[{"day":"03","alternative_title":["ISTA Thesis"],"date_updated":"2026-06-18T18:02:13Z","_id":"20563","title":"Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures","OA_place":"publisher","supervisor":[{"first_name":"Jan","last_name":"Maas","full_name":"Maas, Jan","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87"}],"file":[{"date_created":"2025-11-17T21:04:15Z","relation":"main_file","access_level":"open_access","creator":"fquattro","content_type":"application/pdf","file_size":4326411,"embargo":"2026-01-01","checksum":"6f55275bdf99992be3a6457d949dd664","file_id":"20653","file_name":"2025_quattrocchi_filippo_thesis.pdf","date_updated":"2026-01-01T23:30:03Z"},{"date_created":"2025-11-17T21:05:43Z","relation":"source_file","access_level":"closed","embargo_to":"open_access","creator":"fquattro","file_size":11726509,"content_type":"application/zip","checksum":"707e580f5d993a214c0dba456b75837b","file_id":"20654","file_name":"2025_quattrocchi_thesis.zip","date_updated":"2026-01-01T23:30:03Z"}],"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"publisher":"Institute of Science and Technology Austria","has_accepted_license":"1","keyword":["optimal transport","kinetic equations","boundary value problems","quantization","gradient flows","homogenization"],"citation":{"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>","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>","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>.","ieee":"F. Quattrocchi, “Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures,” Institute of Science and Technology Austria, 2025.","short":"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>."},"article_processing_charge":"No","type":"dissertation","degree_awarded":"PhD","ddc":["515","519"],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa_version":"Published Version","page":"240","related_material":{"record":[{"id":"20569","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"20571","relation":"part_of_dissertation"},{"id":"20570","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"18706"}]},"status":"public","publication_identifier":{"issn":["2663-337X"]},"doi":"10.15479/AT-ISTA-20563","month":"11","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"}],"date_created":"2025-10-28T13:10:49Z","date_published":"2025-11-03T00:00:00Z","file_date_updated":"2026-01-01T23:30:03Z","year":"2025","oa":1,"acknowledgement":"The research contained in this thesis has received funding from the Austrian Science\r\nFund (FWF) project 10.55776/F65.","language":[{"iso":"eng"}],"publication_status":"published","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)"},"corr_author":"1","author":[{"orcid":"0009-0000-9773-1931","full_name":"Quattrocchi, Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","first_name":"Filippo","last_name":"Quattrocchi"}],"project":[{"name":"Taming Complexity in Partial Differential Systems","call_identifier":"FWF","grant_number":"F06504","_id":"260482E2-B435-11E9-9278-68D0E5697425"}]},{"date_updated":"2026-06-28T22:30:41Z","OA_type":"green","day":"10","publication":"arXiv","keyword":["optimal transport","kinetic theory","second-order discrepancy","Vlasov equation","Wasserstein splines."],"citation":{"short":"G. Brigati, J. Maas, F. Quattrocchi, ArXiv (n.d.).","ista":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv, 2502.15665.","chicago":"Brigati, Giovanni, Jan Maas, and Filippo Quattrocchi. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2502.15665\">https://doi.org/10.48550/arXiv.2502.15665</a>.","ama":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2502.15665\">10.48550/arXiv.2502.15665</a>","apa":"Brigati, G., Maas, J., &#38; Quattrocchi, F. (n.d.). Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2502.15665\">https://doi.org/10.48550/arXiv.2502.15665</a>","mla":"Brigati, Giovanni, et al. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.” <i>ArXiv</i>, 2502.15665, doi:<a href=\"https://doi.org/10.48550/arXiv.2502.15665\">10.48550/arXiv.2502.15665</a>.","ieee":"G. Brigati, J. Maas, and F. Quattrocchi, “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures,” <i>arXiv</i>. ."