Stochastic Homogenization Research Articles

Generating high-quality phase-only holograms of binary images using global loss and stochastic homogenization training strategy

Deep learning presents a highly effective for computer-generated holography (CGH). However, existing learning-based CGH algorithms may encounter significant obstacles in generating high-quality holograms for binary images. This study proposes a novel diffraction model-driven neural network, termed load sharing yielding holography (LSY-Holo), which leverages a dual-branch architecture and a global structural similarity index measure (Global_SSIM) loss function to generate phase-only holograms (POHs). LSY-Holo framework facilitates capturing global information, enhancing the network’s performance to address the ill-posed inverse problem. To further optimize the model for binary images, the research introduces a custom Binary dataset alongside a stochastic homogenization preprocessing method (SHPM), which mitigates the prevalent issue of excessive noise in reconstructed images. Additionally, this deep learning approach is integrated with an iterative algorithm, which employs holograms rapidly generated by LSY-Holo as the initial phase of the AdamW Holography (AH) algorithm. This strategy significantly accelerates the convergence of the iterative process. LSY-Holo is further extended to 3D scenes, enabling dynamic defocusing and focusing effects at varying depths. Results from numerical simulations and full-color optical reconstruction experiments demonstrate that the proposed method can enhance image quality and effectively reduce the artifacts in reconstructed images.

An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics

We prove a multidimensional ergodic theorem with weighted averages for the action of the group Zd on a probability space. At level n weights are of the form n−dψ(j/n), j∈Zd, for real functions ψ decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from Faggionato (2023) [Theorem 4.4]. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in Faggionato (2022) [Theorem 4.1] remains valid when removing the above mentioned Assumption (A9).

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L ≫ l around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of l and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that l ≫ 1 ). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion. This work extends, the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of Gloria and Otto.

The time horizon for stochastic homogenization of the one-dimensional wave equation

The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 .

Operator learning for homogenizing hyperelastic materials, without PDE data

In this work, we address operator learning for stochastic homogenization in nonlinear elasticity. A Fourier neural operator is employed to learn the map between the input field describing the material at fine scale and the deformation map. We propose a variationally-consistent loss function that does not involve solution field data. The methodology is tested on materials described either by piecewise constant fields at microscale, or by random fields at mesoscale. High prediction accuracy is obtained for both the solution field and the homogenized response. We show, in particular, that the accuracy achieved with the proposed strategy is comparable to that obtained with the conventional data-driven training method.

Extension of the a posteriori finite element method (APFEM) to geometrical alterations and application to stochastic homogenisation

AbstractWe recently proposed an efficient method facilitating the parametric study of a finite element mechanical simulation as a postprocessing step, that is, without the need to run multiple simulations: the a posteriori finite element method (APFEM). APFEM only requires the knowledge of the vertices of the parameter space and is able to predict accurately how the degrees of freedom of a simulation, i.e., nodal displacements, and other outputs of interests, for example, element stress tensors, evolve when simulation parameters vary within their predefined ranges. In our previous work, these parameters were restricted to material properties and loading conditions. Here, we extend the APFEM to additionally account for changes in the original geometry. This is achieved by defining an intermediary reference frame whose mapping is defined stochastically in the weak form. Subsequent deformation is then reached by correcting for this stochastic variation in the reference frame through multiplicative decomposition of the deformation gradient tensor. The resulting framework is shown here to provide accurate mechanical predictions for relevant applications of increasing complexity: (i) quantifying the stress concentration factor of a plate under uniaxial loading with one and two elliptical holes of varying eccentricities, and (ii) performing the stochastic homogenisation of a composite plate with uncertain mechanical properties and geometry inclusion. This extension of APFEM completes our original approach to account parametrically for geometrical alterations, in addition to boundary conditions and material properties. The advantages of this approach in our original work in terms of stochastic prediction, uncertainty quantification, structural and material optimisation and Bayesian inferences are all naturally conserved.

Stochastic Homogenization and Geometric Singularities: A Study on Corners

Stochastic Homogenization and Geometric Singularities: A Study on Corners

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization

We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field a. Extending the work of Bella et al. (Ann Appl Probab 28(3):1379–1422, 2018), who established the large-scale C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius r∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_*$$\\end{document} describing the minimal scale for this C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} regularity. As an application to stochastic homogenization, we partially generalize results by Gloria et al. (Anal PDE 14(8):2497–2537, 2021) on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on a and a-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a^{-1}$$\\end{document}. We also introduce the ellipticity radius re\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_e$$\\end{document} which encodes the minimal scale where these moments are close to their positive expectation value.

Quantitative stochastic homogenization of nonlinearly elastic, random laminates

In this paper we study quantitative stochastic homogenization of a nonlinearly elastic composite material with a laminate microstructure. We prove that for deformations close to the set of rotations, the homogenized stored energy function W_{\hom} is C^{3} and that W_{\hom} , the stress tensor DW_{\hom} , and the tangent-moduli D^{2}W_{\hom} can be represented with the help of stochastic correctors. Furthermore, we study the error of an approximation of these quantities via representative volume elements. More precisely, we consider periodic representative volume elements (RVEs) obtained by periodizing the distribution of the random material. For materials with a fast decay of correlations on scales larger than a unit scale, we establish error estimates on the random and systematic error of the RVE with optimal scaling in the size of the RVE and with a multiplicative random constant that has exponential moments.

