--- _id: '14797' 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. 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." article_processing_charge: Yes (in subscription journal) article_type: original author: - first_name: Nicolas full_name: Clozeau, Nicolas id: fea1b376-906f-11eb-847d-b2c0cf46455b last_name: Clozeau - first_name: Francesco full_name: Mattesini, Francesco last_name: Mattesini citation: ama: Clozeau N, Mattesini F. Annealed quantitative estimates for the quadratic 2D-discrete random matching problem. Probability Theory and Related Fields. 2024. doi:10.1007/s00440-023-01254-0 apa: Clozeau, N., & Mattesini, F. (2024). Annealed quantitative estimates for the quadratic 2D-discrete random matching problem. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-023-01254-0 chicago: Clozeau, Nicolas, and Francesco Mattesini. “Annealed Quantitative Estimates for the Quadratic 2D-Discrete Random Matching Problem.” Probability Theory and Related Fields. Springer Nature, 2024. https://doi.org/10.1007/s00440-023-01254-0. ieee: N. Clozeau and F. Mattesini, “Annealed quantitative estimates for the quadratic 2D-discrete random matching problem,” Probability Theory and Related Fields. Springer Nature, 2024. ista: Clozeau N, Mattesini F. 2024. Annealed quantitative estimates for the quadratic 2D-discrete random matching problem. Probability Theory and Related Fields. mla: Clozeau, Nicolas, and Francesco Mattesini. “Annealed Quantitative Estimates for the Quadratic 2D-Discrete Random Matching Problem.” Probability Theory and Related Fields, Springer Nature, 2024, doi:10.1007/s00440-023-01254-0. short: N. Clozeau, F. Mattesini, Probability Theory and Related Fields (2024). date_created: 2024-01-14T23:00:57Z date_published: 2024-01-04T00:00:00Z date_updated: 2024-01-17T11:18:34Z day: '04' ddc: - '510' department: - _id: JuFi doi: 10.1007/s00440-023-01254-0 ec_funded: 1 external_id: arxiv: - '2303.00353' has_accepted_license: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.1007/s00440-023-01254-0 month: '01' oa: 1 oa_version: Published Version project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication: Probability Theory and Related Fields publication_identifier: eissn: - 1432-2064 issn: - 0178-8051 publication_status: epub_ahead publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Annealed quantitative estimates for the quadratic 2D-discrete random matching problem 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) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- _id: '13129' abstract: - lang: eng 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." acknowledgement: Open access funding provided by Institute of Science and Technology (IST Austria). article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Nicolas full_name: Clozeau, Nicolas id: fea1b376-906f-11eb-847d-b2c0cf46455b last_name: Clozeau - first_name: Marc full_name: Josien, Marc last_name: Josien - first_name: Felix full_name: Otto, Felix last_name: Otto - first_name: Qiang full_name: Xu, Qiang last_name: Xu citation: ama: 'Clozeau N, Josien M, Otto F, Xu Q. Bias in the representative volume element method: Periodize the ensemble instead of its realizations. Foundations of Computational Mathematics. 2023. doi:10.1007/s10208-023-09613-y' apa: '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. Springer Nature. https://doi.org/10.1007/s10208-023-09613-y' chicago: 'Clozeau, Nicolas, Marc Josien, Felix Otto, and Qiang Xu. “Bias in the Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.” Foundations of Computational Mathematics. Springer Nature, 2023. https://doi.org/10.1007/s10208-023-09613-y.' ieee: 'N. Clozeau, M. Josien, F. Otto, and Q. Xu, “Bias in the representative volume element method: Periodize the ensemble instead of its realizations,” Foundations of Computational Mathematics. Springer Nature, 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.' mla: 'Clozeau, Nicolas, et al. “Bias in the Representative Volume Element Method: Periodize the Ensemble Instead of Its Realizations.” Foundations of Computational Mathematics, Springer Nature, 