr","year":"2023","unstructured":"Bachmayr, M.: Low-rank tensor methods for partial differential equations. Acta Numer. 32, 1\u2013121 (2023)","journal-title":"Acta Numer."},{"issue":"330","key":"1465_CR3","doi-asserted-by":"publisher","first-page":"1799","DOI":"10.1090\/mcom\/3626","volume":"90","author":"M Bachmayr","year":"2021","unstructured":"Bachmayr, M., Eisenmann, H., Kieri, E., Uschmajew, A.: Existence of dynamical low-rank approximations to parabolic problems. Math. Comp. 90(330), 1799\u20131830 (2021)","journal-title":"Math. Comp."},{"issue":"6","key":"1465_CR4","doi-asserted-by":"publisher","first-page":"1423","DOI":"10.1007\/s10208-016-9317-9","volume":"16","author":"M Bachmayr","year":"2016","unstructured":"Bachmayr, M., Schneider, R., Uschmajew, A.: Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations. Found. Comput. Math. 16(6), 1423\u20131472 (2016)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"1465_CR5","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s00205-010-0308-8","volume":"198","author":"C Bardos","year":"2010","unstructured":"Bardos, C., Catto, I., Mauser, N., Trabelsi, S.: Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations. Arch. Ration. Mech. Anal. 198(1), 273\u2013330 (2010)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"1465_CR6","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/j.aml.2007.12.033","volume":"22","author":"C Bardos","year":"2009","unstructured":"Bardos, C., Catto, I., Mauser, N.J., Trabelsi, S.: Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree-Fock equations: a sufficient condition. Appl. Math. Lett. 22(2), 147\u2013152 (2009)","journal-title":"Appl. Math. Lett."},{"issue":"1","key":"1465_CR7","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1137\/20M1321838","volume":"59","author":"G Ceruti","year":"2021","unstructured":"Ceruti, G., Lubich, C., Walach, H.: Time integration of tree tensor networks. SIAM J. Numer. Anal. 59(1), 289\u2013313 (2021)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1465_CR8","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1051\/m2an\/2010018","volume":"44","author":"D Conte","year":"2010","unstructured":"Conte, D., Lubich, C.: An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics. M2AN Math. Model. Numer. Anal. 44(4), 759\u2013780 (2010)","journal-title":"M2AN Math. Model. Numer. Anal."},{"issue":"3","key":"1465_CR9","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1017\/S0305004100016108","volume":"26","author":"PAM Dirac","year":"1930","unstructured":"Dirac, P.A.M.: Note on exchange phenomena in the Thomas atom. Math. Proc. Cambridge Philos. Soc. 26(3), 376\u2013385 (1930)","journal-title":"Math. Proc. Cambridge Philos. Soc."},{"issue":"5","key":"1465_CR10","doi-asserted-by":"publisher","first-page":"B1330","DOI":"10.1137\/18M116383X","volume":"40","author":"L Einkemmer","year":"2018","unstructured":"Einkemmer, L., Lubich, C.: A low-rank projector-splitting integrator for the Vlasov-Poisson equation. SIAM J. Sci. Comput. 40(5), B1330\u2013B1360 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1465_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.109063","volume":"403","author":"L Einkemmer","year":"2020","unstructured":"Einkemmer, L., Ostermann, A., Piazzola, C.: A low-rank projector-splitting integrator for the Vlasov-Maxwell equations with divergence correction. J. Comput. Phys. 403, 109063 (2020)","journal-title":"J. Comput. Phys."},{"key":"1465_CR12","unstructured":"Eisenmann, H.: Multilinear Optimization in Low-rank Models. PhD thesis, Universit\u00e4t Leipzig (2023)"},{"issue":"1","key":"1465_CR13","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s10208-018-9381-4","volume":"19","author":"A Falc\u00f3","year":"2019","unstructured":"Falc\u00f3, A., Hackbusch, W., Nouy, A.: On the Dirac-Frenkel variational principle on tensor Banach spaces. Found. Comput. Math. 19(1), 159\u2013204 (2019)","journal-title":"Found. Comput. Math."