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We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern (Numerical Algorithms, 83, 1441-1487, 2020). The price to pay is that the regularity analysis required for QMC methods becomes much more technical. Nevertheless we obtain a complete analysis with error comprising stochastic dimension truncation error, finite element error and cubature error, with results comparable to those obtained for the diffusion problem.

In nonclassical transport, the free-path length variable s is modeled as an independent variable, and a nonclassical linear Boltzmann transport equation incorporating s has been derived. To model t...

This paper presents a theory of the time-domain radar cross section (RCS) of objects in discrete random media. The time-domain formula is obtained by applying the inverse Fourier transform of the t...

In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well kn...

An investigation of transient second sound phenomena due to moving heat sources on planar random media is conducted. The spatial material randomness of the relaxation time is modeled by Cauchy or Dagum random fields allowing for decoupling of fractal and Hurst effects. The Maxwell–Cattaneo model is solved by a second-order central differencing. The resulting stochastic fluctuations of Mach wedges are examined and compared to unperturbed Mach wedges resulting from the heat source traveling in a homogeneous domain. All the examined cases are illustrated by simulation movies linked to this paper.

This paper is concerned with the well-posedness and regularity of the distributional solutions for the stochastic acoustic and elastic scattering problems. We show that the regularity of the solutions depends on the regularity of both the random medium and the random source.

In the paper, we consider an elliptic problem in a domain with singular stochastic perturbation of the density located near the boundary, depending on a small parameter. Using the boundary homogenization methods, we prove the compactness theorem and study the behavior of eigenelements to the initial problem as the small parameter tends to zero.

This work deals with the measurability of Fourier integral operators (FIOs) with random phase and amplitude functions. The key ingredient is the proof that FIOs depend continuously on their phase and amplitude functions, taken from suitable classes. The results will be applied to the solution FIO of the transport equation with spatially random transport speed as well as to FIOs describing waves in random media.

We consider a discrete and a continuum model for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inhibitory or an acceleratory force. We show that the interface remains bounded for all times even when a small constant external driving force is applied. This phenomenon has already been known when only inhibitory obstacles are present. In this work we extend this result to the case of, for example, a random medium of random zero mean forcing., Comment: 18 pages, 10 figures

International audience; Structures composed of repetitions of multiple identical cells are common and have been the object of a large body of literature on waveguides, periodic media, and cyclic symmetry. Starting from a cell model, possibly with a large number of Degrees of Freedom both inside the cell and on its edges, the objective of this paper is to propose a Ritz-Galerkin reduction procedure retaining the standard second order model form and periodicity properties of the original model, while controlling accuracy in terms of model bandwidth and reproduction of the forced response to applied loads. The procedure is classically decomposed in two phases: subspace learning and basis generation. For the learning phase, Wave Finite Element (WFE) and periodic computations are presented as alternatives. The latter are then preferred for their easier control on the reduced model accuracy using classical modal synthesis and the simple choice of few target wavelengths. For the basis generation phase, constraints needed to generate a periodic superelement are defined and numerical procedures to generate a basis verifying those constraints are proposed. The validity of the reduction is demonstrated for the case of a cell with random elastic properties presenting bandgaps, local modes and wavemode crossing. Accurate predictions of modes and damped forced response are given for both the infinite and finite cases, using frequency and time simulations. The proposed analysis illustrates tracking of waveshapes, evaluation of significant waves by computation of the forced response in the frequency/wavenumber domain and interpretation of the relation between the infinite and finite forced responses. The case of a railway track with edges and transitions between multiple periodic zones is finally used to illustrate scalability issues.

We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in $\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These results are obtained by combining recent bounds on the Helmholtz equation for deterministic $A$ and $n$ and general arguments (i.e. not specific to the Helmholtz equation) presented in this paper for proving a priori bounds and well-posedness of variational formulations of linear elliptic stochastic PDEs. We emphasise that these general results do not rely on either the Lax-Milgram theorem or Fredholm theory, since neither are applicable to the stochastic variational formulation of the Helmholtz equation., 36 pages, 2 figures

We develop the p-adic model of propagation of fluids (e.g., oil or water) in capillary networks in a porous random medium. The hierarchic structure of a system of capillaries is mathematically modeled by endowing trees of capillaries with the structure of an ultrametric space. Considerations are restricted to the case of idealized networks represented by homogeneous p-trees with p branches leaving each vertex, where p >1 is a prime number. Such trees are realized as the fields of p-adic numbers. We introduce and study an inhomogeneous Markov process describing the penetration of fluid into a porous random medium.