},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2502.15665","open_access":"1"}],"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"OA_place":"repository","title":"Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures","_id":"20569","type":"preprint","article_processing_charge":"No","arxiv":1,"doi":"10.48550/arXiv.2502.15665","status":"public","ec_funded":1,"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","related_material":{"record":[{"status":"public","id":"20563","relation":"dissertation_contains"}]},"abstract":[{"lang":"eng","text":"This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy d between two probability distributions in position and velocity states, which is reminiscent of the 2-Wasserstein distance, but of second-order nature. We construct d in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon T. Second, we further optimise over the time horizon T > 0. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of d. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of d holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by d, and identify solutions to Vlasov's equations with curves of measures satisfying a certain d-absolute continuity condition. One consequence is an explicit formula for the d-derivative of such curves."}],"month":"08","acknowledgement":"This work was partially inspired by an unpublished note from 2014 by Guillaume Carlier,\r\nJean Dolbeault, and Bruno Nazaret. GB deeply thanks Jean Dolbeault for proposing\r\nthis problem to him, guiding him into the subject, and sharing the aforementioned note.\r\nWe are grateful to Karthik Elamvazhuthi for making us aware of the work [20].\r\nThe work of GB has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement\r\nNo 101034413.\r\nJM and FQ gratefully acknowledge support from the Austrian Science Fund (FWF)\r\nproject 10.55776/F65.","oa":1,"language":[{"iso":"eng"}],"date_published":"2025-08-10T00:00:00Z","year":"2025","date_created":"2025-10-28T13:12:08Z","author":[{"last_name":"Brigati","first_name":"Giovanni","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","full_name":"Brigati, Giovanni"},{"full_name":"Maas, Jan","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","last_name":"Maas"},{"last_name":"Quattrocchi","first_name":"Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","orcid":"0009-0000-9773-1931","full_name":"Quattrocchi, Filippo"}],"external_id":{"arxiv":["2502.15665"]},"corr_author":"1","publication_status":"draft","article_number":"2502.15665","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"},{"call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}]},{"date_updated":"2026-06-18T17:37:10Z","OA_type":"hybrid","day":"11","publication":"Foundations of Computational Mathematics","keyword":["Optimal transport","Hamilton-Jacobi equation","convex optimization"],"citation":{"chicago":"Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation.” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s10208-024-09686-3\">https://doi.org/10.1007/s10208-024-09686-3</a>.","ista":"Ishida S, Lavenant H. 2024. Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation. Foundations of Computational Mathematics.","short":"S. Ishida, H. Lavenant, Foundations of Computational Mathematics (2024).","ieee":"S. Ishida and H. Lavenant, “Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation,” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2024.","mla":"Ishida, Sadashige, and Hugo Lavenant. “Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation.” <i>Foundations of Computational Mathematics</i>, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s10208-024-09686-3\">10.1007/s10208-024-09686-3</a>.","apa":"Ishida, S., &#38; Lavenant, H. (2024). Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation. <i>Foundations of Computational Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10208-024-09686-3\">https://doi.org/10.1007/s10208-024-09686-3</a>","ama":"Ishida S, Lavenant H. Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation. <i>Foundations of Computational Mathematics</i>. 2024. doi:<a href=\"https://doi.org/10.1007/s10208-024-09686-3\">10.1007/s10208-024-09686-3</a>"},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s10208-024-09686-3"}],"department":[{"_id":"GradSch"},{"_id":"ChWo"}],"publisher":"Springer Nature","OA_place":"publisher","_id":"14703","title":"Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation","ddc":["000"],"type":"journal_article","article_processing_charge":"Yes (via OA deal)","arxiv":1,"publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"doi":"10.1007/s10208-024-09686-3","quality_controlled":"1","status":"public","scopus_import":"1","oa_version":"Published Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"lang":"eng","text":"We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton-Jacobi equation."