Abstract:
This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius-Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions.

Abstract:
This article studies some numerical approximations of the homogenized matrix for stochastic linear elliptic partial differential equations in divergence form. We focus on the case when the underlying random field is a small perturbation of a reference periodic tensor. The size of such a perturbation is encoded by a real parameter eta. In this case, it has already been theoretically shown in the literature that the exact homogenized matrix possesses an expansion in powers of the parameter eta for both models considered in this article, the coefficients of which are deterministic. In practice, one cannot manipulate the exact terms of such an expansion. All objects are subjected to a discretization approach. Thus we need to derive a similar expansion for the approximated random homogenized matrix. In contrast to the expansion of the exact homogenized matrix, the expansion of the approximated homogenized matrix contains intrinsically random coefficients. In particular, the second order term is random in nature. The purpose of this work is to derive and study this expansion in function of the parameters of the approximation procedure (size of the truncated computational domain used, meshsize of the finite elements approximation).

Abstract:
This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.

Abstract:
This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.

Abstract:
This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. In [11] rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincar\'{e} inequality were obtained. Here these results are extended to certain environments in which correlations can have arbitrarily small power law decay. Similar results for discrete elliptic equations were obtained in [12].

Abstract:
We consider a diffusion equation with highly oscillatory coefficients that admits a homogenized limit. As an alternative to standard corrector problems, we introduce here an embedded corrector problem, written as a diffusion equation in the whole space in which the diffusion matrix is uniform outside some ball of radius $R$. Using that problem, we next introduce three approximations of the homogenized coefficients. These approximations, which are variants of the standard approximations obtained using truncated (supercell) corrector problems, are shown to converge when $R \to \infty$. We also discuss efficient numerical methods to solve the embedded corrector problem.

Abstract:
Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged Green's function together with its first and second differences, are bounded by the corresponding quantities for the constant coefficient discrete elliptic equation. It has also been shown that if the random environment is ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized elliptic PDE on $\R^d$. In this paper point-wise estimates are obtained on the difference between the averaged Green's function and the homogenized Green's function for certain random environments which are strongly mixing.

Abstract:
Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. In [3] rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincar\'{e} inequality were obtained. Here these results are extended to certain environments with long range correlations. These environments are simply related via a convolution to environments which do satisfy a Poincar\'{e} inequality.

Abstract:
Based on the differential equation of the deflection curve for the beam, the equation of the deflection curve for the simple beam is obtained by integral. The equation of the deflection curve for the simple beam carrying the linear load is generalized, and then it is expanded into the corresponding Fourier series. With the obtained summation results of the infinite series, it is found that they are related to Bernoulli numbers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam, Bernoulli numbers and Euler numbers are found, and the relative mathematical formulas are presented. Based on the differential equation of the deflection curve for the beam, the equation of the deflection curve for the simple beam is obtained by integral. The equation of the deflection curve for the simple beam carrying the linear load is generalized, and then it is expanded into the corresponding Fourier series. With the obtained summation results of the infinite series, it is found that they are related to Bernoulli numbers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam, Bernoulli numbers and Euler numbers are found, and the relative mathematical formulas are presented.

Abstract:
We consider polynomials whose coefficients are operators belonging to the Schatten-von Neumann ideals of compact operators in a Hilbert space. Bounds for the spectra of perturbed pencils are established. Applications to differential and difference equations are also discussed. 1. Introduction and Preliminaries Numerous mathematical and physical problems lead to polynomial operator pencils (polynomials with operator coefficients); cf. [1] and references therein. Recently, the spectral theory of operator pencils attracts the attention of many mathematicians. In particular, in the paper [2], spectral properties of the quadratic operator pencil of Schr？dinger operators on the whole real axis are studied. The author of the paper [3] establishes sufficient conditions for the finiteness of the discrete spectrum of linear pencils. The paper [4] deals with the spectral analysis of a class of second-order indefinite nonself-adjoint differential operator pencils. In that paper, a method for solving the inverse spectral problem for the Schr？dinger operator with complex periodic potentials is proposed. In [5, 6], certain classes of analytic operator valued functions in a Hilbert space are studied, and bounds for the spectra of these functions are suggested. The results of papers [5, 6] are applied to second-order differential operators and functional differential equations. The paper [7] considers polynomial pencils whose coefficients are compact operators. Besides, inequalities for the sums of absolute values and real and imaginary parts of characteristic values are derived. The paper [8] is devoted to the variational theory of the spectra of operator pencils with self-adjoint operators. A Banach algebra associated with a linear operator pencil is explored in [9]. A functional calculus generated by a quadratic operator pencil is investigated in [10]. A quadratic pencil of differential operators with periodic generalized potential is considered in [11]. The fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces is explored in [12]. Certainly, we could not survey the whole subject here and refer the reader to the pointed papers and references cited therein. Note that perturbations of pencils with nonself-adjoint operator coefficients, to the best of our knowledge, were not investigated in the available literature, although in many applications, for example, in numerical mathematics and stability analysis, bounds for the spectra of perturbed pencils are very important; cf. [13]. In the present paper, we