{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T19:30:01Z","timestamp":1783107001820,"version":"3.54.6"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,10,8]],"date-time":"2020-10-08T00:00:00Z","timestamp":1602115200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,10,8]],"date-time":"2020-10-08T00:00:00Z","timestamp":1602115200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,11]]},"DOI":"10.1007\/s00211-020-01154-8","type":"journal-article","created":{"date-parts":[[2020,10,8]],"date-time":"2020-10-08T04:02:43Z","timestamp":1602129763000},"page":"513-544","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["A fast approach to optimal transport: the back-and-forth method"],"prefix":"10.1007","volume":"146","author":[{"given":"Matt","family":"Jacobs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Flavien","family":"L\u00e9ger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,10,8]]},"reference":[{"key":"1154_CR1","unstructured":"Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: Proceedings of the 34th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol.\u00a070, pp. 214\u2013223. PMLR (2017). http:\/\/proceedings.mlr.press\/v70\/arjovsky17a.html"},{"issue":"1","key":"1154_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2140\/pjm.1966.16.1","volume":"16","author":"L Armijo","year":"1966","unstructured":"Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16(1), 1\u20133 (1966)","journal-title":"Pac. J. Math."},{"issue":"3","key":"1154_CR3","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s002110050002","volume":"84","author":"JD Benamou","year":"2000","unstructured":"Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge\u2013Kantorovich mass transfer problem. Numer. Math. 84(3), 375\u2013393 (2000). https:\/\/doi.org\/10.1007\/s002110050002","journal-title":"Numer. Math."},{"key":"1154_CR4","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.jcp.2013.12.015","volume":"260","author":"JD Benamou","year":"2014","unstructured":"Benamou, J.D., Froese, B.D., Oberman, A.M.: Numerical solution of the optimal transportation problem using the Monge\u2013Amp\u00e8re equation. J. Comput. Phys. 260, 107\u2013126 (2014). https:\/\/doi.org\/10.1016\/j.jcp.2013.12.015","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1154_CR5","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/BF02216923","volume":"20","author":"DP Bertsekas","year":"1989","unstructured":"Bertsekas, D.P., Castanon, D.A.: The auction algorithm for the transportation problem. Ann. Oper. Res. 20(1), 67\u201396 (1989). https:\/\/doi.org\/10.1007\/BF02216923","journal-title":"Ann. Oper. Res."},{"key":"1154_CR6","doi-asserted-by":"publisher","unstructured":"Brenier, Y.: A combinatorial algorithm for the Euler equations of incompressible flows. In: Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987), vol.\u00a075, pp. 325\u2013332 (1989). https:\/\/doi.org\/10.1016\/0045-7825(89)90033-9","DOI":"10.1016\/0045-7825(89)90033-9"},{"issue":"4","key":"1154_CR7","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1002\/cpa.3160440402","volume":"44","author":"Y Brenier","year":"1991","unstructured":"Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4), 375\u2013417 (1991). https:\/\/doi.org\/10.1002\/cpa.3160440402","journal-title":"Comm. Pure Appl. Math."},{"issue":"21\u201324","key":"1154_CR8","first-page":"1071","volume":"3","author":"R Chartrand","year":"2009","unstructured":"Chartrand, R., Wohlberg, B., Vixie, K.R., Bollt, E.M.: A gradient descent solution to the Monge\u2013Kantorovich problem. Appl. Math. Sci. (Ruse) 3(21\u201324), 1071\u20131080 (2009)","journal-title":"Appl. Math. Sci. (Ruse)"},{"issue":"4","key":"1154_CR9","doi-asserted-by":"publisher","first-page":"1534","DOI":"10.1137\/S0036142993260208","volume":"33","author":"L Corrias","year":"1996","unstructured":"Corrias, L.: Fast Legendre\u2013Fenchel transform and applications to Hamilton\u2013Jacobi equations and conservation laws. SIAM J. Numer. Anal. 33(4), 1534\u20131558 (1996). https:\/\/doi.org\/10.1137\/S0036142993260208","journal-title":"SIAM J. Numer. Anal."},{"key":"1154_CR10","unstructured":"Cuturi, M.: Sinkhorn distances: Lightspeed computation of optimal transport. In: Proceedings of the 26th International Conference on Neural Information Processing Systems\u2014Volume 2, NIPS\u201913, pp. 2292\u20132300. Curran Associates Inc., USA (2013). http:\/\/dl.acm.org\/citation.cfm?id=2999792.2999868"},{"issue":"4","key":"1154_CR11","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/BF00387715","volume":"128","author":"W Gangbo","year":"1994","unstructured":"Gangbo, W.: An elementary proof of the polar factorization of vector-valued functions. Arch. Rat. Mech. Anal. 128(4), 381\u2013399 (1994). https:\/\/doi.org\/10.1007\/BF00387715","journal-title":"Arch. Rat. Mech. Anal."