[{"external_id":{"isi":["000990804300001"],"arxiv":["2106.11217"]},"title":"A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature","ec_funded":1,"doi":"10.1016/j.jfa.2023.109963","year":"2023","related_material":{"record":[{"relation":"earlier_version","id":"9792","status":"public"}]},"article_number":"109963","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2106.11217","open_access":"1"}],"isi":1,"author":[{"orcid":"0000-0003-0754-8530","last_name":"Feliciangeli","full_name":"Feliciangeli, Dario","first_name":"Dario","id":"41A639AA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Gerolin, Augusto","last_name":"Gerolin","first_name":"Augusto"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","full_name":"Portinale, Lorenzo","last_name":"Portinale","first_name":"Lorenzo"}],"abstract":[{"text":"This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.","lang":"eng"}],"citation":{"ama":"Feliciangeli D, Gerolin A, Portinale L. A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature. <i>Journal of Functional Analysis</i>. 2023;285(4). doi:<a href=\"https://doi.org/10.1016/j.jfa.2023.109963\">10.1016/j.jfa.2023.109963</a>","mla":"Feliciangeli, Dario, et al. “A Non-Commutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature.” <i>Journal of Functional Analysis</i>, vol. 285, no. 4, 109963, Elsevier, 2023, doi:<a href=\"https://doi.org/10.1016/j.jfa.2023.109963\">10.1016/j.jfa.2023.109963</a>.","ista":"Feliciangeli D, Gerolin A, Portinale L. 2023. A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature. Journal of Functional Analysis. 285(4), 109963.","short":"D. Feliciangeli, A. Gerolin, L. Portinale, Journal of Functional Analysis 285 (2023).","apa":"Feliciangeli, D., Gerolin, A., &#38; Portinale, L. (2023). A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature. <i>Journal of Functional Analysis</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jfa.2023.109963\">https://doi.org/10.1016/j.jfa.2023.109963</a>","ieee":"D. Feliciangeli, A. Gerolin, and L. Portinale, “A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature,” <i>Journal of Functional Analysis</i>, vol. 285, no. 4. Elsevier, 2023.","chicago":"Feliciangeli, Dario, Augusto Gerolin, and Lorenzo Portinale. “A Non-Commutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature.” <i>Journal of Functional Analysis</i>. Elsevier, 2023. <a href=\"https://doi.org/10.1016/j.jfa.2023.109963\">https://doi.org/10.1016/j.jfa.2023.109963</a>."},"publication_status":"published","oa_version":"Preprint","project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117"},{"grant_number":"694227","call_identifier":"H2020","name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425"},{"grant_number":" F06504","_id":"260482E2-B435-11E9-9278-68D0E5697425","name":"Taming Complexity in Partial Di erential Systems","call_identifier":"FWF"}],"quality_controlled":"1","acknowledgement":"This work started when A.G. was visiting the Erwin Schrödinger Institute and then continued when D.F. and L.P visited the Theoretical Chemistry Department of the Vrije Universiteit Amsterdam. The authors thank the hospitality of both places and, especially, P. Gori-Giorgi and K. Giesbertz for fruitful discussions and literature suggestions in the early state of the project. The authors also thank J. Maas and R. Seiringer for their feedback and useful comments to a first draft of the article. Finally, we acknowledge the high quality review done by the anonymous referee of our paper, who we would like to thank for the excellent work and constructive feedback.\r\nD.F acknowledges support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No 716117 and No 694227). A.G. acknowledges funding by the HORIZON EUROPE European Research Council under H2020/MSCA-IF “OTmeetsDFT” [grant ID: 795942] as well as partial support of his research by the Canada Research Chairs Program (ID 2021-00234) and Natural Sciences and Engineering Research Council of Canada, RGPIN-2022-05207. L.P. acknowledges support by the Austrian Science Fund (FWF), grants No W1245 and No F65, and by the Deutsche Forschungsgemeinschaft (DFG) - Project number 390685813.