Narrow Results

We develop a qualitative homogenization and spectral theory for elliptic systems of partial differential equations in divergence form with highly contrasting (i.e. non-uniformly elliptic) random coefficients. The focus of this paper is on the behavior of the spectrum as the heterogeneity parameter tends to zero; in particular, we show that in general one does not have Hausdorff convergence of spectra. The theoretical analysis is complemented by several explicit examples, showcasing the wider range of applications and physical effects of systems with random coefficients, when compared with systems with periodic coefficients or with scalar operators (both random and periodic).

In this paper we consider the problem of a polymer chain in random media with finite correlation. We show that the mean square end-to-end distance of a polymer chain can be obtained using the Feynman path integral developed by Feynman for treating the polaron problem and successfuly applied to the theory of heavily doped semiconductor. We show that for short-range correlation or the white Gaussian model we derive the results obtained by Edwards and Muthukumar using the replica method and for long-range correlation we obtain the result of Yohannes Shiferaw and Yadin Y. Goldschimidt.

The main idea of this paper is to generalize the model proposed by Edwards and Muthukumar for short-range correlation to finite correlation. Instead of using a replica method, we employ the Feynman path integral by modeling the polymer Hamiltonian as a model of non-local quadratic trial Hamiltonian. This non-local trial Hamiltonian is essential as it will reflect the translation invariant of the original Hamiltonian. The calculation is proceeded by considering the differences between the polymer propagator and the trial propagator as the first cumulant approximation. The variational principle is used to find the optimal values of the variational parameters and the mean square end-to-end distance is obtained. Several asymptotic limits are considered and a comparison between this approaches and replica approach will be discussed.

In this paper we consider the model of a flexible polymer chain embedded in a quenched random medium with long-range disorder correlations. Using the Feynman path integral approach we show that for the case of long-range quadratic correlations, we obtain an analytical result. The result is , where 〈R²〉 is the mean square end-to-end distance of the polymer chain, ξ is the correlation length of disorder, Δ is an unknown parameter, b is the Kuhn step length, ρ is the density of random obstacles and N is the number of links. It is shown that for a polymer chain in a random media with long-range quadratic correlations, where ρ is not too high, the behavior of the polymer chain is like that of a free chain. This result agrees with the calculation using the replica method. However, in a medium where ρ is very high, the variation of the mean square end-to-end distance with disorder and its distance depending on ρ are found in our approach.

We consider the imaging of objects buried in unknown heterogeneous media. The medium is probed by using classical (e.g. acoustic or electromagnetic) waves. When heterogeneities in the medium become too strong, inversion methodologies based on a microscopic description of wave propagation (e.g. a wave equation or Maxwell's equations) become strongly dependent on the unknown details of the heterogeneous medium. In some situations, it is preferable to use a macroscopic model for a quantity that is quadratic in the wave fields. Here, such macroscopic models take the form of radiative transfer equations also referred to as transport equations. They can model either the energy density of the propagating wave fields or more generally the correlation of two wave fields propagating in possibly different media. In particular, we consider the correlation of the two fields propagating in the heterogeneous medium when the inclusion is absent and present, respectively. We present theoretical and numerical results showing that reconstructions based on this correlation are more accurate than reconstructions based on measurements of the energy density.

In this paper we analyze a time reversal experiment in a random underwater acoustic channel. In this kind of waveguide with a semi-infinite cross-section a propagating field can be decomposed over three kinds of modes: the propagating modes, the radiating modes and the evanescent modes. Using an asymptotic analysis based on a separation of scales technique we derive the asymptotic form of the the coupled mode power equation for the propagating modes. This approximation is used to compute the transverse profile of the refocused field and show that random inhomogeneities inside the waveguide deteriorate the spatial focusing. This result, in an underwater acoustic channel context, is a counterexample of the classical results about time reversal experiment in other configurations for which randomness in the propagation medium enhances the refocusing.

We consider a class of stochastic linear elasticity problems whose elastic moduli depend linearly on a countable set of random variables. The stochastic equation is studied via a deterministic parametric problem on an infinite-dimensional parameter space. We first study the best N-term approximation of the generalized polynomial chaos (gpc) expansion of the solution to the displacement formula by considering a Galerkin projection onto the space obtained by truncating the gpc expansion. We provide sufficient conditions on the coefficients of the elastic moduli's expansion so that a rate of convergence for this approximation holds. We then consider two classes of stochastic and parametric mixed elasticity problems. The first one is the Hellinger–Reissner formula for approximating directly the gpc expansion of the stress. For isotropic problems, the multiplying constant of the best N-term convergence rate for the displacement formula grows with the ratio of the Lamé constants. We thus consider stochastic and parametric mixed problems for nearly incompressible isotropic materials whose best N-term approximation rate is uniform with respect to the ratio of the Lamé constants.

We consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner–Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on ℝ^ℕ. It is shown that the weak solution can be represented as Wiener–Itô Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in ℝ¹.

