{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T18:29:14Z","timestamp":1783621754982,"version":"3.55.0"},"reference-count":38,"publisher":"American Mathematical Society (AMS)","issue":"297","license":[{"start":{"date-parts":[[2016,5,26]],"date-time":"2016-05-26T00:00:00Z","timestamp":1464220800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["279389"],"award-info":[{"award-number":["279389"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider a numerical scheme for Hamilton\u2013Jacobi equations based on a direct discretization of the Lax\u2013Oleinik semi\u2013group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a\n                    <italic>geometric integrator<\/italic>\n                    satisfying a discrete weak\u2013KAM theorem which allows us to control its long time behavior. Taking advantage of a fast algorithm for computing (min,plus) convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.\n                  <\/p>","DOI":"10.1090\/mcom\/2976","type":"journal-article","created":{"date-parts":[[2015,5,26]],"date-time":"2015-05-26T10:25:29Z","timestamp":1432635929000},"page":"85-117","source":"Crossref","is-referenced-by-count":8,"title":["Fast weak\u2013KAM integrators for separable Hamiltonian systems"],"prefix":"10.1090","volume":"85","author":[{"given":"Anne","family":"Bouillard","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erwan","family":"Faou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maxime","family":"Zavidovique","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,5,26]]},"reference":[{"issue":"12","key":"1","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1002\/(SICI)1097-0312(199612)49:12<1339::AID-CPA5>3.0.CO;2-B","article-title":"Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes","volume":"49","author":"Abgrall, R.","year":"1996","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1137\/060655286","article-title":"The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis","volume":"47","author":"Akian, Marianne","year":"2008","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"397","DOI":"10.2969\/aspm\/04720397","article-title":"Weak KAM pairs and Monge-Kantorovich duality","author":"Bernard, Patrick","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9784931469419"},{"key":"4","series-title":"Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics","isbn-type":"print","volume-title":"Synchronization and linearity","author":"Baccelli, Fran\u00e7ois Louis","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/047193609X"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"540","DOI":"10.1137\/S003614290343815X","article-title":"Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations","volume":"43","author":"Barles, Guy","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"[BJT08] A. Bouillard, L. Jouhet, and E. Thierry, Computation of a (min,+) multi-dimensional convolution for end-to-end performance analysis. In Proceedings of Valuetools\u20192008, 2008.","DOI":"10.4108\/ICST.VALUETOOLS2008.4349"},{"issue":"3-4","key":"7","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1081\/PDE-120030404","article-title":"Convergence to time-periodic solutions in time-periodic Hamilton-Jacobi equations on the circle","volume":"29","author":"Bernard, Patrick","year":"2004","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"issue":"20","key":"8","first-page":"587","article-title":"Un algorithme rapide pour le calcul de transform\u00e9es de Legendre-Fenchel discr\u00e8tes","volume":"308","author":"Brenier, Yann","year":"1989","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0249-6291","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"crossref","first-page":"271","DOI":"10.3233\/ASY-1991-4305","article-title":"Convergence of approximation schemes for fully nonlinear second order equations","volume":"4","author":"Barles, G.","year":"1991","journal-title":"Asymptotic Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0921-7134","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10626-007-0028-x","article-title":"An algorithmic toolbox for network calculus","volume":"18","author":"Bouillard, Anne","year":"2008","journal-title":"Discrete Event Dyn. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/0924-6703","issn-type":"print"},{"key":"11","unstructured":"[CISM00] G. Contreras, R. Iturriaga, and H. Sanchez-Morgado, Weak solutions of the Hamilton-Jacobi equation for time periodic Lagrangians, preprint, 2000."},{"issue":"167","key":"12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2007396","article-title":"Two approximations of solutions of Hamilton-Jacobi equations","volume":"43","author":"Crandall, M. G.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01442644","article-title":"A numerical approach to the infinite horizon problem of deterministic control theory","volume":"15","author":"Falcone, M.","year":"1987","journal-title":"Appl. Math. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0095-4616","issn-type":"print"},{"issue":"9","key":"14","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1016\/S0764-4442(97)87883-4","article-title":"Th\u00e9or\u00e8me KAM faible et th\u00e9orie de Mather sur les syst\u00e8mes lagrangiens","volume":"324","author":"Fathi, Albert","year":"1997","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/S0764-4442(98)80144-4","article-title":"Sur la convergence du semi-groupe de Lax-Oleinik","volume":"327","author":"Fathi, Albert","year":"1998","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"16","unstructured":"[Fat05] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, preliminary version, Pisa, 16 f\u00e9vrier 2005."