[{"related_material":{"record":[{"id":"20563","relation":"dissertation_contains","status":"public"}]},"title":"Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs","intvolume":"        37","has_accepted_license":"1","year":"2026","abstract":[{"text":"We prove discrete-to-continuum convergence for dynamical optimal transport on  Zd\r\n -periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of  1 -Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.","lang":"eng"}],"date_updated":"2026-10-27T23:31:03Z","researchdata_availability":"no","month":"06","file":[{"relation":"main_file","success":1,"creator":"dernst","checksum":"d038f4d00cbfbde2672c17138eab21c9","content_type":"application/pdf","date_created":"2026-07-23T05:55:03Z","file_size":612317,"access_level":"open_access","file_id":"22386","file_name":"2026_EuropJourAppliedMath_Portinale.pdf","date_updated":"2026-07-23T05:55:03Z"}],"publication":"European Journal of Applied Mathematics","das_tickbox":"0","supplementarymaterial":"no","issue":"3","acknowledgement":"L.P. gratefully acknowledges fundings from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – GZ 2047/1, Projekt-ID 390685813. F.Q. gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.","OA_type":"gold","article_processing_charge":"Yes","type":"journal_article","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"doi":"10.1017/s0956792524000810","file_date_updated":"2026-07-23T05:55:03Z","keyword":["optimal transport","discrete-to-continuum","homogenisation","linear growth","gamma-convergence"],"publication_identifier":{"eissn":["1469-4425"],"issn":["0956-7925"]},"external_id":{"isi":["001381435800001"]},"date_published":"2026-06-01T00:00:00Z","article_type":"original","DOAJ_listed":"1","PlanS_conform":"1","publisher":"Cambridge University Press","status":"public","OA_place":"publisher","author":[{"full_name":"Portinale, Lorenzo","first_name":"Lorenzo","last_name":"Portinale","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Filippo","full_name":"Quattrocchi, Filippo","orcid":"0009-0000-9773-1931","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","last_name":"Quattrocchi"}],"page":"614-642","citation":{"ieee":"L. Portinale and F. Quattrocchi, “Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs,” <i>European Journal of Applied Mathematics</i>, vol. 37, no. 3. Cambridge University Press, pp. 614–642, 2026.","short":"L. Portinale, F. Quattrocchi, European Journal of Applied Mathematics 37 (2026) 614–642.","chicago":"Portinale, Lorenzo, and Filippo Quattrocchi. “Discrete-to-Continuum Limits of Optimal Transport with Linear Growth on Periodic Graphs.” <i>European Journal of Applied Mathematics</i>. Cambridge University Press, 2026. <a href=\"https://doi.org/10.1017/s0956792524000810\">https://doi.org/10.1017/s0956792524000810</a>.","apa":"Portinale, L., &#38; Quattrocchi, F. (2026). Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. <i>European Journal of Applied Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0956792524000810\">https://doi.org/10.1017/s0956792524000810</a>","ama":"Portinale L, Quattrocchi F. Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. <i>European Journal of Applied Mathematics</i>. 2026;37(3):614-642. doi:<a href=\"https://doi.org/10.1017/s0956792524000810\">10.1017/s0956792524000810</a>","mla":"Portinale, Lorenzo, and Filippo Quattrocchi. “Discrete-to-Continuum Limits of Optimal Transport with Linear Growth on Periodic Graphs.” <i>European Journal of Applied Mathematics</i>, vol. 37, no. 3, Cambridge University Press, 2026, pp. 614–42, doi:<a href=\"https://doi.org/10.1017/s0956792524000810\">10.1017/s0956792524000810</a>.","ista":"Portinale L, Quattrocchi F. 2026. Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. European Journal of Applied Mathematics. 37(3), 614–642."},"isi":1,"quality_controlled":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","scopus_import":"1","_id":"18706","ddc":["500"],"date_created":"2024-12-23T11:03:59Z","day":"01","project":[{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"volume":37,"publication_status":"published","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","fulldoi":"https://doi.org/10.1017/s0956792524000810"},{"external_id":{"isi":["000889274600001"],"arxiv":["2008.10962"]},"publication_identifier":{"eissn":["1095-7154"],"issn":["0036-1410"]},"date_published":"2022-07-18T00:00:00Z","article_type":"original","department":[{"_id":"JaMa"}],"doi":"10.1137/21M1410968","keyword":["Fokker--Planck equation","gradient flow","evolutionary $\\Gamma$-convergence"],"page":"4297-4333","author":[{"last_name":"Forkert","id":"35C79D68-F248-11E8-B48F-1D18A9856A87","full_name":"Forkert, Dominik L","first_name":"Dominik L"},{"last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","first_name":"Jan"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","last_name":"Portinale","first_name":"Lorenzo","full_name":"Portinale, Lorenzo"}],"citation":{"apa":"Forkert, D. L., Maas, J., &#38; Portinale, L. (2022). Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/21M1410968\">https://doi.org/10.1137/21M1410968</a>","chicago":"Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2022. <a href=\"https://doi.org/10.1137/21M1410968\">https://doi.org/10.1137/21M1410968</a>.","short":"D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis 54 (2022) 4297–4333.","ieee":"D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4. Society for Industrial and Applied Mathematics, pp. 4297–4333, 2022.","ista":"Forkert DL, Maas J, Portinale L. 2022. Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. 