[{"publication":"Probability Theory and Related Fields","status":"public","type":"journal_article","day":"04","date_created":"2024-01-14T23:00:57Z","department":[{"_id":"JuFi"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","article_type":"original","date_published":"2024-01-04T00:00:00Z","month":"01","project":[{"grant_number":"948819","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","name":"Bridging Scales in Random Materials","call_identifier":"H2020"}],"oa_version":"Published Version","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"NC has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No 948819).\r\nFM is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the SPP 2265 Random Geometric Systems. FM has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics–Geometry–Structure. FM has been funded by the Max Planck Institute for Mathematics in the Sciences.","publication_identifier":{"issn":["0178-8051"],"eissn":["1432-2064"]},"_id":"14797","article_processing_charge":"Yes (in subscription journal)","oa":1,"date_updated":"2025-08-12T12:22:41Z","arxiv":1,"keyword":["Troll","Norway","Fjell"],"author":[{"last_name":"Clozeau","full_name":"Clozeau, Nicolas","first_name":"Nicolas","id":"fea1b376-906f-11eb-847d-b2c0cf46455b"},{"full_name":"Mattesini, Francesco","last_name":"Mattesini","first_name":"Francesco"}],"abstract":[{"lang":"eng","text":"We study a random matching problem on closed compact  2 -dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers  n  and  m=m(n)  of points, asymptotically equivalent as  n  goes to infinity, the optimal transport plan between the two empirical measures  μn  and  νm  is quantitatively well-approximated by  (Id,exp(∇hn))#μn  where  hn  solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Ampère equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the  α -mixing coefficient holds and for a class of discrete-time Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure."}],"citation":{"ieee":"N. Clozeau and F. Mattesini, “Annealed quantitative estimates for the quadratic 2D-discrete random matching problem,” <i>Probability Theory and Related Fields</i>. Springer Nature, 2024.","apa":"Clozeau, N., &#38; Mattesini, F. (2024). Annealed quantitative estimates for the quadratic 2D-discrete random matching problem. <i>Probability Theory and Related Fields</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00440-023-01254-0\">https://doi.org/10.1007/s00440-023-01254-0</a>","chicago":"Clozeau, Nicolas, and Francesco Mattesini. “Annealed Quantitative Estimates for the Quadratic 2D-Discrete Random Matching Problem.” <i>Probability Theory and Related Fields</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00440-023-01254-0\">https://doi.org/10.1007/s00440-023-01254-0</a>.","ama":"Clozeau N, Mattesini F. Annealed quantitative estimates for the quadratic 2D-discrete random matching problem. <i>Probability Theory and Related Fields</i>. 2024. doi:<a href=\"https://doi.org/10.1007/s00440-023-01254-0\">10.1007/s00440-023-01254-0</a>","mla":"Clozeau, Nicolas, and Francesco Mattesini. “Annealed Quantitative Estimates for the Quadratic 2D-Discrete Random Matching Problem.” <i>Probability Theory and Related Fields</i>, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s00440-023-01254-0\">10.1007/s00440-023-01254-0</a>.","ista":"Clozeau N, Mattesini F. 2024. Annealed quantitative estimates for the quadratic 2D-discrete random matching problem. Probability Theory and Related Fields.","short":"N. Clozeau, F. Mattesini, Probability Theory and Related Fields (2024)."},"publication_status":"epub_ahead","ddc":["510"],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00440-023-01254-0"}],"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)"},"title":"Annealed quantitative estimates for the quadratic 2D-discrete random matching problem","external_id":{"arxiv":["2303.00353"]},"ec_funded":1,"doi":"10.1007/s00440-023-01254-0","year":"2024"},{"date_created":"2023-06-11T22:00:40Z","has_accepted_license":"1","department":[{"_id":"JuFi"}],"scopus_import":"1","publisher":"Springer Nature","language":[{"iso":"eng"}],"month":"05","article_type":"original","date_published":"2023-05-30T00:00:00Z","publication":"Foundations of Computational Mathematics","status":"public","day":"30","type":"journal_article","ddc":["510"],"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)"},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s10208-023-09613-y"}],"isi":1,"external_id":{"isi":["000999623100001"]},"title":"Bias in the representative volume element method: Periodize the ensemble instead of its realizations","doi":"10.1007/s10208-023-09613-y","year":"2023","publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"_id":"13129","oa_version":"Published Version","quality_controlled":"1","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"Yes (via OA deal)","oa":1,"date_updated":"2023-08-02T06:12:39Z","abstract":[{"text":"We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨⋅⟩ and the corresponding linear elliptic operator −∇⋅a∇. In line with the theory of homogenization, the method proceeds by computing d=3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨⋅⟩. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L−1). In case of a suitable periodization of ⟨⋅⟩\r\n, we rigorously show that it is O(L−d). In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨⋅⟩\r\n of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.","lang":"eng"}],"author":[{"first_name":"Nicolas","full_name":"Clozeau, Nicolas","last_name":"Clozeau","id":"fea1b376-906f-11eb-847d-b2c0cf46455b"},{"full_name":"Josien, Marc","last_name":"Josien","first_name":"Marc"},{"last_name":"Otto","full_name":"Otto, Felix","first_name":"Felix"},{"first_name":"Qiang","last_name":"Xu","full_name":"Xu, Qiang"}],"citation":{"chicago":"Clozeau, Nicolas, Marc Josien, Felix Otto, and Qiang Xu. “Bias in the Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s10208-023-09613-y\">https://doi.org/10.1007/s10208-023-09613-y</a>.","apa":"Clozeau, N., Josien, M., Otto, F., &#38; Xu, Q. (2023). Bias in the representative volume element method: Periodize the ensemble instead of its realizations. <i>Foundations of Computational Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10208-023-09613-y\">https://doi.org/10.1007/s10208-023-09613-y</a>","ieee":"N. Clozeau, M. Josien, F. Otto, and Q. Xu, “Bias in the representative volume element method: Periodize the ensemble instead of its realizations,” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2023.","short":"N. Clozeau, M. Josien, F. Otto, Q. Xu, Foundations of Computational Mathematics (2023).","ista":"Clozeau N, Josien M, Otto F, Xu Q. 2023. Bias in the representative volume element method: Periodize the ensemble instead of its realizations. Foundations of Computational Mathematics.","ama":"Clozeau N, Josien M, Otto F, Xu Q. Bias in the representative volume element method: Periodize the ensemble instead of its realizations. <i>Foundations of Computational Mathematics</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s10208-023-09613-y\">10.1007/s10208-023-09613-y</a>","mla":"Clozeau, Nicolas, et al. “Bias in the Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.” <i>Foundations of Computational Mathematics</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s10208-023-09613-y\">10.1007/s10208-023-09613-y</a>."},"publication_status":"epub_ahead"},{"status":"public","intvolume":"        11","type":"journal_article","day":"01","page":"1254–1378","file_date_updated":"2023-08-14T11:51:04Z","publication":"Stochastics and Partial Differential Equations: Analysis and Computations","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2023-09-01T00:00:00Z","article_type":"original","month":"09","date_created":"2021-10-23T10:50:22Z","file":[{"success":1,"file_id":"14052","creator":"dernst","content_type":"application/pdf","relation":"main_file","date_created":"2023-08-14T11:51:04Z","checksum":"f83dcaecdbd3ace862c4ed97a20e8501","file_name":"2023_StochPartialDiffEquations_Clozeau.pdf","file_size":1635193,"date_updated":"2023-08-14T11:51:04Z","access_level":"open_access"}],"department":[{"_id":"JuFi"}],"has_accepted_license":"1","author":[{"last_name":"Clozeau","full_name":"Clozeau, Nicolas","first_name":"Nicolas","id":"fea1b376-906f-11eb-847d-b2c0cf46455b"}],"abstract":[{"text":"We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R larger than 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.","lang":"eng"}],"publication_status":"published","citation":{"short":"N. Clozeau, Stochastics and Partial Differential Equations: Analysis and Computations 11 (2023) 1254–1378.","ista":"Clozeau N. 2023. Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields. Stochastics and Partial Differential Equations: Analysis and Computations. 11, 1254–1378.","mla":"Clozeau, Nicolas. “Optimal Decay of the Parabolic Semigroup in Stochastic Homogenization  for Correlated Coefficient Fields.” <i>Stochastics and Partial Differential Equations: Analysis and Computations</i>, vol. 11, Springer Nature, 2023, pp. 1254–1378, doi:<a href=\"https://doi.org/10.1007/s40072-022-00254-w\">10.1007/s40072-022-00254-w</a>.","ama":"Clozeau N. Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields. <i>Stochastics and Partial Differential Equations: Analysis and Computations</i>. 