2023, doi:10.1007/s10208-023-09613-y.' short: N. Clozeau, M. Josien, F. Otto, Q. Xu, Foundations of Computational Mathematics (2023). date_created: 2023-06-11T22:00:40Z date_published: 2023-05-30T00:00:00Z date_updated: 2023-08-02T06:12:39Z day: '30' ddc: - '510' department: - _id: JuFi doi: 10.1007/s10208-023-09613-y external_id: isi: - '000999623100001' has_accepted_license: '1' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.1007/s10208-023-09613-y month: '05' oa: 1 oa_version: Published Version publication: Foundations of Computational Mathematics publication_identifier: eissn: - 1615-3383 issn: - 1615-3375 publication_status: epub_ahead publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: 'Bias in the representative volume element method: Periodize the ensemble instead of its realizations' 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) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 year: '2023' ... --- _id: '10173' abstract: - lang: eng 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. 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)." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Nicolas full_name: Clozeau, Nicolas id: fea1b376-906f-11eb-847d-b2c0cf46455b last_name: Clozeau citation: ama: 'Clozeau N. Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields. Stochastics and Partial Differential Equations: Analysis and Computations. 2023;11:1254–1378. doi:10.1007/s40072-022-00254-w' apa: 'Clozeau, N. (2023). Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields. Stochastics and Partial Differential Equations: Analysis and Computations. Springer Nature. https://doi.org/10.1007/s40072-022-00254-w' chicago: 'Clozeau, Nicolas. “Optimal Decay of the Parabolic Semigroup in Stochastic Homogenization  for Correlated Coefficient Fields.” Stochastics and Partial Differential Equations: Analysis and Computations. Springer Nature, 2023. https://doi.org/10.1007/s40072-022-00254-w.' ieee: 'N. Clozeau, “Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields,” Stochastics and Partial Differential Equations: Analysis and Computations, vol. 11. Springer Nature, pp. 1254–1378, 2023.' 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.” Stochastics and Partial Differential Equations: Analysis and Computations, vol. 11, Springer Nature, 2023, pp. 1254–1378, doi:10.1007/s40072-022-00254-w.' short: 'N. Clozeau, Stochastics and Partial Differential Equations: Analysis and Computations 11 (2023) 1254–1378.' date_created: 2021-10-23T10:50:22Z date_published: 2023-09-01T00:00:00Z date_updated: 2023-08-14T11:51:47Z day: '01' ddc: - '510' department: - _id: JuFi doi: 10.1007/s40072-022-00254-w external_id: arxiv: - '2102.07452' isi: - '000799715600001' file: - access_level: open_access checksum: f83dcaecdbd3ace862c4ed97a20e8501 content_type: application/pdf creator: dernst date_created: 2023-08-14T11:51:04Z date_updated: 2023-08-14T11:51:04Z file_id: '14052' file_name: 2023_StochPartialDiffEquations_Clozeau.pdf file_size: 1635193 relation: main_file success: 1 file_date_updated: 2023-08-14T11:51:04Z has_accepted_license: '1' intvolume: ' 11' isi: 1 language: - iso: eng month: '09' oa: 1 oa_version: Published Version page: 1254–1378 publication: 'Stochastics and Partial Differential Equations: Analysis and Computations' publication_identifier: issn: - 2194-0401 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields 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) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 11 year: '2023' ... --- _id: '10174' abstract: - lang: eng 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. 