54(4), 4297–4333.","mla":"Forkert, Dominik L., et al. “Evolutionary $\\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4, Society for Industrial and Applied Mathematics, 2022, pp. 4297–333, doi:<a href=\"https://doi.org/10.1137/21M1410968\">10.1137/21M1410968</a>.","ama":"Forkert DL, Maas J, Portinale L. Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(4):4297-4333. doi:<a href=\"https://doi.org/10.1137/21M1410968\">10.1137/21M1410968</a>"},"isi":1,"quality_controlled":"1","publisher":"Society for Industrial and Applied Mathematics","status":"public","ec_funded":1,"day":"18","date_created":"2022-08-07T22:01:59Z","corr_author":"1","oa_version":"Preprint","language":[{"iso":"eng"}],"scopus_import":"1","_id":"11739","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","fulldoi":"https://doi.org/10.1137/21M1410968","volume":54,"project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117"},{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"},{"call_identifier":"FWF","_id":"260788DE-B435-11E9-9278-68D0E5697425","name":"Dissipation and dispersion in nonlinear partial differential equations","grant_number":"W1245"}],"publication_status":"published","oa":1,"year":"2022","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2008.10962"}],"abstract":[{"lang":"eng","text":"We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality."}],"related_material":{"record":[{"id":"10022","status":"public","relation":"earlier_version"}]},"title":"Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","intvolume":"        54","date_updated":"2025-04-15T08:31:31Z","arxiv":1,"month":"07","publication":"SIAM Journal on Mathematical Analysis","issue":"4","acknowledgement":"This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme grant 716117 and by the AustrianScience Fund (FWF) through grants F65 and W1245.","type":"journal_article","article_processing_charge":"No"},{"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","fulldoi":"https://doi.org/10.1137/15M1035379","volume":54,"publication_status":"published","date_created":"2018-12-11T11:51:19Z","day":"02","language":[{"iso":"eng"}],"oa_version":"None","_id":"1315","author":[{"full_name":"Brunner, Fabian","first_name":"Fabian","last_name":"Brunner"},{"orcid":"0000-0002-0479-558X","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","last_name":"Fischer","first_name":"Julian L","full_name":"Fischer, Julian L"},{"last_name":"Knabner","full_name":"Knabner, Peter","first_name":"Peter"}],"page":"2359 - 2378","citation":{"ama":"Brunner F, Fischer JL, Knabner P. Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. <i>Journal on Numerical Analysis</i>. 2016;54(4):2359-2378. doi:<a href=\"https://doi.org/10.1137/15M1035379\">10.1137/15M1035379</a>","mla":"Brunner, Fabian, et al. “Analysis of a Modified Second-Order Mixed Hybrid BDM1 Finite Element Method for Transport Problems in Divergence Form.” <i>Journal on Numerical Analysis</i>, vol. 54, no. 4, Society for Industrial and Applied Mathematics , 2016, pp. 2359–78, doi:<a href=\"https://doi.org/10.1137/15M1035379\">10.1137/15M1035379</a>.","ista":"Brunner F, Fischer JL, Knabner P. 2016. Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. Journal on Numerical Analysis. 54(4), 2359–2378.","ieee":"F. Brunner, J. L. Fischer, and P. Knabner, “Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form,” <i>Journal on Numerical Analysis</i>, vol. 54, no. 4. Society for Industrial and Applied Mathematics , pp. 2359–2378, 2016.","short":"F. Brunner, J.L. Fischer, P. Knabner, Journal on Numerical Analysis 54 (2016) 2359–2378.","chicago":"Brunner, Fabian, Julian L Fischer, and Peter Knabner. “Analysis of a Modified Second-Order Mixed Hybrid BDM1 Finite Element Method for Transport Problems in Divergence Form.” <i>Journal on Numerical Analysis</i>. Society for Industrial and Applied Mathematics , 2016. <a href=\"https://doi.org/10.1137/15M1035379\">https://doi.org/10.1137/15M1035379</a>.","apa":"Brunner, F., Fischer, J. L., &#38; Knabner, P. (2016). Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. <i>Journal on Numerical Analysis</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/15M1035379\">https://doi.org/10.1137/15M1035379</a>"},"quality_controlled":"1","publisher":"Society for Industrial and Applied Mathematics ","status":"public","publication_identifier":{"issnl":["0036-1429"],"eissn":["1095-7170"]},"date_published":"2016-08-02T00:00:00Z","article_type":"original","doi":"10.1137/15M1035379","keyword":["advection-diffusion problem","mixed finite element methods","suboptimal conver-gence","optimal convergence"],"extern":"1","type":"journal_article","OA_type":"closed access","article_processing_charge":"No","issue":"4","date_updated":"2026-05-18T09:48:39Z","month":"08","publication":"Journal on Numerical Analysis","year":"2016","publist_id":"5954","abstract":[{"lang":"eng","text":"We prove optimal second order convergence of a modified lowest-order Brezzi-Douglas-Marini (BDM1) mixed finite element scheme for advection-diffusion problems in divergence form. If advection is present, it is known that the total flux is approximated only with first-order accuracy by the classical BDM1 mixed method, which is suboptimal since the same order of convergence is obtained if the computationally less expensive Raviart-Thomas (RT0) element is used. The modification that was first proposed by Brunner et al. [Adv. Water Res., 35 (2012),pp. 163-171] is based on the hybrid problem formulation and consists in using the Lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown."}],"intvolume":"        54","title":"Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form"}]