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["0022-1236"],"eissn":["1096-0783"]},"_id":"12911","article_processing_charge":"No","volume":285,"oa":1,"date_updated":"2023-11-14T13:21:01Z","arxiv":1,"language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Elsevier","date_published":"2023-08-15T00:00:00Z","article_type":"original","month":"08","date_created":"2023-05-07T22:01:02Z","department":[{"_id":"RoSe"},{"_id":"JaMa"}],"status":"public","intvolume":"       285","type":"journal_article","day":"15","publication":"Journal of Functional Analysis","issue":"4"},{"language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","date_published":"2023-04-28T00:00:00Z","article_type":"original","month":"04","file":[{"success":1,"content_type":"application/pdf","relation":"main_file","creator":"dernst","file_id":"14393","file_name":"2023_CalculusEquations_Gladbach.pdf","file_size":1240995,"date_created":"2023-10-04T11:34:10Z","checksum":"359bee38d94b7e0aa73925063cb8884d","date_updated":"2023-10-04T11:34:10Z","access_level":"open_access"}],"date_created":"2023-05-14T22:01:00Z","department":[{"_id":"JaMa"}],"has_accepted_license":"1","status":"public","intvolume":"        62","type":"journal_article","day":"28","file_date_updated":"2023-10-04T11:34:10Z","publication":"Calculus of Variations and Partial Differential Equations","issue":"5","title":"Homogenisation of dynamical optimal transport on periodic graphs","external_id":{"isi":["000980588900001"],"arxiv":["2110.15321"]},"ec_funded":1,"doi":"10.1007/s00526-023-02472-z","year":"2023","ddc":["510"],"isi":1,"article_number":"143","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":[{"last_name":"Gladbach","full_name":"Gladbach, Peter","first_name":"Peter"},{"first_name":"Eva","last_name":"Kopfer","full_name":"Kopfer, Eva"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","orcid":"0000-0002-0845-1338","last_name":"Maas","full_name":"Maas, Jan"},{"full_name":"Portinale, Lorenzo","last_name":"Portinale","first_name":"Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"This paper deals with the large-scale behaviour of dynamical optimal transport on Zd\r\n-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ\r\n-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs."}],"citation":{"short":"P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Calculus of Variations and Partial Differential Equations 62 (2023).","ista":"Gladbach P, Kopfer E, Maas J, Portinale L. 2023. Homogenisation of dynamical optimal transport on periodic graphs. Calculus of Variations and Partial Differential Equations. 62(5), 143.","ama":"Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of dynamical optimal transport on periodic graphs. <i>Calculus of Variations and Partial Differential Equations</i>. 2023;62(5). doi:<a href=\"https://doi.org/10.1007/s00526-023-02472-z\">10.1007/s00526-023-02472-z</a>","mla":"Gladbach, Peter, et al. “Homogenisation of Dynamical Optimal Transport on Periodic Graphs.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 62, no. 5, 143, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00526-023-02472-z\">10.1007/s00526-023-02472-z</a>.","chicago":"Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation of Dynamical Optimal Transport on Periodic Graphs.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00526-023-02472-z\">https://doi.org/10.1007/s00526-023-02472-z</a>.","apa":"Gladbach, P., Kopfer, E., Maas, J., &#38; Portinale, L. (2023). Homogenisation of dynamical optimal transport on periodic graphs. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-023-02472-z\">https://doi.org/10.1007/s00526-023-02472-z</a>","ieee":"P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of dynamical optimal transport on periodic graphs,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 62, no. 5. Springer Nature, 2023."},"publication_status":"published","oa_version":"Published Version","quality_controlled":"1","project":[{"grant_number":"716117","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics"},{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","name":"Dissipation and Dispersion in Nonlinear Partial Differential Equations","call_identifier":"FWF"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"J.M. gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 716117). J.M and L.P. also acknowledge support from the Austrian Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support by the German Research Foundation through the Hausdorff Center for Mathematics and the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—350398276. We thank the anonymous reviewer for the careful reading and for useful suggestions. Open access funding provided by Austrian