We continue the study of the homogenization of coercive non-convex Hamilton–Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton–Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize.

Fractional calculus is entering the field of nonlinear optics to describe unconventional regimes, as disorder biological media and soft-matter. Here we investigate spatiotemporal modulational instability (MI) in a fractional nonlinear Schrodinger equation. We derive the MI gain spectrum in terms of the Levy indexes and a varying number of spatial dimensions. We show theoretically and numerically that the Levy indexes affect fastest growth frequencies and MI bandwidth and gain. Our results unveil a very rich scenario that may occur in the propagation of ultrashort pulses in random media and metamaterials, and may sustain novel kinds of propagation invariant optical bullets.

Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Holder estimate in ω , ϵ , λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ ∈ ( 0 , 1 ] is the length scale of the interfaces, and λ 2 ∈ ( 0 , 1 ] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Holder estimate in ω , ϵ , λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.

We consider a large number of randomly dispersed spherical, identical, perfectly conducting inclusions (of infinite conductivity) in a bounded domain. The host medium's conductivity is finite and can be inhomogeneous. In the dilute limit, with some boundedness assumption on a large number (proportional to the global volume fraction raised to the power of $-1/2$) of marginal probability densities, we prove convergence in $H^1$ norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, where the conductivity is given by the Clausius--Mossotti formula. Error estimates are provided as well.

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize wave propagation through an inhomogeneous material, the most crucial parameter is the effective wavenumber . For this reason, there are many published studies on how to calculate a single effective wavenumber. Here, we present a proof that there does not exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half-space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener–Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.

In this paper we present the vector radiative transfer theory for a discrete random medium illuminated by a Gaussian beam. The analysis is based on a plane wave spectrum representation for a Gaussian beam and uses an approach developed previously for a discrete random medium illuminated by a plane electromagnetic wave. Specifically, we establish an integral representation for the coherent field, define an approximate coherent field that satisfies the differential equation fulfilled by the coherent field corresponding to a plane electromagnetic wave and matches the Gaussian beam at the interface of the particulate medium, and finally, derive the vector radiative transfer equation. For weakly focused Gaussian beams, the resulting equation is the traditional radiative transfer equation.

We show that the “intensity bump” cross-spectral density used in Opt. Lett. 45, 698 (2020)OPLEDP0146-959210.1364/OL.385246 does not satisfy the non-negative definiteness requirement and cannot be used as a cross-spectral density model.

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit., 30 pages

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity., Added references

Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter., 32 pages. V3: revised introduction, some corrections and clarifications

We consider a discretized version of the quenched Edwards-Wilkinson model for the propagation of a driven interface through a random field of obstacles. Our model consists of a system of ordinary differential equations on a $d$-dimensional lattice coupled by the discrete Laplacian. At each lattice point, the system is subject to a constant driving force and a random obstacle force impeding free propagation. The obstacle force depends on the current state of the solution and thus renders the problem non-linear. For independent and identically distributed obstacle strengths with exponential moment we prove ballistic propagation (i.e., propagation with a positive velocity) of the interface if the driving force is large enough. For a specific case of dependent obstacles, we show that no stationary solution exists, but still the propagation of the front is not ballistic., 13 pages, 1 figure

© American Nuclear Society. We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path length s), and models particle transport in homogenized random media in which the distance to collision of a particle is not exponentially distributed. To solve the nonclassical equation, one needs to know the s-dependent ensemble-averaged total cross section Σ t (μ, s) or its corresponding path-length distribution function p(μ, s). We consider a one-dimensional (1-D) spatially periodic system consisting of alternating solid and void layers, randomly placed along the x-axis. We obtain an analytical expression for p(μ, s) and use this result to compute the corresponding Σ t (μ, s). Then, we proceed to solve numerically the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions μ = ±1. To assess the accuracy of these solutions, we produce benchmark results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely used atomic mix model in most problems.