}],"month":"11","oa":1,"language":[{"iso":"eng"}],"isi":1,"acknowledgement":"The authors would like to thank Chris Wojtan for his continuous support and several interesting discussions. Part of this research was performed during two visits: one of SI to the BIDSA research center at Bocconi University, and one of HL to the Institute of Science and Technology Austria. Both host institutions are warmly acknowledged for the hospitality. HL is partially supported by the MUR-Prin 2022-202244A7YL “Gradient Flows and Non-Smooth Geometric Structures with Applications to Optimization and Machine Learning”, funded by the European Union - Next Generation EU. SI is supported in part by ERC Consolidator Grant 101045083 “CoDiNA” funded by the European Research Council.","date_published":"2024-11-11T00:00:00Z","year":"2024","article_type":"original","date_created":"2023-12-21T10:14:37Z","author":[{"first_name":"Sadashige","last_name":"Ishida","full_name":"Ishida, Sadashige","orcid":"0000-0002-3121-3100","id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425"},{"last_name":"Lavenant","first_name":"Hugo","full_name":"Lavenant, Hugo"}],"external_id":{"isi":["001352503300001"],"arxiv":["2312.12213"]},"corr_author":"1","publication_status":"epub_ahead","project":[{"_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","grant_number":"101045083","name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena"}]},{"month":"04","abstract":[{"text":"We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\\Omega \\Subset \\mathbb{R}^d$ with general, positive and temporally constant, Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum.   In the special case where $\\Omega$ is an interval in $\\mathbb{R}^1$, we prove that such a solution is a gradient flow -- curve of maximal slope -- within a suitable space of measures, endowed with a modified Wasserstein distance.\r\nOur discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\\partial \\Omega$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure $\\overline \\Omega$. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $\\Omega$ is an interval in $\\mathbb{R}^1$, we find a formula for the descending slope of this geodesically nonconvex functional. ","lang":"eng"}],"acknowledgement":"The author would like to thank Jan Maas for suggesting this project and for many helpful\r\ncomments, Antonio Agresti, Lorenzo Dello Schiavo and Julian Fischer for several fruitful discussions, and Oliver Tse for pointing out the reference [15]. He also gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.\r\n","language":[{"iso":"eng"}],"oa":1,"date_created":"2025-10-28T13:12:56Z","date_published":"2024-04-09T00:00:00Z","year":"2024","author":[{"first_name":"Filippo","last_name":"Quattrocchi","full_name":"Quattrocchi, Filippo","orcid":"0009-0000-9773-1931","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308"}],"publication_status":"draft","article_number":"2403.07803","external_id":{"arxiv":["2403.07803"]},"corr_author":"1","project":[{"_id":"260482E2-B435-11E9-9278-68D0E5697425","name":"Taming Complexity in Partial Differential Systems","grant_number":"F06504","call_identifier":"FWF"}],"OA_type":"green","date_updated":"2026-06-28T22:30:41Z","day":"09","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2403.07803"}],"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"publication":"arXiv","citation":{"ista":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. arXiv, 2403.07803.","chicago":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2403.07803\">https://doi.org/10.48550/arXiv.2403.07803</a>.","short":"F. Quattrocchi, ArXiv (n.d.).","apa":"Quattrocchi, F. (n.d.). Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2403.07803\">https://doi.org/10.48550/arXiv.2403.07803</a>","mla":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>ArXiv</i>, 2403.07803, doi:<a href=\"https://doi.org/10.48550/arXiv.2403.07803\">10.48550/arXiv.2403.07803</a>.","ieee":"F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” <i>arXiv</i>. .","ama":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2403.07803\">10.48550/arXiv.2403.07803</a>"},"keyword":["gradient flows","Jordan–Kinderlehrer–Otto scheme","curves of maximal slope","optimal transport","Dirichlet boundary conditions","Fokker–Planck equation"],"_id":"20571","title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions","OA_place":"repository","arxiv":1,"article_processing_charge":"No","type":"preprint","status":"public","doi":"10.48550/arXiv.2403.07803","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","related_material":{"record":[{"status":"public","id":"20865","relation":"later_version"},{"relation":"dissertation_contains","id":"20563","status":"public"}]}},{"project":[{"name":"Taming Complexity in Partial Differential Systems","grant_number":"F06504","call_identifier":"FWF","_id":"260482E2-B435-11E9-9278-68D0E5697425"}],"corr_author":"1","external_id":{"arxiv":["2408.12924"]},"article_number":"2408.12924","publication_status":"draft","author":[{"last_name":"Quattrocchi","first_name":"Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","orcid":"0009-0000-9773-1931","full_name":"Quattrocchi, Filippo"}],"year":"2024","date_published":"2024-08-23T00:00:00Z","date_created":"2025-10-28T13:12:22Z","language":[{"iso":"eng"}],"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.","oa":1,"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."}],"month":"08","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"20563"}]},"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.48550/arXiv.2408.12924","status":"public","type":"preprint","article_processing_charge":"No","arxiv":1,"OA_place":"repository","_id":"20570","title":"Asymptotics for optimal empirical quantization of measures","citation":{"ista":"Quattrocchi F. Asymptotics for optimal empirical quantization of measures. arXiv, 2408.12924.","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>.","short":"F. Quattrocchi, ArXiv (n.d.).","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>","ieee":"F. Quattrocchi, “Asymptotics for optimal empirical quantization of measures,” <i>arXiv</i>. .","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>"},"keyword":["optimal empirical quantization","vector quantization","Wasserstein distance","semidiscrete optimal transport","Zador’s Theorem","Pierce’s Lemma"],"publication":"arXiv","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2408.12924","open_access":"1"}],"day":"23","date_updated":"2026-06-28T22:30:41Z","OA_type":"green"},{"day":"21","date_updated":"2023-09-04T11:42:10Z","_id":"10553","title":"Brief announcement: Be prepared when network goes bad: An asynchronous view-change protocol","publisher":"Association for Computing Machinery","department":[{"_id":"ElKo"}],"main_file_link":[{"url":"https://arxiv.org/abs/2103.03181","open_access":"1"}],"keyword":["optimal","state machine replication","fallback","asynchrony","byzantine faults"],"citation":{"ista":"Gelashvili R, Kokoris Kogias E, Spiegelman A, Xiang Z. 2021. Brief announcement: Be prepared when network goes bad: An asynchronous view-change protocol. Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing. PODC: Principles of Distributed Computing, 187–190.","chicago":"Gelashvili, Rati, Eleftherios Kokoris Kogias, Alexander Spiegelman, and Zhuolun Xiang. “Brief Announcement: Be Prepared When Network Goes Bad: An Asynchronous View-Change Protocol.” In <i>Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing</i>, 187–90. Association for Computing Machinery, 2021. <a href=\"https://doi.org/10.1145/3465084.3467941\">https://doi.org/10.1145/3465084.3467941</a>.","short":"R. Gelashvili, E. Kokoris Kogias, A. Spiegelman, Z. Xiang, in:, Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing, Association for Computing Machinery, 2021, pp. 187–190.","mla":"Gelashvili, Rati, et al. “Brief Announcement: Be Prepared When Network Goes Bad: An Asynchronous View-Change Protocol.” <i>Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing</i>, Association for Computing Machinery, 2021, pp. 187–90, doi:<a href=\"https://doi.org/10.1145/3465084.3467941\">10.1145/3465084.3467941</a>.","apa":"Gelashvili, R., Kokoris Kogias, E., Spiegelman, A., &#38; Xiang, Z. (2021). Brief announcement: Be prepared when network goes bad: An asynchronous view-change protocol. In <i>Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing</i> (pp. 