},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1006\/jcph.2001.6954","article-title":"Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods","volume":"175","author":"Falcone, M.","year":"2002","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1-2","key":"18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00030-007-2047-6","article-title":"Weak KAM theorem on non compact manifolds","volume":"14","author":"Fathi, Albert","year":"2007","journal-title":"NoDEA Nonlinear Differential Equations Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1021-9722","issn-type":"print"},{"key":"19","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Geometric numerical integration","volume":"31","author":"Hairer, Ernst","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/3540306633","edition":"2"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1112\/plms\/s3-73.1.216","article-title":"Minimizing measures for time-dependent Lagrangians","volume":"73","author":"Iturriaga, Renato","year":"1996","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1137\/S003614290036823X","article-title":"On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms","volume":"39","author":"Jakobsen, Espen Robstad","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"22","doi-asserted-by":"publisher","first-page":"2126","DOI":"10.1137\/S106482759732455X","article-title":"Weighted ENO schemes for Hamilton-Jacobi equations","volume":"21","author":"Jiang, Guang-Shan","year":"2000","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1006\/jcph.1996.0130","article-title":"Efficient implementation of weighted ENO schemes","volume":"126","author":"Jiang, Guang-Shan","year":"1996","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"24","doi-asserted-by":"publisher","first-page":"2385","DOI":"10.1137\/S0036142996314366","article-title":"Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, relaxation schemes","volume":"35","author":"Jin, Shi","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"25","series-title":"Lecture Notes in Computer Science","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45318-0","volume-title":"Network calculus","volume":"2050","author":"Le Boudec, Jean-Yves","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/354042184X"},{"key":"26","series-title":"Research Notes in Mathematics","isbn-type":"print","volume-title":"Generalized solutions of Hamilton-Jacobi equations","volume":"69","author":"Lions, Pierre-Louis","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0273085565"},{"key":"27","unstructured":"[LPV87] P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint, 1987."},{"key":"28","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","volume-title":"Simulating Hamiltonian dynamics","volume":"14","author":"Leimkuhler, Benedict","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0521772907"},{"issue":"4","key":"29","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/s002110050102","article-title":"Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations","volume":"69","author":"Lions, P.-L.","year":"1995","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"30","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1007\/PL00005430","article-title":"\ud835\udc3f\u00b9-stability and error estimates for approximate Hamilton-Jacobi solutions","volume":"87","author":"Lin, Chi-Tien","year":"2001","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"31","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1023\/A:1019191114493","article-title":"Faster than the fast Legendre transform, the linear-time Legendre transform","volume":"16","author":"Lucet, Yves","year":"1997","journal-title":"Numer. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/1017-1398","issn-type":"print"},{"issue":"2","key":"32","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/BF02571383","article-title":"Action minimizing invariant measures for positive definite Lagrangian systems","volume":"207","author":"Mather, John N.","year":"1991","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"1","key":"33","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","article-title":"Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations","volume":"79","author":"Osher, Stanley","year":"1988","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"4","key":"34","doi-asserted-by":"publisher","first-page":"907","DOI":"10.1137\/0728049","article-title":"High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations","volume":"28","author":"Osher, Stanley","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"9","key":"35","doi-asserted-by":"publisher","first-page":"1238","DOI":"10.1016\/j.apnum.2006.03.006","article-title":"An approximation scheme for the effective Hamiltonian and applications","volume":"56","author":"Rorro, M.","year":"2006","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"292","key":"36","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1090\/S0025-5718-2014-02863-9","article-title":"Stochastic and variational approach to the Lax-Friedrichs scheme","volume":"84","author":"Soga, Kohei","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-0396(85)90136-6","article-title":"Approximation schemes for viscosity solutions of Hamilton-Jacobi equations","volume":"59","author":"Souganidis, Panagiotis E.","year":"1985","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"1","key":"38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4171\/CMH\/247","article-title":"Strict sub-solutions and Ma\u00f1\u00e9 potential in discrete weak KAM theory","volume":"87","author":"Zavidovique, Maxime","year":"2012","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-297\/S0025-5718-2015-02976-7\/S0025-5718-2015-02976-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-297\/S0025-5718-2015-02976-7\/S0025-5718-2015-02976-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:37:53Z","timestamp":1776796673000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-297\/S0025-5718-2015-02976-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,26]]},"references-count":38,"journal-issue":{"issue":"297","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["S0025-5718-2015-02976-7"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/2976","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,5,26]]}}}