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Nicolas Clozeau

Publications and source records attributed to Nicolas Clozeau.

8 recordsLinked to original sources
arXiv2026

Annealed quantitative estimates for the quadratic 2D-discrete random matching problem

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 $μ^n$ and $ν^{m}$ is quantitatively well-approximated by $\big(\mathrm{Id},\exp(\nabla h^{n})\big)_\#μ^n$ where $h^{n}$ 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.

math.PR↗
arXiv2026

Renormalisation of Singular SPDEs with Correlated Coefficients

We show local well-posedness of the g-PAM and the $ϕ^{K+1}_2$-equation for $K\geq 1$ on the two-dimensional torus when the coefficient field is random and correlated to the driving noise. In the setting considered here, even when the model in the sense of Hairer (2014) is stationary, naive use of renormalisation constants in general leads to variance blow-up. Instead, we prove convergence of renormalised models choosing random renormalisation functions analogous to the deterministic variable coefficient setting. The main technical contribution are stochastic estimates on the model in this correlated setting which are obtained by a combination of heat kernel asymptotics, Gaussian integration by parts formulae and Hairer-Quastel type bounds.

math.AP↗
arXiv2024

Quantitative homogenization for log-normal coefficients

We establish quantitative homogenization results for the popular log-normal coefficients. Since the coefficients are neither bounded nor uniformly elliptic, standard proofs do not apply directly. Instead, we take inspiration from the approach developed for the nonlinear setting by the first two authors and capitalize on large-scale regularity results by Bella, Fehrmann, and Otto for degenerate coefficients in order to leverage an optimal control (in terms of scaling and stochastic integrability) of oscillations and fluctuations.

math.AP↗
arXiv2023

Artificial boundary conditions for random ellitpic systems with correlated coefficient field

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.

math.AP↗
arXiv2023

Quantitative nonlinear homogenization: control of oscillations

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 work is dedicated to a quantitative two-scale expansion result. By treating the range of exponents $2\le p <\infty$ in dimensions $d\le 3$, we are able to consider genuinely nonlinear elliptic equations and systems such as $-\nabla \cdot A(x)(1+|\nabla u|^{p-2})\nabla u=f$ (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 optimal quantitative two-scale expansion result we derive (this is also new in the periodic setting).

math.AP↗
arXiv2022

Bias in the representative volume element method: periodize the ensemble instead of its realizations

We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior $a_{\text{hom}}$ of a stationary random medium. The latter is described by a coefficient field $a(x)$ generated from a given ensemble $\langle\cdot\rangle$ and the corresponding linear elliptic operator $-\nabla\cdot a\nabla$. 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 $\langle\cdot\rangle$. We make this point by investigating the bias (or systematic error), i. e. the difference between $a_{\text{hom}}$ 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 $\langle\cdot\rangle$, 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 $\langle\cdot\rangle$ of Gaussian type with integrable covariance, which allow for a straightforward periodization and which make the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors.

math.AP↗
arXiv2022

Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields

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.

math.AP↗