Science Fund (FWF).","publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"_id":"12959","article_processing_charge":"Yes (via OA deal)","oa":1,"date_updated":"2023-10-04T11:34:49Z","volume":62,"arxiv":1},{"language":[{"iso":"eng"}],"publisher":"Society for Industrial and Applied Mathematics","scopus_import":"1","date_published":"2022-07-18T00:00:00Z","article_type":"original","month":"07","date_created":"2022-08-07T22:01:59Z","department":[{"_id":"JaMa"}],"status":"public","intvolume":"        54","type":"journal_article","day":"18","page":"4297-4333","issue":"4","publication":"SIAM Journal on Mathematical Analysis","title":"Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","external_id":{"arxiv":["2008.10962"],"isi":["000889274600001"]},"ec_funded":1,"doi":"10.1137/21M1410968","year":"2022","related_material":{"record":[{"status":"public","id":"10022","relation":"earlier_version"}]},"isi":1,"main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2008.10962"}],"author":[{"last_name":"Forkert","full_name":"Forkert, Dominik L","first_name":"Dominik L","id":"35C79D68-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Jan","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","full_name":"Portinale, Lorenzo","last_name":"Portinale","first_name":"Lorenzo"}],"keyword":["Fokker--Planck equation","gradient flow","evolutionary $\\Gamma$-convergence"],"abstract":[{"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.","lang":"eng"}],"publication_status":"published","citation":{"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.","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>.","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>","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.","short":"D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis 54 (2022) 4297–4333."},"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.","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","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"}],"quality_controlled":"1","oa_version":"Preprint","_id":"11739","publication_identifier":{"issn":["0036-1410"],"eissn":["1095-7154"]},"volume":54,"oa":1,"date_updated":"2023-08-03T12:37:21Z","article_processing_charge":"No","arxiv":1},{"ddc":["515"],"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"10022"},{"status":"public","id":"9792","relation":"part_of_dissertation"},{"status":"public","id":"7573","relation":"part_of_dissertation"}]},"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)"},"alternative_title":["ISTA Thesis"],"title":"Discrete-to-continuum limits of transport problems and gradient flows in the space of measures","year":"2021","doi":"10.15479/at:ista:10030","acknowledged_ssus":[{"_id":"M-Shop"},{"_id":"NanoFab"}],"oa_version":"Published Version","project":[{"name":"Dissipation and Dispersion in Nonlinear Partial Differential Equations","_id":"260788DE-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"acknowledgement":"The author gratefully acknowledges support by the Austrian Science Fund (FWF), grants No W1245.","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publication_identifier":{"issn":["2663-337X"]},"_id":"10030","article_processing_charge":"No","date_updated":"2023-09-07T13:31:06Z","oa":1,"author":[{"first_name":"Lorenzo","last_name":"Portinale","full_name":"Portinale, Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"This PhD thesis is primarily focused on the study of discrete transport problems, introduced for the first time in the seminal works of Maas [Maa11] and Mielke [Mie11] on finite state Markov chains and reaction-diffusion equations, respectively. More in detail, my research focuses on the study of transport costs on graphs, in particular the convergence and the stability of such problems in the discrete-to-continuum limit. This thesis also includes some results concerning\r\nnon-commutative optimal transport. The first chapter of this thesis consists of a general introduction to the optimal transport problems, both in the discrete, the continuous, and the non-commutative setting. Chapters 2 and 3 present the content of two works, obtained in collaboration with Peter Gladbach, Eva Kopfer, and Jan Maas, where we have been able to show the convergence of discrete transport costs on periodic graphs to suitable continuous ones, which can be described by means of a homogenisation result. We first focus on the particular case of quadratic costs