We establish sufficient conditions on the random inputs for weighted sequence norms of the Wiener–Itô decomposition coefficients of the random solution to be p-summable for some 0 < p < 1. For random inputs with additional spatial regularity, stronger norms of the weighted coefficient sequence in the random solutions' Wiener–Itô decomposition are shown to be p-summable for the same value of 0 < p < 1.

We prove rates of nonlinear, best N-term Wiener Polynomial Chaos approximations of the random field, as well as of Finite Element discretizations of these approximations from a dense, nested family V₀ ⊂ V₁ ⊂ V₂ ⊂ ⋯ V of finite element spaces of continuous, piecewise linear Finite Elements.

We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media.

A generalization of acoustic propagation in an uncertain ocean waveguide environment is described using a probabilistic formulation in terms of stochastic basis expansions. The problem is studied in the context of wave propagation in random media, where environmental uncertainty and its interaction with the acoustic field are described by stochastic, rather than deterministic parameters and fields. This representation, constructed explicitly in terms of Karhunen-Loève (KL) and polynomial chaos (PC) expansions, leads to coupled differential equations for the expansion coefficients from which the stochastic acoustic field can be obtained as a random process. The equations are solved in the narrow-angle parabolic approximation using a split-step method to compute moments of the random acoustic field at any point in the waveguide. Results are compared with Monte-Carlo computations of the acoustic field in the same environment to study the convergence of the truncated stochastic basis expansion representing the acoustic field. The rate of convergence of the truncated chaos expansion was found to be dependent on the particular moment computed. For the first and second moments corresponding to the mean field and the field intensity, convergence was achieved rapidly, only requiring low order expansions. Another second moment, the acoustic spatial coherence, converged more slowly due to the relative phase information that, in this formulation, is described by polynomial approximation. While stochastic basis expansions show promise for the development of compact representations of the acoustic field in the presence of environmental uncertainty, accelerated convergence schemes will be needed to allow for practical applications.

The birefringence and natural ability to form periodic structures make cholesteric liquid crystalline (CLC — chiral nematics) materials particularly attractive as 1D photonic band gap systems. If a CLC is doped with dye fluorescent molecules, in such a way that the maximum peak of fluorescence matches one of the edges of the selective stop band, laser action is expected at that spectral position. By confining the helical super-structure of chiral liquid crystals in polymeric micro-cavity channels, a tunable microcavity laser array was achieved. In multiple scattering systems, the propagation of the light waves is quite different, as optical scattering may induce a phase transition in the photon transport behavior. Beyond a critical scattering level, the system makes a transition into a strongly localized state and light transmission is inhibited. This effect can be used as a photon trapping mechanism to obtain laser action in the presence of a gain medium. Random lasing modes come from interference effects which survive in disordered systems and open a particular chapter in the study of the interplay between localization and amplification. Here, experiments performed on systems having different order degree and confinement are presented and possible technological implications are discussed.

We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a prioriL²-bounds available in the time-reversal setting, we prove that the Wigner transform in the high frequency limit converges in probability to its deterministic limit, which is the solution of a transport equation.

We show that the propagation of pulses in a one-dimensional random medium can be characterized using a new complex martingale representation of the transmission coefficient. This representation holds in the frequency domain and in a certain asymptotic limit where diffusion approximations can be used.

Initial boundary value problems of linear second-order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform stability with respect to the support of the resulting coupled hyperbolic systems, and provide sufficient smoothness and compatibility conditions on the data for the solution to exhibit analytic, respectively, Gevrey regularity with respect to the countably many parameters. Sufficient conditions for the p-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best N-term polynomial chaos type approximations of the parametric solution are given. In addition, regularity both in space and time for the parametric family of solutions is proved for data satisfying certain compatibility conditions. The results allow obtaining convergence rates and stability of sparse space-time tensor product Galerkin discretizations in the parameter space.

A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and n known, separated microscopic length scales ε_i, i = 1, …, n in a bounded domain D ⊂ ℝ^d is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge ℙ-a.s, as ε_i → 0, to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension (n + 1)d. It is shown that this stochastic limit problem admits best N-term "polynomial chaos" type approximations which converge at a rate σ > 0 that is determined by the summability of the random inputs' Karhúnen–Loève expansion. The convergence of the polynomial chaos expansion is shown to hold ℙ-a.s. and uniformly with respect to the scale parameters ε_i. Regularity results for the stochastic, one-scale limiting problem are established. An error bound for the approximation of the random solution at finite, positive values of the scale parameters ε_i is established in the case of two scales, and in the case of n > 2, scales convergence is shown, albeit without giving a convergence rate in this case.

We present a microscopical approach to the electromagnetic vacuum energy of a random medium made of non-polar molecules. We evaluate the contribution of statistical fluctuations to the average total vacuum energy. While the Lamb shift is a function of the electrical susceptibility only, the vacuum energy is generally not, except in the quasicrystalline approximation. We comment on the possibility of testing experimentally our results. We clarify why the effective medium vacuum energy (i.e., that of long-wavelengths) does not account for the total vacuum energy of a molecular dielectric. Consequently, the Lamb shift does not derive from the effective medium vacuum energy except at leading order in the molecular density.