},{"issue":"2","key":"1465_CR14","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s40324-018-0177-x","volume":"78","author":"A Falc\u00f3","year":"2021","unstructured":"Falc\u00f3, A., Hackbusch, W., Nouy, A.: Tree-based tensor formats. SeMA J. 78(2), 159\u2013173 (2021)","journal-title":"SeMA J."},{"key":"1465_CR15","volume-title":"Wave Mechanics","author":"J Frenkel","year":"1934","unstructured":"Frenkel, J.: Advanced general theory. In: Wave Mechanics. Clarendon Press, Oxford (1934)"},{"key":"1465_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111089","volume":"458","author":"W Guo","year":"2022","unstructured":"Guo, W., Qiu, J.-M.: A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps. J. Comput. Phys. 458, 111089, 24 (2022)","journal-title":"J. Comput. Phys."},{"key":"1465_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-35554-8","volume-title":"Tensor Spaces and Numerical Tensor Calculus","author":"W Hackbusch","year":"2019","unstructured":"Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus, 2nd edn. Springer, Cham (2019)","edition":"2"},{"issue":"4","key":"1465_CR18","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1007\/s00211-011-0419-7","volume":"120","author":"S Holtz","year":"2012","unstructured":"Holtz, S., Rohwedder, T., Schneider, R.: On manifolds of tensors of fixed TT-rank. Numer. Math. 120(4), 701\u2013731 (2012)","journal-title":"Numer. Math."},{"issue":"3","key":"1465_CR19","first-page":"603","volume":"9","author":"Y Kazashi","year":"2021","unstructured":"Kazashi, Y., Nobile, F.: Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval. Stoch. Part. Differ. Equ. Anal. Comput. 9(3), 603\u2013629 (2021)","journal-title":"Stoch. Part. Differ. Equ. Anal. Comput."},{"issue":"2","key":"1465_CR20","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(2), 434\u2013454 (2007)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"5","key":"1465_CR21","doi-asserted-by":"publisher","first-page":"2360","DOI":"10.1137\/09076578X","volume":"31","author":"O Koch","year":"2010","unstructured":"Koch, O., Lubich, C.: Dynamical tensor approximation. SIAM J. Matrix Anal. Appl. 31(5), 2360\u20132375 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1465_CR22","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","volume":"51","author":"TG Kolda","year":"2009","unstructured":"Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455\u2013500 (2009)","journal-title":"SIAM Rev."},{"key":"1465_CR23","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1093\/amrx\/abv006","volume":"2","author":"C Lubich","year":"2015","unstructured":"Lubich, C.: Time integration in the multiconfiguration time-dependent Hartree method of molecular quantum dynamics. Appl. Math. Res. Express. AMRX 2, 311\u2013328 (2015)","journal-title":"Appl. Math. Res. Express. AMRX"},{"issue":"1","key":"1465_CR24","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(1), 171\u2013188 (2014)","journal-title":"BIT"},{"issue":"2","key":"1465_CR25","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1137\/140976546","volume":"53","author":"C Lubich","year":"2015","unstructured":"Lubich, C., Oseledets, I.V., Vandereycken, B.: Time integration of tensor trains. SIAM J. Numer. Anal. 53(2), 917\u2013941 (2015)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"1465_CR26","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1137\/120885723","volume":"34","author":"C Lubich","year":"2013","unstructured":"Lubich, C., Rohwedder, T., Schneider, R., Vandereycken, B.: Dynamical approximation by hierarchical Tucker and tensor-train tensors. SIAM J. Matrix Anal. Appl. 34(2), 470\u2013494 (2013)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"1465_CR27","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0009-2614(90)87014-I","volume":"165","author":"H-D Meyer","year":"1990","unstructured":"Meyer, H.-D., Manthe, U., Cederbaum, L.S.: The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165(1), 73\u201378 (1990)","journal-title":"Chem. Phys. Lett."},{"issue":"4","key":"1465_CR28","doi-asserted-by":"publisher","first-page":"1346","DOI":"10.1016\/j.matcom.2008.03.007","volume":"79","author":"Achim Nonnenmacher","year":"2008","unstructured":"Nonnenmacher, Achim, Lubich, Christian: Dynamical low-rank approximation: applications and numerical experiments. Math. Comput. Simul. 