on the real line and then extending the result to more general costs in arbitrary dimension. Our results are the first complete characterisation of limits of transport costs on periodic graphs in arbitrary dimension which do not rely on any additional symmetry. In Chapter 4 we turn our attention to one of the intriguing connection between evolution equations and optimal transport, represented by the theory of gradient flows. We show that discrete gradient flow structures associated to a finite volume approximation of a certain class of diffusive equations (Fokker–Planck) is stable in the limit of vanishing meshes, reproving the convergence of the scheme via the method of evolutionary Γ-convergence and exploiting a more variational point of view on the problem. This is based on a collaboration with Dominik Forkert and Jan Maas. Chapter 5 represents a change of perspective, moving away from the discrete world and reaching the non-commutative one. As in the discrete case, we discuss how classical tools coming from the commutative optimal transport can be translated into the setting of density matrices. In particular, in this final chapter we present a non-commutative version of the Schrödinger problem (or entropic regularised optimal transport problem) and discuss existence and characterisation of minimisers, a duality result, and present a non-commutative version of the well-known Sinkhorn algorithm to compute the above mentioned optimisers. This is based on a joint work with Dario Feliciangeli and Augusto Gerolin. Finally, Appendix A and B contain some additional material and discussions, with particular attention to Harnack inequalities and the regularity of flows on discrete spaces."}],"citation":{"short":"L. Portinale, Discrete-to-Continuum Limits of Transport Problems and Gradient Flows in the Space of Measures, Institute of Science and Technology Austria, 2021.","ista":"Portinale L. 2021. Discrete-to-continuum limits of transport problems and gradient flows in the space of measures. Institute of Science and Technology Austria.","ama":"Portinale L. Discrete-to-continuum limits of transport problems and gradient flows in the space of measures. 2021. doi:<a href=\"https://doi.org/10.15479/at:ista:10030\">10.15479/at:ista:10030</a>","mla":"Portinale, Lorenzo. <i>Discrete-to-Continuum Limits of Transport Problems and Gradient Flows in the Space of Measures</i>. Institute of Science and Technology Austria, 2021, doi:<a href=\"https://doi.org/10.15479/at:ista:10030\">10.15479/at:ista:10030</a>.","chicago":"Portinale, Lorenzo. “Discrete-to-Continuum Limits of Transport Problems and Gradient Flows in the Space of Measures.” Institute of Science and Technology Austria, 2021. <a href=\"https://doi.org/10.15479/at:ista:10030\">https://doi.org/10.15479/at:ista:10030</a>.","ieee":"L. Portinale, “Discrete-to-continuum limits of transport problems and gradient flows in the space of measures,” Institute of Science and Technology Austria, 2021.","apa":"Portinale, L. (2021). <i>Discrete-to-continuum limits of transport problems and gradient flows in the space of measures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:10030\">https://doi.org/10.15479/at:ista:10030</a>"},"publication_status":"published","file":[{"file_id":"10032","creator":"cchlebak","relation":"source_file","content_type":"application/x-zip-compressed","access_level":"closed","date_updated":"2022-03-10T12:14:42Z","checksum":"8cd60dcb8762e8f21867e21e8001e183","date_created":"2021-09-21T09:17:34Z","file_size":3876668,"file_name":"tex_and_pictures.zip"},{"date_created":"2021-09-27T11:14:31Z","checksum":"9789e9d967c853c1503ec7f307170279","file_name":"thesis_portinale_Final (1).pdf","file_size":2532673,"date_updated":"2021-09-27T11:14:31Z","access_level":"open_access","creator":"cchlebak","file_id":"10047","content_type":"application/pdf","relation":"main_file"}],"date_created":"2021-09-21T09:14:15Z","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"has_accepted_license":"1","degree_awarded":"PhD","language":[{"iso":"eng"}],"publisher":"Institute of Science and Technology Austria","date_published":"2021-09-22T00:00:00Z","month":"09","file_date_updated":"2022-03-10T12:14:42Z","supervisor":[{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","last_name":"Maas"}],"status":"public","type":"dissertation","day":"22"},{"article_processing_charge":"No","date_updated":"2023-11-14T13:21:01Z","oa":1,"arxiv":1,"oa_version":"Preprint","project":[{"call_identifier":"H2020","name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","grant_number":"694227"},{"call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"},{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This