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty-Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully threedimensional situation.

We consider a medium composed of randomly dispersed spherical, identical inclusions in a bounded domain, with conductivity different than that of the host medium. We study the limit where the number inclusions tends to infinity but their volume fraction remains fixed. For small volume fractions, we prove convergence, in the $W^{1,p}$ norm ($1

International audience; The intact grains of the dead leaves model enables us to generate random media with non overlapping grains. Using the time non homogeneous sequential model with convex grains, theoretically very dense packings can be generated, up to a full covering of space. For these models, the theroretical volume fraction, the size distribution of grains, and the pair correlation function of centers of grains are given.

Some research on the wave propagation in random medium with Von Karman correlation has been developed in this paper. It focuses on the seismic record of circular disturbance in random medium with Von Karman autocorrelation function. Six different kinds of random medium become the background of the dielectric object. The study of the impact to the responds of the dielectric objects can be measured by applying the FDTD to random background medium model. The numerical results show that the random media make the most obvious effect when the scale of imhomogeneity is close to the wave length.

SUMMARY This paper presents a numerical strategy that allows to lower the costs associated to the prediction of the value of homogenized tensors in elliptic problems. This is performed by solving a coupled problem, in which the complex microstructure is confined to a small region and surrounded by a tentative homogenized medium. The characteristics of this homogenized medium are updated using a self-consistent approach and are shown to converge to the actual solution. The main feature of the coupling strategy is that it really couples the random microstructure with the deterministic homogenized model, and not one (deterministic) realization of the random medium with a homogenized model. The advantages of doing so are twofold: (a) the influence of the boundary conditions is significantly mitigated, and (b) the ergodicity of the random medium can be used in full through appropriate definition of the coupling operator. Both of these advantages imply that the resulting coupled problem is less expensive to solve, for a given bias, than the computation of homogenized tensor using classical approaches. Examples of 1-D and 2-D problems with continuous properties, as well as a 2-D matrix-inclusion problem, illustrate the effectiveness and potential of the method. Copyright © 2013 John Wiley & Sons, Ltd.

In order to analyze the depolarization due to random medium quantitatively, we have derived an integral equation using the dyadic Green's function on the assumption that there exists a random medium layer of which the dielectric constant is fluctuating randomly. On the integral equation, the depolarized EM wave has been analyzed by using a perturbation method, and we have shown a far-field expression of the first order perturbed EM wave depolarized by random medium.

We consider a large number of randomly dispersed spherical, identical, inclusions in a bounded domain, with conductivity different than that of the host medium. In the dilute limit, with some mild assumptions on the first few marginal probability densities (no periodicity or stationarity is assumed), we prove convergence in the H1 norm of the expectation of the solution of the steady state heat equation to the solution of an effective medium problem, where the conductivity is given by the Clausius–Mossotti formula. Error estimates are provided as well.

A methodology is presented for bounding the higher L p norms, 2 ⩽ p ⩽ ∞ , of the local strain inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

In this paper we study the asymptotic phase space energy distribution of solution of the Schr\"{o}dinger equation with a time-dependent random potential. The random potential is assumed to be with slowly decaying correlations. We show that the Wigner transform of a solution of the random Schr\"{o}dinger equation converges in probability to the solution of a radiative transfer equation. Moreover, we show that this radiative transfer equation with long-range coupling has a regularizing effect on its solutions. Finally, we give an approximation of this equation in term of a fractional Laplacian. The derivations of these results are based on an asymptotic analysis using perturbed-test-functions, martingale techniques, and probabilistic representations., Comment: 33 pages

In this article,by incorporating equivalent linearization method and the orthogonal expansion method into the wave finite element analysis of scattering problem, an analytical methodology for the evaluation of seismic response of nonlinear soil site with uncertain properties is proposed . Example is given to show the applicability of the methodology. The results show that the randomness of the site media has important effect on seismic site response , the randomness has greater influence on the variation of accelerations than on displacements. The coupling of the nonlinearity and the randomness of soil enhances the effect of randomness on the soil site.

