[{"citation":{"short":"F. Quattrocchi, Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures, Institute of Science and Technology Austria, 2025.","ieee":"F. Quattrocchi, “Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures,” Institute of Science and Technology Austria, 2025.","ista":"Quattrocchi F. 2025. Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. Institute of Science and Technology Austria.","chicago":"Quattrocchi, Filippo. “Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/AT-ISTA-20563\">https://doi.org/10.15479/AT-ISTA-20563</a>.","ama":"Quattrocchi F. Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. 2025. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-20563\">10.15479/AT-ISTA-20563</a>","mla":"Quattrocchi, Filippo. <i>Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-20563\">10.15479/AT-ISTA-20563</a>.","apa":"Quattrocchi, F. (2025). <i>Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-20563\">https://doi.org/10.15479/AT-ISTA-20563</a>"},"related_material":{"record":[{"id":"20569","relation":"part_of_dissertation","status":"public"},{"relation":"part_of_dissertation","id":"20571","status":"public"},{"relation":"part_of_dissertation","id":"20570","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18706"}]},"project":[{"grant_number":"F06504","name":"Taming Complexity in Partial Differential Systems","_id":"260482E2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"alternative_title":["ISTA Thesis"],"publication_identifier":{"issn":["2663-337X"]},"degree_awarded":"PhD","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"date_published":"2025-11-03T00:00:00Z","page":"240","publisher":"Institute of Science and Technology Austria","date_updated":"2026-06-18T18:02:13Z","publication_status":"published","month":"11","supervisor":[{"orcid":"0000-0002-0845-1338","first_name":"Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","last_name":"Maas"}],"oa_version":"Published Version","title":"Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures","type":"dissertation","file_date_updated":"2026-01-01T23:30:03Z","_id":"20563","abstract":[{"text":"The theory of optimal transport provides an elegant and powerful description of many evolution\r\nequations as gradient flows. The primary objective of this thesis is to adapt and extend the\r\ntheory to deal with important equations that are not covered by the classical framework,\r\nspecifically boundary value problems and kinetic equations. Additionally, we establish new\r\nresults in periodic homogenization for discrete dynamical optimal transport and in quantization\r\nof measures.\r\nSection 1.1 serves as an invitation to the classical theory of optimal transport, including the\r\nmain definitions and a selection of well-established theorems. Sections 1.2-1.5 introduce the\r\nmain results of this thesis, outline the motivations, and review the current state of the art.\r\nIn Chapter 2, we consider the Fokker–Planck equation on a bounded set with positive Dirichlet\r\nboundary conditions. We construct a time-discrete scheme involving a modification of the\r\nWasserstein distance and, under weak assumptions, prove its convergence to a solution of this\r\nboundary value problem. In dimension 1, we show that this solution is a gradient flow in a\r\nsuitable space of measures.\r\nChapter 3 presents joint work with Giovanni Brigati and Jan Maas. We introduce a new theory\r\nof optimal transport to describe and study particle systems at the mesoscopic scale. We prove\r\nadapted versions of some fundamental theorems, including the Benamou–Brenier formula and\r\nthe identification of absolutely continuous curves of measures.\r\nChapter 4 presents joint work with Lorenzo Portinale. We prove convergence of dynamical\r\ntransportation functionals on periodic graphs in the large-scale limit when the cost functional\r\nis asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances converge\r\nto 1-Wasserstein distances constructed from crystalline norms on R\r\nd\r\n.\r\nChapter 5 concerns optimal empirical quantization: the problem of approximating a measure\r\nby the sum of n equally weighted Dirac deltas, so as to minimize the error in the p-Wasserstein\r\ndistance. Our main result is an analog of Zador’s theorem, providing asymptotic bounds for\r\nthe minimal error as n tends to infinity.