},{"key":"1154_CR12","volume-title":"Quelques probl\u00e8mes d\u2019analyse non convexe, Habilitation \u00e0 diriger des recherches en math\u00e9matiques. Habilitation","author":"W Gangbo","year":"1995","unstructured":"Gangbo, W.: Quelques probl\u00e8mes d\u2019analyse non convexe, Habilitation \u00e0 diriger des recherches en math\u00e9matiques. Habilitation. Universit\u00e9 de Metz, Metz (1995)"},{"issue":"2","key":"1154_CR13","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/BF02392620","volume":"177","author":"W Gangbo","year":"1996","unstructured":"Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113\u2013161 (1996). https:\/\/doi.org\/10.1007\/BF02392620","journal-title":"Acta Math."},{"issue":"3","key":"1154_CR14","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1023\/B:VISI.0000036836.66311.97","volume":"60","author":"S Haker","year":"2004","unstructured":"Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. J. Comput. Vis. 60(3), 225\u2013240 (2004)","journal-title":"Int. J. Comput. Vis."},{"issue":"3","key":"1154_CR15","doi-asserted-by":"crossref","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","volume":"6","author":"M Huang","year":"2006","unstructured":"Huang, M., Malham\u00e9, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221\u2013251 (2006)","journal-title":"Commun. Inf. Syst."},{"key":"1154_CR16","doi-asserted-by":"crossref","unstructured":"Jacobs, M., Kim, I., M\u00e9sz\u00e1ros, A.R.: Weak solutions to the Muskat problem with surface tension via optimal transport. arXiv e-prints (2019)","DOI":"10.1007\/s00205-020-01579-3"},{"issue":"1","key":"1154_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/S0036141096303359","volume":"29","author":"R Jordan","year":"1998","unstructured":"Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker\u2013Planck equation. SIAM J. Math. Anal. 29(1), 1\u201317 (1998). https:\/\/doi.org\/10.1137\/S0036141096303359","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"1154_CR18","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s11537-007-0657-8","volume":"2","author":"JM Lasry","year":"2007","unstructured":"Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2(1), 229\u2013260 (2007). https:\/\/doi.org\/10.1007\/s11537-007-0657-8","journal-title":"Jpn. J. Math."},{"issue":"2","key":"1154_CR19","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1023\/A:1019191114493","volume":"16","author":"Y Lucet","year":"1997","unstructured":"Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16(2), 171\u2013185 (1997). https:\/\/doi.org\/10.1023\/A:1019191114493","journal-title":"Numer. Algorithms"},{"key":"1154_CR20","doi-asserted-by":"publisher","unstructured":"Nesterov, Y.: Introductory lectures on convex optimization. In: Applied Optimization, vol.\u00a087. Kluwer Academic Publishers, Boston, MA (2004). https:\/\/doi.org\/10.1007\/978-1-4419-8853-9","DOI":"10.1007\/978-1-4419-8853-9"},{"issue":"1\u20132","key":"1154_CR21","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1081\/PDE-100002243","volume":"26","author":"F Otto","year":"2001","unstructured":"Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26(1\u20132), 101\u2013174 (2001). https:\/\/doi.org\/10.1081\/PDE-100002243","journal-title":"Comm. Partial Differ. Equ."},{"key":"1154_CR22","doi-asserted-by":"publisher","unstructured":"Santambrogio, F.: Optimal transport for applied mathematicians. Progress in Nonlinear Differential Equations and their Applications, vol.\u00a087. Birkh\u00e4user\/Springer, Cham (2015). https:\/\/doi.org\/10.1007\/978-3-319-20828-2","DOI":"10.1007\/978-3-319-20828-2"},{"key":"1154_CR23","doi-asserted-by":"publisher","unstructured":"Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol.\u00a058. American Mathematical Society, Providence, RI (2003). https:\/\/doi.org\/10.1007\/b12016","DOI":"10.1007\/b12016"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01154-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-020-01154-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01154-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T14:25:12Z","timestamp":1669127112000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-020-01154-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,8]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["1154"],"URL":"https:\/\/doi.org\/10.1007\/s00211-020-01154-8","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,8]]},"assertion":[{"value":"29 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 May 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 September 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}