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). article_number: '2104.04263' article_processing_charge: No author: - first_name: Nicolas full_name: Clozeau, Nicolas id: fea1b376-906f-11eb-847d-b2c0cf46455b last_name: Clozeau - first_name: Antoine full_name: Gloria, Antoine last_name: Gloria citation: ama: 'Clozeau N, Gloria A. Quantitative nonlinear homogenization: control of oscillations. arXiv.' apa: 'Clozeau, N., & Gloria, A. (n.d.). Quantitative nonlinear homogenization: control of oscillations. arXiv.' chicago: 'Clozeau, Nicolas, and Antoine Gloria. “Quantitative Nonlinear Homogenization: Control of Oscillations.” ArXiv, n.d.' ieee: 'N. Clozeau and A. Gloria, “Quantitative nonlinear homogenization: control of oscillations,” arXiv. .' ista: 'Clozeau N, Gloria A. Quantitative nonlinear homogenization: control of oscillations. arXiv, 2104.04263.' mla: 'Clozeau, Nicolas, and Antoine Gloria. “Quantitative Nonlinear Homogenization: Control of Oscillations.” ArXiv, 2104.04263.' short: N. Clozeau, A. Gloria, ArXiv (n.d.). date_created: 2021-10-23T10:50:55Z date_published: 2021-04-09T00:00:00Z date_updated: 2021-10-28T15:44:05Z day: '09' department: - _id: JuFi external_id: arxiv: - '2104.04263' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2104.04263 month: '04' oa: 1 oa_version: Preprint publication: arXiv publication_status: submitted status: public title: 'Quantitative nonlinear homogenization: control of oscillations' type: preprint user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425 year: '2021' ... --- _id: '10175' abstract: - lang: eng 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. article_processing_charge: No article_type: original author: - first_name: Omar full_name: Anza Hafsa, Omar last_name: Anza Hafsa - first_name: Nicolas full_name: Clozeau, Nicolas id: fea1b376-906f-11eb-847d-b2c0cf46455b last_name: Clozeau - first_name: Jean-Philippe full_name: Mandallena, Jean-Philippe last_name: Mandallena citation: ama: Anza Hafsa O, Clozeau N, Mandallena J-P. Homogenization of nonconvex unbounded singular integrals. Annales mathématiques Blaise Pascal. 2017;24(2):135-193. doi:10.5802/ambp.367 apa: Anza Hafsa, O., Clozeau, N., & Mandallena, J.-P. (2017). Homogenization of nonconvex unbounded singular integrals. Annales Mathématiques Blaise Pascal. Université Clermont Auvergne. https://doi.org/10.5802/ambp.367 chicago: Anza Hafsa, Omar, Nicolas Clozeau, and Jean-Philippe Mandallena. “Homogenization of Nonconvex Unbounded Singular Integrals.” Annales Mathématiques Blaise Pascal. Université Clermont Auvergne, 2017. https://doi.org/10.5802/ambp.367. ieee: O. Anza Hafsa, N. Clozeau, and J.-P. Mandallena, “Homogenization of nonconvex unbounded singular integrals,” Annales mathématiques Blaise Pascal, vol. 24, no. 2. Université Clermont Auvergne, pp. 135–193, 2017. ista: Anza Hafsa O, Clozeau N, Mandallena J-P. 2017. Homogenization of nonconvex unbounded singular integrals. Annales mathématiques Blaise Pascal. 24(2), 135–193. mla: Anza Hafsa, Omar, et al. “Homogenization of Nonconvex Unbounded Singular Integrals.” Annales Mathématiques Blaise Pascal, vol. 24, no. 2, Université Clermont Auvergne, 2017, pp. 135–93, doi:10.5802/ambp.367. short: O. Anza Hafsa, N. Clozeau, J.-P. Mandallena, Annales Mathématiques Blaise Pascal 24 (2017) 135–193. date_created: 2021-10-23T10:54:23Z date_published: 2017-11-20T00:00:00Z date_updated: 2021-10-28T15:16:25Z day: '20' ddc: - '510' doi: 10.5802/ambp.367 extern: '1' file: - access_level: open_access checksum: 18f40d13dc5d1e24438260b1875b886f content_type: application/pdf creator: cziletti date_created: 2021-10-28T15:02:56Z date_updated: 2021-10-28T15:02:56Z file_id: '10194' file_name: 2017_AMBP_AnzaHafsa.pdf file_size: 850726 relation: main_file success: 1 file_date_updated: 2021-10-28T15:02:56Z has_accepted_license: '1' intvolume: ' 24' issue: '2' language: - iso: eng license: https://creativecommons.org/licenses/by-nd/3.0/ month: '11' oa: 1 oa_version: Published Version page: 135-193 publication: Annales mathématiques Blaise Pascal publication_identifier: eissn: - 2118-7436 issn: - 1259-1734 publication_status: published publisher: Université Clermont Auvergne quality_controlled: '1' status: public title: Homogenization of nonconvex unbounded singular integrals tmp: image: /images/cc_by_nd.png legal_code_url: https://creativecommons.org/licenses/by-nd/3.0/legalcode name: Creative Commons Attribution-NoDerivs 3.0 Unported (CC BY-ND 3.0) short: CC BY-ND (3.0) type: journal_article user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425 volume: 24 year: '2017' ...