Stochastic Homogenization of Micromagnetic Energies and Emergence of Magnetic Skyrmions

Stochastic Homogenization of Micromagnetic Energies and Emergence of Magnetic Skyrmions

Stochastic homogenization with space-time ergodic divergence-free drift

Stochastic homogenization with space-time ergodic divergence-free drift

Mini-Workshop: New Horizons in Motions in Random Media

The general topic of the mini-workshop New Horizons in Motions in Random Media was the study of random walks in random environments, both in their own right and in relation to stochastic homogenization and to models in statistical mechanics, in particular spin system. This is a subject at the intersection of probability, analysis and mathematical physics, and the workshop brought together leading researchers from those areas. While each of these areas has been quite active for decades with many remarkable breakthroughs obtained throughout the years, the workshop provided a unique opportunity to identify principal new objectives and initiate new collaborations.

Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains

Abstract In this paper, we study the asymptotic behaviour of a family of random free-discontinuity energies E ε {E_{\varepsilon}} defined in a randomly perforated domain, as ε goes to zero. The functionals E ε {E_{\varepsilon}} model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to overcome the lack of equi-coerciveness of the functionals. We do so by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The limit energy is then obtained in two steps. As a first step, we apply a general result of stochastic convergence of free-discontinuity functionals to a modified, coercive version of E ε {E_{\varepsilon}} . Then the effective volume and surface energy densities are identified by means of a careful limit procedure.

Large-scale regularity in stochastic homogenization with divergence-free drift

Large-scale regularity in stochastic homogenization with divergence-free drift

Arbeitsgemeinschaft: Quantitative Stochastic Homogenization

Homogenization means approximating the effective, i. e. macroscopic, behavior of a heterogeneous medium by a homogeneous one, which amounts to a substantial conceptual and practical reduction of complexity. Stochastic homogenization means that one is considering an ensemble of, i. e. a probability measure on, such heterogeneities (typically expressing a lack of knowledge of the details); and that the effective behavior is also deterministic next to being homogeneous. The aim of this Arbeitsgemeinschaft is to present the recent progress in this field.

Gamma -convergence and stochastic homogenisation of singularly-perturbed elliptic functionals

We study the limit behaviour of singularly-perturbed elliptic functionals of the form Fk(u,v)=∫Av2fk(x,∇u)dx+1εk∫Agk(x,v,εk∇v)dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathscr {F}}_k(u,v)=\\int _A v^2\\,f_k(x,\ abla u)\\, \ extrm{d}x+\\frac{1}{\\varepsilon _k}\\int _A g_k(x,v,\\varepsilon _k\ abla v)\\, \ extrm{d}x, \\end{aligned}$$\\end{document}where u is a vector-valued Sobolev function, v in [0,1] a phase-field variable, and varepsilon _k>0 a singular-perturbation parameter; i.e., varepsilon _k rightarrow 0, as krightarrow +infty . Under mild assumptions on the integrands f_k and g_k, we show that if f_k grows superlinearly in the gradient-variable, then the functionals {mathscr {F}}_kGamma -converge (up to subsequences) to a brittle energy-functional; i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of u. This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in {mathscr {F}}_kdecouple in the limit. The abstract Gamma -convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

Stochastic Homogenization of Gaussian Fields on Random Media

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξg,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{g,\ extbf{a}} $$\\end{document} and bi-Laplacian fields Ξb,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{b,\ extbf{a}}$$\\end{document}. They can be characterized as follows: for f=δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=\\delta $$\\end{document} the solution u of ∇·a∇u=f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla \\cdot \ extbf{a} \ abla u =f$$\\end{document}, a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{a}$$\\end{document} is a uniformly elliptic random environment, is the covariance of Ξg,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{g,\ extbf{a}}$$\\end{document}. When f is the white noise, the field Ξb,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{b,\ extbf{a}}$$\\end{document} can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain D⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\subset \\mathbb {R}^d$$\\end{document}, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator a¯Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\bar{\ extbf{a}}}\\,}}\\Delta $$\\end{document}, with constant a¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\bar{\ extbf{a}}}\\,}}$$\\end{document} depending on the law of the environment a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{a}$$\\end{document}. The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014).

Macroscopic wave propagation for 2D lattice with random masses

AbstractWe consider a simple two‐dimensional harmonic lattice with random, independent, and identically distributed masses. Using the methods of stochastic homogenization, we prove that solutions with initial data, which varies slowly relative to the lattice spacing, converge in an appropriate sense to solutions of an effective wave equation. The convergence is strong and almost sure. In addition, the role of the lattice's dimension in the rate of convergence is discussed. The technique combines energy estimates with powerful classical results about sub‐Gaussian random variables.

Homogenization of the linearized ionic transport equations in random porous media

In this paper we obtain the homogenization results for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid random disperse porous medium. We present a study of the nonlinear Poisson–Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scales, we prove that the effective tensor satisfies the so-called Onsager properties, that is the tensor is symmetric and positive definite. This result shows that the Onsager theory applies to random porous media. The strong convergence of the fluxes is also established. In the periodic case homogenization results for the mentioned system have been obtained in Allaire et al (2010 J. Math. Phys. 51 123103).