work started when A.G. was visiting the Erwin Schrödinger Institute and then continued when D.F. and L.P visited the Theoretical Chemistry Department of the Vrije Universiteit Amsterdam. The authors thanks the hospitality of both places and, especially, P. Gori-Giorgi and K. Giesbertz for fruitful discussions and literature suggestions in the early state of the project. Finally, the authors also thanks J. Maas and R. Seiringer for their feedback and useful comments to a first draft of the article.  L.P. acknowledges support by the Austrian Science Fund (FWF), grants No W1245 and NoF65. D.F acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 716117 and No 694227). A.G. acknowledges funding by the European Research Council under H2020/MSCA-IF “OTmeetsDFT” [grant ID: 795942].","_id":"9792","citation":{"mla":"Feliciangeli, Dario, et al. “A Non-Commutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature.” <i>ArXiv</i>, 2106.11217, doi:<a href=\"https://doi.org/10.48550/arXiv.2106.11217\">10.48550/arXiv.2106.11217</a>.","ama":"Feliciangeli D, Gerolin A, Portinale L. A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2106.11217\">10.48550/arXiv.2106.11217</a>","short":"D. Feliciangeli, A. Gerolin, L. Portinale, ArXiv (n.d.).","ista":"Feliciangeli D, Gerolin A, Portinale L. A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature. arXiv, 2106.11217.","apa":"Feliciangeli, D., Gerolin, A., &#38; Portinale, L. (n.d.). A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2106.11217\">https://doi.org/10.48550/arXiv.2106.11217</a>","ieee":"D. Feliciangeli, A. Gerolin, and L. Portinale, “A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature,” <i>arXiv</i>. .","chicago":"Feliciangeli, Dario, Augusto Gerolin, and Lorenzo Portinale. “A Non-Commutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2106.11217\">https://doi.org/10.48550/arXiv.2106.11217</a>."},"publication_status":"submitted","author":[{"full_name":"Feliciangeli, Dario","last_name":"Feliciangeli","orcid":"0000-0003-0754-8530","first_name":"Dario","id":"41A639AA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Augusto","full_name":"Gerolin, Augusto","last_name":"Gerolin"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","first_name":"Lorenzo","last_name":"Portinale","full_name":"Portinale, Lorenzo"}],"abstract":[{"text":"This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on careful a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.","lang":"eng"}],"article_number":"2106.11217","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2106.11217"}],"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)"},"ddc":["510"],"related_material":{"record":[{"relation":"dissertation_contains","id":"9733","status":"public"},{"status":"public","relation":"dissertation_contains","id":"10030"},{"id":"12911","relation":"later_version","status":"public"}]},"ec_funded":1,"year":"2021","doi":"10.48550/arXiv.2106.11217","external_id":{"arxiv":["2106.11217"]},"title":"A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature","publication":"arXiv","type":"preprint","day":"21","status":"public","department":[{"_id":"RoSe"},{"_id":"JaMa"}],"has_accepted_license":"1","date_created":"2021-08-06T09:07:12Z","date_published":"2021-07-21T00:00:00Z","month":"07","language":[{"iso":"eng"}]},{"issue":"7","publication":"Journal de Mathematiques Pures et Appliquees","page":"204-234","day":"01","type":"journal_article","intvolume":"       139","status":"public","department":[{"_id":"JaMa"}],"date_created":"2020-03-08T23:00:47Z","month":"07","date_published":"2020-07-01T00:00:00Z","article_type":"original","publisher":"Elsevier","scopus_import":"1","language":[{"iso":"eng"}],"arxiv":1,"date_updated":"2023-09-07T13:31:05Z","volume":139,"oa":1,"article_processing_charge":"No","_id":"7573","publication_identifier":{"issn":["00217824"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","acknowledgement":"J.M. gratefully acknowledges support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 716117). J.M. and L.P. also acknowledge support from the Austrian Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support by the German Research