79(4), 1346\u20131357 (2008)","journal-title":"Math. Comput. Simul."},{"issue":"5","key":"1465_CR29","doi-asserted-by":"publisher","first-page":"2295","DOI":"10.1137\/090752286","volume":"33","author":"IV Oseledets","year":"2011","unstructured":"Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295\u20132317 (2011)","journal-title":"SIAM J. Sci. Comput."},{"key":"1465_CR30","unstructured":"Steinlechner, M.M.: Riemannian Optimization for Solving High-Dimensional Problems with Low-Rank Tensor Structure. PhD thesis, MATHICSE, EPFL, Lausanne (2016)"},{"key":"1465_CR31","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/140","volume-title":"Ordinary Differential Equations and Dynamical Systems","author":"G Teschl","year":"2012","unstructured":"Teschl, G.: Ordinary Differential Equations and Dynamical Systems. American Mathematical Society, Providence, RI (2012)"},{"issue":"3","key":"1465_CR32","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s00365-010-9125-4","volume":"34","author":"A Uschmajew","year":"2011","unstructured":"Uschmajew, A.: Regularity of tensor product approximations to square integrable functions. Constr. Approx. 34(3), 371\u2013391 (2011)","journal-title":"Constr. Approx."},{"issue":"1","key":"1465_CR33","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/j.laa.2013.03.016","volume":"439","author":"A Uschmajew","year":"2013","unstructured":"Uschmajew, A., Vandereycken, B.: The geometry of algorithms using hierarchical tensors. Linear Algebra Appl. 439(1), 133\u2013166 (2013)","journal-title":"Linear Algebra Appl."},{"key":"1465_CR34","doi-asserted-by":"crossref","unstructured":"Uschmajew, A., Vandereycken, B.: Geometric methods on low-rank matrix and tensor manifolds. In: Handbook of variational methods for nonlinear geometric data, pp. 261\u2013313. Springer, Cham (2020)","DOI":"10.1007\/978-3-030-31351-7_9"},{"key":"1465_CR35","volume-title":"Nonlinear Functional Analysis and Its Applications. III. Variational Methods and Optimization","author":"E Zeidler","year":"1985","unstructured":"Zeidler, E.: Nonlinear Functional Analysis and Its Applications. III. Variational Methods and Optimization. Springer-Verlag, New York (1985)"},{"key":"1465_CR36","volume-title":"Nonlinear Functional Analysis and Its Applications. IV. Applications to Mathematical Physics","author":"E Zeidler","year":"1988","unstructured":"Zeidler, E.: Nonlinear Functional Analysis and Its Applications. IV. Applications to Mathematical Physics. Springer-Verlag, New York (1988)"},{"key":"1465_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0981-2","volume-title":"Nonlinear Functional Analysis and Its Applications. II\/A. Linear monotone operators","author":"E Zeidler","year":"1990","unstructured":"Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II\/A. Linear monotone operators. Springer-Verlag, New York (1990)"},{"key":"1465_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0981-2","volume-title":"Nonlinear Functional Analysis and Its Applications. II\/B. Nonlinear Monotone Operators","author":"E Zeidler","year":"1990","unstructured":"Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II\/B. Nonlinear Monotone Operators. Springer-Verlag, New York (1990)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01465-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01465-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01465-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T10:57:55Z","timestamp":1748516275000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01465-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,23]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["1465"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01465-8","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"type":"print","value":"0029-599X"},{"type":"electronic","value":"0945-3245"}],"subject":[],"published":{"date-parts":[[2025,4,23]]},"assertion":[{"value":"14 September 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}