\r\n","lang":"eng"}],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","OA_place":"publisher","ddc":["515","519"],"oa":1,"author":[{"orcid":"0009-0000-9773-1931","first_name":"Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","full_name":"Quattrocchi, Filippo","last_name":"Quattrocchi"}],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"article_processing_charge":"No","year":"2025","file":[{"access_level":"open_access","date_updated":"2026-01-01T23:30:03Z","creator":"fquattro","file_name":"2025_quattrocchi_filippo_thesis.pdf","file_size":4326411,"embargo":"2026-01-01","checksum":"6f55275bdf99992be3a6457d949dd664","date_created":"2025-11-17T21:04:15Z","file_id":"20653","content_type":"application/pdf","relation":"main_file"},{"relation":"source_file","checksum":"707e580f5d993a214c0dba456b75837b","content_type":"application/zip","file_id":"20654","embargo_to":"open_access","date_created":"2025-11-17T21:05:43Z","file_name":"2025_quattrocchi_thesis.zip","file_size":11726509,"access_level":"closed","date_updated":"2026-01-01T23:30:03Z","creator":"fquattro"}],"status":"public","keyword":["optimal transport","kinetic equations","boundary value problems","quantization","gradient flows","homogenization"],"day":"03","has_accepted_license":"1","language":[{"iso":"eng"}],"doi":"10.15479/AT-ISTA-20563","acknowledgement":"The research contained in this thesis has received funding from the Austrian Science\r\nFund (FWF) project 10.55776/F65.","corr_author":"1","date_created":"2025-10-28T13:10:49Z"},{"date_published":"2024-08-23T00:00:00Z","month":"08","date_updated":"2026-07-07T22:30:39Z","publication_status":"draft","title":"Asymptotics for optimal empirical quantization of measures","oa_version":"Preprint","_id":"20570","type":"preprint","arxiv":1,"citation":{"chicago":"Quattrocchi, Filippo. “Asymptotics for Optimal Empirical Quantization of Measures.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2408.12924\">https://doi.org/10.48550/arXiv.2408.12924</a>.","ama":"Quattrocchi F. Asymptotics for optimal empirical quantization of measures. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2408.12924\">10.48550/arXiv.2408.12924</a>","mla":"Quattrocchi, Filippo. “Asymptotics for Optimal Empirical Quantization of Measures.” <i>ArXiv</i>, 2408.12924, doi:<a href=\"https://doi.org/10.48550/arXiv.2408.12924\">10.48550/arXiv.2408.12924</a>.","apa":"Quattrocchi, F. (n.d.). Asymptotics for optimal empirical quantization of measures. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2408.12924\">https://doi.org/10.48550/arXiv.2408.12924</a>","short":"F. Quattrocchi, ArXiv (n.d.).","ista":"Quattrocchi F. Asymptotics for optimal empirical quantization of measures. arXiv, 2408.12924.","ieee":"F. Quattrocchi, “Asymptotics for optimal empirical quantization of measures,” <i>arXiv</i>. ."},"project":[{"grant_number":"F06504","name":"Taming Complexity in Partial Differential Systems","call_identifier":"FWF","_id":"260482E2-B435-11E9-9278-68D0E5697425"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"20563"}]},"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"year":"2024","day":"23","OA_type":"green","keyword":["optimal empirical quantization","vector quantization","Wasserstein distance","semidiscrete optimal transport","Zador’s Theorem","Pierce’s Lemma"],"status":"public","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2408.12924"}],"acknowledgement":"The author is thankful to Nicolas Clozeau, Lorenzo Dello Schiavo, Jan Maas, Dejan Slepčev,\r\nand Dario Trevisan for many fruitful discussions and comments. The author gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.","corr_author":"1","date_created":"2025-10-28T13:12:22Z","publication":"arXiv","doi":"10.48550/arXiv.2408.12924","abstract":[{"lang":"eng","text":"We investigate the minimal error in approximating a general probability\r\nmeasure $\\mu$ on $\\mathbb{R}^d$ by the uniform measure on a finite set with\r\nprescribed cardinality $n$. The error is measured in the $p$-Wasserstein\r\ndistance. In particular, when $1\\le p<d$, we establish asymptotic upper and\r\nlower bounds as $n \\to \\infty$ on the rescaled minimal error that have the\r\nsame, explicit dependency on $\\mu$.\r\n  In some instances, we prove that the rescaled minimal error has a limit.\r\nThese include general measures in dimension $d = 2$ with $1 \\le p < 2$, and\r\nuniform measures in arbitrary dimension with $1 \\le p < d$. For some uniform\r\nmeasures, we prove the limit existence for $p \\ge d$ as well.\r\n  For a class of compactly supported measures with H\\\"older densities, we\r\ndetermine the convergence speed of the minimal error for every $p \\ge 1$.\r\n  Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper\r\nestimate of the minimal error when $1 \\le p < d$.\r\n  In the initial sections, we survey the state of the art and draw connections\r\nwith similar problems, such as classical and random quantization."}],"external_id":{"arxiv":["2408.12924"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"repository","author":[{"orcid":"0009-0000-9773-1931","first_name":"Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","full_name":"Quattrocchi, Filippo","last_name":"Quattrocchi"}],"article_number":"2408.12924","oa":1,"article_processing_charge":"No"}]