Foundation through the Hausdorff Center for Mathematics and the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 350398276.","project":[{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":" F06504","name":"Taming Complexity in Partial Di erential Systems","_id":"260482E2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","name":"Dissipation and Dispersion in Nonlinear Partial Differential Equations","call_identifier":"FWF"}],"oa_version":"Preprint","quality_controlled":"1","publication_status":"published","citation":{"chicago":"Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation of One-Dimensional Discrete Optimal Transport.” <i>Journal de Mathematiques Pures et Appliquees</i>. Elsevier, 2020. <a href=\"https://doi.org/10.1016/j.matpur.2020.02.008\">https://doi.org/10.1016/j.matpur.2020.02.008</a>.","apa":"Gladbach, P., Kopfer, E., Maas, J., &#38; Portinale, L. (2020). Homogenisation of one-dimensional discrete optimal transport. <i>Journal de Mathematiques Pures et Appliquees</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.matpur.2020.02.008\">https://doi.org/10.1016/j.matpur.2020.02.008</a>","ieee":"P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of one-dimensional discrete optimal transport,” <i>Journal de Mathematiques Pures et Appliquees</i>, vol. 139, no. 7. Elsevier, pp. 204–234, 2020.","short":"P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Journal de Mathematiques Pures et Appliquees 139 (2020) 204–234.","ista":"Gladbach P, Kopfer E, Maas J, Portinale L. 2020. Homogenisation of one-dimensional discrete optimal transport. Journal de Mathematiques Pures et Appliquees. 139(7), 204–234.","ama":"Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of one-dimensional discrete optimal transport. <i>Journal de Mathematiques Pures et Appliquees</i>. 2020;139(7):204-234. doi:<a href=\"https://doi.org/10.1016/j.matpur.2020.02.008\">10.1016/j.matpur.2020.02.008</a>","mla":"Gladbach, Peter, et al. “Homogenisation of One-Dimensional Discrete Optimal Transport.” <i>Journal de Mathematiques Pures et Appliquees</i>, vol. 139, no. 7, Elsevier, 2020, pp. 204–34, doi:<a href=\"https://doi.org/10.1016/j.matpur.2020.02.008\">10.1016/j.matpur.2020.02.008</a>."},"abstract":[{"lang":"eng","text":"This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such metrics appear naturally in discretisations of -gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to , unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport."}],"author":[{"last_name":"Gladbach","full_name":"Gladbach, Peter","first_name":"Peter"},{"first_name":"Eva","last_name":"Kopfer","full_name":"Kopfer, Eva"},{"first_name":"Jan","full_name":"Maas, Jan","last_name":"Maas","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","first_name":"Lorenzo","full_name":"Portinale, Lorenzo","last_name":"Portinale"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1905.05757"}],"isi":1,"related_material":{"record":[{"status":"public","id":"10030","relation":"dissertation_contains"}]},"doi":"10.1016/j.matpur.2020.02.008","year":"2020","ec_funded":1,"external_id":{"isi":["000539439400008"],"arxiv":["1905.05757"]},"title":"Homogenisation of one-dimensional discrete optimal transport"},{"article_processing_charge":"No","page":"33","date_updated":"2023-09-07T13:31:05Z","oa":1,"publication":"arXiv","arxiv":1,"project":[{"grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"oa_version":"Preprint","acknowledgement":"This work is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117) and by the Austrian Science Fund (FWF), grants No F65 and W1245.","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","_id":"10022","type":"preprint","citation":{"ista":"Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv, 2008.10962.","short":"D.L. Forkert, J. Maas, L. Portinale, ArXiv (n.d.).","ama":"Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>arXiv</i>.","mla":"Forkert, Dominik L., et al. “Evolutionary Γ-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” <i>ArXiv</i>, 2008.10962.","chicago":"Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary Γ-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” <i>ArXiv</i>, n.d.","ieee":"D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” <i>arXiv</i>. .","apa":"Forkert, D. L., Maas, J., &#38; Portinale, L. (n.d.). Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>arXiv</i>."