2023;11:1254–1378. doi:<a href=\"https://doi.org/10.1007/s40072-022-00254-w\">10.1007/s40072-022-00254-w</a>","chicago":"Clozeau, Nicolas. “Optimal Decay of the Parabolic Semigroup in Stochastic Homogenization  for Correlated Coefficient Fields.” <i>Stochastics and Partial Differential Equations: Analysis and Computations</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s40072-022-00254-w\">https://doi.org/10.1007/s40072-022-00254-w</a>.","ieee":"N. Clozeau, “Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields,” <i>Stochastics and Partial Differential Equations: Analysis and Computations</i>, vol. 11. Springer Nature, pp. 1254–1378, 2023.","apa":"Clozeau, N. (2023). Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields. <i>Stochastics and Partial Differential Equations: Analysis and Computations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40072-022-00254-w\">https://doi.org/10.1007/s40072-022-00254-w</a>"},"acknowledgement":"I would like to thank my advisor Antoine Gloria for suggesting this problem to me, as well for many interesting discussions and suggestions.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","oa_version":"Published Version","_id":"10173","publication_identifier":{"issn":["2194-0401"]},"volume":11,"oa":1,"date_updated":"2023-08-14T11:51:47Z","article_processing_charge":"Yes (via OA deal)","arxiv":1,"external_id":{"arxiv":["2102.07452"],"isi":["000799715600001"]},"title":"Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields","doi":"10.1007/s40072-022-00254-w","year":"2023","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)"}},{"department":[{"_id":"JuFi"}],"article_number":"2104.04263","main_file_link":[{"url":"https://arxiv.org/abs/2104.04263","open_access":"1"}],"date_created":"2021-10-23T10:50:55Z","date_published":"2021-04-09T00:00:00Z","month":"04","year":"2021","language":[{"iso":"eng"}],"title":"Quantitative nonlinear homogenization: control of oscillations","external_id":{"arxiv":["2104.04263"]},"date_updated":"2021-10-28T15:44:05Z","oa":1,"article_processing_charge":"No","arxiv":1,"publication":"arXiv","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","acknowledgement":"The authors warmly thank Mitia Duerinckx for discussions on annealed estimates, and Mathias Schäffner for pointing out that the conditions of [14] apply to  ̄a in the setting of Theorem 2.2 and for discussions on regularity theory for operators with non-standard growth conditions. The authors received financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement n◦ 864066).","oa_version":"Preprint","_id":"10174","type":"preprint","publication_status":"submitted","day":"09","citation":{"ama":"Clozeau N, Gloria A. Quantitative nonlinear homogenization: control of oscillations. <i>arXiv</i>.","mla":"Clozeau, Nicolas, and Antoine Gloria. “Quantitative Nonlinear Homogenization: Control of Oscillations.” <i>ArXiv</i>, 2104.04263.","ista":"Clozeau N, Gloria A. Quantitative nonlinear homogenization: control of oscillations. arXiv, 2104.04263.","short":"N. Clozeau, A. Gloria, ArXiv (n.d.).","ieee":"N. Clozeau and A. Gloria, “Quantitative nonlinear homogenization: control of oscillations,” <i>arXiv</i>. .","apa":"Clozeau, N., &#38; Gloria, A. (n.d.). Quantitative nonlinear homogenization: control of oscillations. <i>arXiv</i>.","chicago":"Clozeau, Nicolas, and Antoine Gloria. “Quantitative Nonlinear Homogenization: Control of Oscillations.” <i>ArXiv</i>, n.d."},"author":[{"id":"fea1b376-906f-11eb-847d-b2c0cf46455b","last_name":"Clozeau","full_name":"Clozeau, Nicolas","first_name":"Nicolas"},{"full_name":"Gloria, Antoine","last_name":"Gloria","first_name":"Antoine"}],"status":"public","abstract":[{"text":"Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with p-growth. This first work is dedicated to a quantitative two-scale expansion result. Fluctuations will be addressed in companion articles. By treating the range of exponents 2≤p<∞ in dimensions d≤3, we are able to consider genuinely nonlinear elliptic equations and systems such as −∇⋅A(x)(1+|∇u|p−2)∇u=f (with A random, non-necessarily symmetric) for the first time. When going from p=2 to p>2, the main difficulty is to analyze the associated linearized operator, whose coefficients are degenerate, unbounded, and depend on the random input A via the solution of a nonlinear equation. One of our main achievements is the control of this intricate nonlinear dependence, leading to annealed Meyers' estimates for the linearized operator, which are key to the quantitative two-scale expansion