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is, when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations.

We consider a stationary and isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with value 0 or 1 according to whether a regular stationary process X is less or greater than a given level. The intervals corresponding to the points at which X is in a given phase are named chords. We are interested in obtaining information on the chord-length distribution functions. Working with the Palm probability measure and using level crossings techniques, in particular Rice methods, we can obtain not only the exact analytical formula of the chord-length distribution function but also the joint distribution function of the lengths of two successive chords. Finally, we indicate some concrete applications for the computation of usual stereological parameters.

The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived.The results show that the total mass of the solution concentrates on a small set in the space of configuration., Comment: 18 pages

A class of random composite materials with statistically inhomogeneous microstructure, for example, functionally graded materials is considered in this paper. The microstructures inside a component are gradually varying in the statistical sense. In view of this particularity, a novel statistical second-order two-scale (SSOTS) method is presented to predict the mechanical properties, including stiffness, and elastic limit. To develop this method, the microstructures of statistically homogeneous, and inhomogeneous materials are represented. In addition the SSOTS formulas are derived based on normalized cell depending on the position variables by a constructing way, and the algorithm procedure is described. The mechanical properties of the different inhomogeneous materials are evaluated. The numerical results are compared with the experimental findings. It shows that the SSTOS method is effective. Copyright © 2010 John Wiley & Sons, Ltd.

Dimensionless conductance for light propagating through a random medium with amplification tends to diverge with an increase of gain. This raises questions on the applicability of the localization criteria based on this quantity. To circumvent this problem, we study the properties of the ratio between the transmission (conductance) and the energy stored in the random medium. We argue that the generalized conductance gG ¼ g �ð E0=EÞFconductance normalized by the energy buildup (ratio between energy stored in the medium with gain E to that in the passive system E0)—may be a convenient quantity on which a localization criterion can be built.

This paper concerns the asymptotic structure of the scintillation function in the simplified setting of wave propagation modeled by an Ito–Schrodinger equation. We show that the size of the scintil...

We investigate the homogenization of Stefan-type problems with oscillating diffusion coefficients. Both cases of periodic and random (stationary ergodic) mediums are considered. The proof relies on the coincidence of viscosity solutions and weak solutions (which are the time derivatives of the solutions of an obstacle problem) for the Stefan problem. This coincidence result is of independent interest.

We consider a random variable X satisfying almost-sure conditions involving G : = 〈 D X , − D L − 1 X 〉 where D X is X ’s Malliavin derivative and L − 1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. A lower- (resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P [ X > z ] . Bounds of other natures are also given. A key ingredient is the use of Stein’s lemma, including the explicit form of the solution of Stein’s equation relative to the function 1 x > z , and its relation to G . Another set of comparable results is established, without the use of Stein’s lemma, using instead a formula for the density of a random variable based on G , recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G , we show that the Brownian polymer in a Gaussian environment, which is white-noise in time and positively correlated in space, has deviations of Gaussian type and a fluctuation exponent χ = 1 / 2 . We also show this exponent remains 1 / 2 after a non-linear transformation of the polymer’s Hamiltonian.

International audience; The overall mechanical response as well as strain and stress field statistics of an heterogeneous material made of two randomly distributed, linear elastic phases, are investigated numerically. The Boolean model of spheres is used to generate microstructures consisting of either porous or rigid inclusions, at any volume fraction of the phases. Stress and strain field integral ranges, or equivalently the representative volume element, are computed and linked to features of the field statistics, and to the microstructure geometry.

A methodology is presented for bounding the higher $L^p$ norms, $2\leq p\leq\infty$, of the local stress inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

We describe an approximate method for the calculation of all characteristics of coherent backscattering for a homogeneous, semi-infinite particulate medium. The method allows one to transform a system of integral equations describing coherent backscattering exactly into a system of linear algebraic equations affording an efficient numerical solution. Comparisons of approximate theoretical results with experimental data as well as with benchmark numerical results for a medium composed of nonabsorbing Rayleigh scatterers have shown that the method can be expected to give a good accuracy.