},"day":"25","publication_status":"submitted","status":"public","author":[{"full_name":"Forkert, Dominik L","last_name":"Forkert","first_name":"Dominik L","id":"35C79D68-F248-11E8-B48F-1D18A9856A87"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","full_name":"Maas, Jan","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"last_name":"Portinale","full_name":"Portinale, Lorenzo","first_name":"Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in 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 Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising 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."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2008.10962"}],"article_number":"2008.10962","department":[{"_id":"JaMa"}],"date_created":"2021-09-17T10:57:27Z","related_material":{"record":[{"relation":"later_version","id":"11739","status":"public"},{"id":"10030","relation":"dissertation_contains","status":"public"}]},"date_published":"2020-08-25T00:00:00Z","ec_funded":1,"month":"08","year":"2020","language":[{"iso":"eng"}],"title":"Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","external_id":{"arxiv":["2008.10962"]}},{"date_created":"2020-02-28T10:54:41Z","department":[{"_id":"JaMa"}],"language":[{"iso":"eng"}],"publisher":"Gakko Tosho","article_type":"original","date_published":"2019-10-22T00:00:00Z","month":"10","page":"425-447","issue":"2","publication":"Advances in Mathematical Sciences and Applications","status":"public","intvolume":"        28","type":"journal_article","day":"22","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.1910.10050","open_access":"1"}],"title":"Penalization via global functionals of optimal-control problems for dissipative evolution","external_id":{"arxiv":["1910.10050"]},"year":"2019","acknowledgement":"This work is supported by Vienna Science and Technology Fund (WWTF) through Project MA14-009 and by the Austrian Science Fund (FWF) projects F 65 and I 2375.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","project":[{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"}],"oa_version":"Preprint","_id":"7550","publication_identifier":{"issn":["1343-4373"]},"oa":1,"volume":28,"date_updated":"2022-06-17T07:52:41Z","article_processing_charge":"No","arxiv":1,"author":[{"first_name":"Lorenzo","last_name":"Portinale","full_name":"Portinale, Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Ulisse","full_name":"Stefanelli, Ulisse","last_name":"Stefanelli"}],"abstract":[{"lang":"eng","text":"We consider an optimal control problem for an abstract nonlinear dissipative evolution equation. The differential constraint is penalized by augmenting the target functional by a nonnegative global-in-time functional which is null-minimized in the evolution equation is satisfied. Different variational settings are presented, leading to the convergence of the penalization method for gradient flows, noncyclic and semimonotone flows, doubly nonlinear evolutions, and GENERIC systems. "}],"publication_status":"published","citation":{"short":"L. Portinale, U. Stefanelli, Advances in Mathematical Sciences and Applications 28 (2019) 425–447.","ista":"Portinale L, Stefanelli U. 2019. Penalization via global functionals of optimal-control problems for dissipative evolution. Advances in Mathematical Sciences and Applications. 28(2), 425–447.","ama":"Portinale L, Stefanelli U. Penalization via global functionals of optimal-control problems for dissipative evolution. <i>Advances in Mathematical Sciences and Applications</i>. 2019;28(2):425-447.","mla":"Portinale, Lorenzo, and Ulisse Stefanelli. “Penalization via Global Functionals of Optimal-Control Problems for Dissipative Evolution.” <i>Advances in Mathematical Sciences and Applications</i>, vol. 28, no. 2, Gakko Tosho, 2019, pp. 425–47.","chicago":"Portinale, Lorenzo, and Ulisse Stefanelli. “Penalization via Global Functionals of Optimal-Control Problems for Dissipative Evolution.” <i>Advances in Mathematical Sciences and Applications</i>. Gakko Tosho, 2019.","ieee":"L. Portinale and U. Stefanelli, “Penalization via global functionals of optimal-control problems for dissipative evolution,” <i>Advances in Mathematical Sciences and Applications</i>, vol. 28, no. 2. Gakko Tosho, pp. 425–447, 2019.","apa":"Portinale, L., &#38; Stefanelli, U. (2019). Penalization via global functionals of optimal-control problems for dissipative evolution. <i>Advances in Mathematical Sciences and Applications</i>. Gakko Tosho."}}]