result.","lang":"eng"}]},{"ddc":["510"],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nd/3.0/legalcode","image":"/images/cc_by_nd.png","short":"CC BY-ND (3.0)","name":"Creative Commons Attribution-NoDerivs 3.0 Unported (CC BY-ND 3.0)"},"title":"Homogenization of nonconvex unbounded singular integrals","doi":"10.5802/ambp.367","year":"2017","publication_identifier":{"issn":["1259-1734"],"eissn":["2118-7436"]},"extern":"1","_id":"10175","quality_controlled":"1","oa_version":"Published Version","user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","article_processing_charge":"No","oa":1,"volume":24,"date_updated":"2021-10-28T15:16:25Z","abstract":[{"text":"We study periodic homogenization by Γ-convergence of integral functionals with integrands W(x,ξ) having no polynomial growth and which are both not necessarily continuous with respect to the space variable and not necessarily convex with respect to the matrix variable. This allows to deal with homogenization of composite hyperelastic materials consisting of two or more periodic components whose the energy densities tend to infinity as the volume of matter tends to zero, i.e., W(x,ξ)=∑j∈J1Vj(x)Hj(ξ) where {Vj}j∈J is a finite family of open disjoint subsets of RN, with |∂Vj|=0 for all j∈J and ∣∣RN∖⋃j∈JVj|=0, and, for each j∈J, Hj(ξ)→∞ as detξ→0. In fact, our results apply to integrands of type W(x,ξ)=a(x)H(ξ) when H(ξ)→∞ as detξ→0 and a∈L∞(RN;[0,∞[) is 1-periodic and is either continuous almost everywhere or not continuous. When a is not continuous, we obtain a density homogenization formula which is a priori different from the classical one by Braides–Müller. Although applications to hyperelasticity are limited due to the fact that our framework is not consistent with the constraint of noninterpenetration of the matter, our results can be of technical interest to analysis of homogenization of integral functionals.","lang":"eng"}],"author":[{"first_name":"Omar","full_name":"Anza Hafsa, Omar","last_name":"Anza Hafsa"},{"id":"fea1b376-906f-11eb-847d-b2c0cf46455b","full_name":"Clozeau, Nicolas","last_name":"Clozeau","first_name":"Nicolas"},{"full_name":"Mandallena, Jean-Philippe","last_name":"Mandallena","first_name":"Jean-Philippe"}],"citation":{"ama":"Anza Hafsa O, Clozeau N, Mandallena J-P. Homogenization of nonconvex unbounded singular integrals. <i>Annales mathématiques Blaise Pascal</i>. 2017;24(2):135-193. doi:<a href=\"https://doi.org/10.5802/ambp.367\">10.5802/ambp.367</a>","mla":"Anza Hafsa, Omar, et al. “Homogenization of Nonconvex Unbounded Singular Integrals.” <i>Annales Mathématiques Blaise Pascal</i>, vol. 24, no. 2, Université Clermont Auvergne, 2017, pp. 135–93, doi:<a href=\"https://doi.org/10.5802/ambp.367\">10.5802/ambp.367</a>.","ista":"Anza Hafsa O, Clozeau N, Mandallena J-P. 2017. Homogenization of nonconvex unbounded singular integrals. Annales mathématiques Blaise Pascal. 24(2), 135–193.","short":"O. Anza Hafsa, N. Clozeau, J.-P. Mandallena, Annales Mathématiques Blaise Pascal 24 (2017) 135–193.","ieee":"O. Anza Hafsa, N. Clozeau, and J.-P. Mandallena, “Homogenization of nonconvex unbounded singular integrals,” <i>Annales mathématiques Blaise Pascal</i>, vol. 24, no. 2. Université Clermont Auvergne, pp. 135–193, 2017.","apa":"Anza Hafsa, O., Clozeau, N., &#38; Mandallena, J.-P. (2017). Homogenization of nonconvex unbounded singular integrals. <i>Annales Mathématiques Blaise Pascal</i>. Université Clermont Auvergne. <a href=\"https://doi.org/10.5802/ambp.367\">https://doi.org/10.5802/ambp.367</a>","chicago":"Anza Hafsa, Omar, Nicolas Clozeau, and Jean-Philippe Mandallena. “Homogenization of Nonconvex Unbounded Singular Integrals.” <i>Annales Mathématiques Blaise Pascal</i>. Université Clermont Auvergne, 2017. <a href=\"https://doi.org/10.5802/ambp.367\">https://doi.org/10.5802/ambp.367</a>."},"publication_status":"published","date_created":"2021-10-23T10:54:23Z","file":[{"access_level":"open_access","date_updated":"2021-10-28T15:02:56Z","checksum":"18f40d13dc5d1e24438260b1875b886f","date_created":"2021-10-28T15:02:56Z","file_size":850726,"file_name":"2017_AMBP_AnzaHafsa.pdf","creator":"cziletti","file_id":"10194","relation":"main_file","content_type":"application/pdf","success":1}],"has_accepted_license":"1","license":"https://creativecommons.org/licenses/by-nd/3.0/","publisher":"Université Clermont Auvergne","language":[{"iso":"eng"}],"month":"11","date_published":"2017-11-20T00:00:00Z","article_type":"original","publication":"Annales mathématiques Blaise Pascal","issue":"2","file_date_updated":"2021-10-28T15:02:56Z","page":"135-193","intvolume":"        24","status":"public","day":"20","type":"journal_article"}]