[{"author":[{"last_name":"Gladbach","full_name":"Gladbach, Peter","first_name":"Peter"},{"first_name":"Eva","last_name":"Kopfer","full_name":"Kopfer, Eva"},{"full_name":"Maas, Jan","last_name":"Maas","orcid":"0000-0002-0845-1338","first_name":"Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87"}],"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"}],"citation":{"short":"P. Gladbach, E. Kopfer, J. Maas, SIAM Journal on Mathematical Analysis 52 (2020) 2759–2802.","ista":"Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 52(3), 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>.","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>","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>"},"publication_status":"published","quality_controlled":"1","oa_version":"Preprint","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publication_identifier":{"eissn":["10957154"],"issn":["00361410"]},"_id":"71","article_processing_charge":"No","oa":1,"date_updated":"2023-09-18T08:13:15Z","volume":52,"publist_id":"7983","arxiv":1,"external_id":{"arxiv":["1809.01092"],"isi":["000546975100017"]},"title":"Scaling limits of discrete optimal transport","doi":"10.1137/19M1243440","year":"2020","main_file_link":[{"url":"https://arxiv.org/abs/1809.01092","open_access":"1"}],"isi":1,"status":"public","intvolume":"        52","type":"journal_article","day":"01","page":"2759-2802","publication":"SIAM Journal on Mathematical Analysis","issue":"3","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Society for Industrial and Applied Mathematics","date_published":"2020-10-01T00:00:00Z","article_type":"original","month":"10","date_created":"2018-12-11T11:44:28Z","department":[{"_id":"JaMa"}]},{"language":[{"iso":"eng"}],"publisher":"Society for Industrial and Applied Mathematics","scopus_import":"1","article_type":"original","date_published":"2020-12-15T00:00:00Z","month":"12","file":[{"checksum":"21aa1cf4c30a86a00cae15a984819b5d","date_created":"2021-01-25T07:48:39Z","file_size":310655,"file_name":"2020_SIAM_Fischer.pdf","access_level":"open_access","date_updated":"2021-01-25T07:48:39Z","success":1,"file_id":"9041","creator":"dernst","relation":"main_file","content_type":"application/pdf"}],"date_created":"2021-01-24T23:01:09Z","department":[{"_id":"JuFi"}],"has_accepted_license":"1","status":"public","intvolume":"        52","type":"journal_article","day":"15","page":"6222-6233","file_date_updated":"2021-01-25T07:48:39Z","issue":"6","publication":"SIAM Journal on Mathematical Analysis","title":"Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies","external_id":{"isi":["000600695200027"]},"ec_funded":1,"year":"2020","doi":"10.1137/20M1322182","ddc":["510"],"isi":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"author":[{"first_name":"Julian L","orcid":"0000-0002-0479-558X","last_name":"Fischer","full_name":"Fischer, Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Laux","full_name":"Laux, Tim","first_name":"Tim"},{"last_name":"Simon","full_name":"Simon, Theresa M.","first_name":"Theresa M."}],"abstract":[{"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.","lang":"eng"}],"publication_status":"published","citation":{"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.","short":"J.L. Fischer, T. Laux, T.M. Simon, SIAM Journal on Mathematical Analysis 52 (2020) 6222–6233.","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>","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>.","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>.","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.","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>"},"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.","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","project":[{"call_identifier":"H2020","name":"International IST Doctoral Program","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","grant_number":"665385"}],"oa_version":"Published Version","quality_controlled":"1","_id":"9039","publication_identifier":{"eissn":["10957154"],"issn":["00361410"]},"volume":52,"oa":1,"date_updated":"2023-08-24T11:15:16Z","article_processing_charge":"No"},{"article_processing_charge":"No","date_updated":"2023-09-22T09:43:36Z","volume":49,"oa":1,"publist_id":"6381","publication_identifier":{"issn":["00361410"]},"extern":"1","_id":"1014","quality_controlled":"1","oa_version":"Submitted Version","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"chicago":"Fischer, Julian L, and Claudia Raithel. “Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics , 2017. <a href=\"https://doi.org/10.1137/16M1070384\">https://doi.org/10.1137/16M1070384</a>.","apa":"Fischer, J. L., &#38; Raithel, C. (2017). Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/16M1070384\">https://doi.org/10.1137/16M1070384</a>","ieee":"J. L. Fischer and C. Raithel, “Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 49, no. 1. Society for Industrial and Applied Mathematics , pp. 82–114, 2017.","ista":"Fischer JL, Raithel C. 2017. Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space. SIAM Journal on Mathematical Analysis. 49(1), 82–114.","short":"J.L. Fischer, C. Raithel, SIAM Journal on Mathematical Analysis 49 (2017) 82–114.","mla":"Fischer, Julian L., and Claudia Raithel. “Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 49, no. 1, Society for Industrial and Applied Mathematics , 2017, pp. 82–114, doi:<a href=\"https://doi.org/10.1137/16M1070384\">10.1137/16M1070384</a>.","ama":"Fischer JL, Raithel C. Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space. <i>SIAM Journal on Mathematical Analysis</i>. 2017;49(1):82-114. doi:<a href=\"https://doi.org/10.1137/16M1070384\">10.1137/16M1070384</a>"},"publication_status":"published","abstract":[{"text":"We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated C1,α-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are based on the existence of homogenization correctors adapted to the half-space setting, which we construct-by an entirely deterministic argument-as a modification of the homogenization corrector on the whole space. This adaption procedure is carried out inductively on larger scales, crucially relying on the regularity theory already established on smaller scales.","lang":"eng"}],"author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","last_name":"Fischer","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","first_name":"Julian L"},{"full_name":"Raithel, Claudia","last_name":"Raithel","first_name":"Claudia"}],"isi":1,"main_file_link":[{"url":"https://arxiv.org/abs/1703.04328","open_access":"1"}],"year":"2017","doi":"10.1137/16M1070384","external_id":{"isi":["000396681800004"]},"title":"Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space","publication":"SIAM Journal on Mathematical Analysis","issue":"1","page":"82 - 114","day":"12","type":"journal_article","intvolume":"        49","status":"public","date_created":"2018-12-11T11:49:41Z","month":"01","date_published":"2017-01-12T00:00:00Z","publisher":"Society for Industrial and Applied Mathematics ","language":[{"iso":"eng"}]}]