187–190). Virtual, Italy: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3465084.3467941\">https://doi.org/10.1145/3465084.3467941</a>","ieee":"R. Gelashvili, E. Kokoris Kogias, A. Spiegelman, and Z. Xiang, “Brief announcement: Be prepared when network goes bad: An asynchronous view-change protocol,” in <i>Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing</i>, Virtual, Italy, 2021, pp. 187–190.","ama":"Gelashvili R, Kokoris Kogias E, Spiegelman A, Xiang Z. Brief announcement: Be prepared when network goes bad: An asynchronous view-change protocol. In: <i>Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing</i>. Association for Computing Machinery; 2021:187-190. doi:<a href=\"https://doi.org/10.1145/3465084.3467941\">10.1145/3465084.3467941</a>"},"publication":"Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing","article_processing_charge":"No","arxiv":1,"type":"conference","conference":{"name":"PODC: Principles of Distributed Computing","start_date":"2021-07-26","location":"Virtual, Italy","end_date":"2021-07-30"},"page":"187-190","scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","status":"public","quality_controlled":"1","publication_identifier":{"isbn":["9-781-4503-8548-0"]},"doi":"10.1145/3465084.3467941","month":"07","abstract":[{"lang":"eng","text":"The popularity of permissioned blockchain systems demands BFT SMR protocols that are efficient under good network conditions (synchrony) and robust under bad network conditions (asynchrony). The state-of-the-art partially synchronous BFT SMR protocols provide optimal linear communication cost per decision under synchrony and good leaders, but lose liveness under asynchrony. On the other hand, the state-of-the-art asynchronous BFT SMR protocols are live even under asynchrony, but always pay quadratic cost even under synchrony. In this paper, we propose a BFT SMR protocol that achieves the best of both worlds -- optimal linear cost per decision under good networks and leaders, optimal quadratic cost per decision under bad networks, and remains always live."}],"date_created":"2021-12-16T13:20:19Z","year":"2021","date_published":"2021-07-21T00:00:00Z","isi":1,"language":[{"iso":"eng"}],"oa":1,"publication_status":"published","external_id":{"isi":["000744439800018"],"arxiv":["2103.03181"]},"author":[{"first_name":"Rati","last_name":"Gelashvili","full_name":"Gelashvili, Rati"},{"full_name":"Kokoris Kogias, Eleftherios","id":"f5983044-d7ef-11ea-ac6d-fd1430a26d30","first_name":"Eleftherios","last_name":"Kokoris Kogias"},{"full_name":"Spiegelman, Alexander","first_name":"Alexander","last_name":"Spiegelman"},{"first_name":"Zhuolun","last_name":"Xiang","full_name":"Xiang, Zhuolun"}]},{"extern":"1","type":"journal_article","article_processing_charge":"No","quality_controlled":"1","doi":"10.1137/15M1035379","publication_identifier":{"issnl":["0036-1429"],"eissn":["1095-7170"]},"status":"public","page":"2359 - 2378","oa_version":"None","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_updated":"2026-05-18T09:48:39Z","OA_type":"closed access","day":"02","keyword":["advection-diffusion problem","mixed finite element methods","suboptimal conver-gence","optimal convergence"],"citation":{"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>","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>.","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.","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>","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.","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>.","short":"F. Brunner, J.L. Fischer, P. Knabner, Journal on Numerical Analysis 54 (2016) 2359–2378."},"publication":"Journal on Numerical Analysis","publisher":"Society for Industrial and Applied Mathematics ","title":"Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form","_id":"1315","author":[{"last_name":"Brunner","first_name":"Fabian","full_name":"Brunner, Fabian"},{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","last_name":"Fischer","first_name":"Julian L"},{"last_name":"Knabner","first_name":"Peter","full_name":"Knabner, Peter"}],"publication_status":"published","volume":54,"abstract":[{"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.","lang":"eng"}],"month":"08","language":[{"iso":"eng"}],"issue":"4","intvolume":"        54","year":"2016","article_type":"original","publist_id":"5954","date_published":"2016-08-02T00:00:00Z","date_created":"2018-12-11T11:51:19Z"}]