[{"volume":52,"abstract":[{"text":"We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric 𝕎2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of 𝕎2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.","lang":"eng"}],"doi":"10.1137/19M1243440","arxiv":1,"day":"01","isi":1,"external_id":{"arxiv":["1809.01092"],"isi":["000546975100017"]},"date_updated":"2023-09-18T08:13:15Z","citation":{"ista":"Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 52(3), 2759–2802.","short":"P. Gladbach, E. Kopfer, J. Maas, SIAM Journal on Mathematical Analysis 52 (2020) 2759–2802.","mla":"Gladbach, Peter, et al. “Scaling Limits of Discrete Optimal Transport.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 52, no. 3, Society for Industrial and Applied Mathematics, 2020, pp. 2759–802, doi:<a href=\"https://doi.org/10.1137/19M1243440\">10.1137/19M1243440</a>.","chicago":"Gladbach, Peter, Eva Kopfer, and Jan Maas. “Scaling Limits of Discrete Optimal Transport.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2020. <a href=\"https://doi.org/10.1137/19M1243440\">https://doi.org/10.1137/19M1243440</a>.","ieee":"P. Gladbach, E. Kopfer, and J. Maas, “Scaling limits of discrete optimal transport,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 52, no. 3. Society for Industrial and Applied Mathematics, pp. 2759–2802, 2020.","apa":"Gladbach, P., Kopfer, E., &#38; Maas, J. (2020). Scaling limits of discrete optimal transport. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/19M1243440\">https://doi.org/10.1137/19M1243440</a>","ama":"Gladbach P, Kopfer E, Maas J. Scaling limits of discrete optimal transport. <i>SIAM Journal on Mathematical Analysis</i>. 2020;52(3):2759-2802. doi:<a href=\"https://doi.org/10.1137/19M1243440\">10.1137/19M1243440</a>"},"year":"2020","article_type":"original","publisher":"Society for Industrial and Applied Mathematics","page":"2759-2802","quality_controlled":"1","title":"Scaling limits of discrete optimal transport","intvolume":"        52","publication_status":"published","date_created":"2018-12-11T11:44:28Z","department":[{"_id":"JaMa"}],"article_processing_charge":"No","author":[{"full_name":"Gladbach, Peter","first_name":"Peter","last_name":"Gladbach"},{"full_name":"Kopfer, Eva","first_name":"Eva","last_name":"Kopfer"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","first_name":"Jan","last_name":"Maas"}],"issue":"3","_id":"71","scopus_import":"1","status":"public","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","main_file_link":[{"url":"https://arxiv.org/abs/1809.01092","open_access":"1"}],"oa":1,"publist_id":"7983","publication_identifier":{"issn":["00361410"],"eissn":["10957154"]},"date_published":"2020-10-01T00:00:00Z","type":"journal_article","language":[{"iso":"eng"}],"month":"10","oa_version":"Preprint","publication":"SIAM Journal on Mathematical Analysis"},{"file_date_updated":"2021-01-25T07:48:39Z","ec_funded":1,"quality_controlled":"1","page":"6222-6233","article_type":"original","publisher":"Society for Industrial and Applied Mathematics","issue":"6","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","last_name":"Fischer","first_name":"Julian L"},{"full_name":"Laux, Tim","first_name":"Tim","last_name":"Laux"},{"full_name":"Simon, Theresa M.","last_name":"Simon","first_name":"Theresa M."}],"license":"https://creativecommons.org/licenses/by/4.0/","scopus_import":"1","_id":"9039","intvolume":"        52","title":"Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies","article_processing_charge":"No","date_created":"2021-01-24T23:01:09Z","department":[{"_id":"JuFi"}],"publication_status":"published","ddc":["510"],"volume":52,"acknowledgement":"This work was supported by the European Union's Horizon 2020 Research and Innovation\r\nProgramme under Marie Sklodowska-Curie grant agreement 665385 and by the Deutsche\r\nForschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy, EXC-2047/1--390685813.","external_id":{"isi":["000600695200027"]},"isi":1,"citation":{"ieee":"J. L. Fischer, T. Laux, and T. M. Simon, “Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 52, no. 6. Society for Industrial and Applied Mathematics, pp. 6222–6233, 2020.","chicago":"Fischer, Julian L, Tim Laux, and Theresa M. Simon. “Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2020. <a href=\"https://doi.org/10.1137/20M1322182\">https://doi.org/10.1137/20M1322182</a>.","ama":"Fischer JL, Laux T, Simon TM. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. <i>SIAM Journal on Mathematical Analysis</i>. 2020;52(6):6222-6233. doi:<a href=\"https://doi.org/10.1137/20M1322182\">10.1137/20M1322182</a>","apa":"Fischer, J. L., Laux, T., &#38; Simon, T. M. (2020). Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/20M1322182\">https://doi.org/10.1137/20M1322182</a>","ista":"Fischer JL, Laux T, Simon TM. 2020. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. 52(6), 6222–6233.","mla":"Fischer, Julian L., et al. “Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 52, no. 6, Society for Industrial and Applied Mathematics, 2020, pp. 6222–33, doi:<a href=\"https://doi.org/10.1137/20M1322182\">10.1137/20M1322182</a>.","short":"J.L. Fischer, T. Laux, T.M. Simon, SIAM Journal on Mathematical Analysis 52 (2020) 6222–6233."},"year":"2020","date_updated":"2023-08-24T11:15:16Z","abstract":[{"lang":"eng","text":"We give a short and self-contained proof for rates of convergence of the Allen--Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen--Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems."}],"day":"15","doi":"10.1137/20M1322182","language":[{"iso":"eng"}],"has_accepted_license":"1","publication":"SIAM Journal on Mathematical Analysis","month":"12","project":[{"call_identifier":"H2020","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","grant_number":"665385","name":"International IST Doctoral Program"}],"oa_version":"Published Version","status":"public","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","file":[{"access_level":"open_access","success":1,"relation":"main_file","file_id":"9041","creator":"dernst","date_created":"2021-01-25T07:48:39Z","checksum":"21aa1cf4c30a86a00cae15a984819b5d","file_size":310655,"date_updated":"2021-01-25T07:48:39Z","file_name":"2020_SIAM_Fischer.pdf","content_type":"application/pdf"}],"type":"journal_article","date_published":"2020-12-15T00:00:00Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"oa":1,"publication_identifier":{"eissn":["10957154"],"issn":["00361410"]}}]