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We consider the perturbation of parabolic operators of the form $\partial_t+P(x,D)$ by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator $P(x,D)$, the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with a multiplicative term that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions that are larger than or equal to the order of the elliptic pseudo-differential operator $P(x,D)$. In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in the companion paper [2]. The stochastic model is therefore valid only for sufficiently small space dimensions in this class of parabolic problems., 21 pages

This paper generalizes well-established derivations of the radiative transfer equation from first principles to model the energy density of time-dependent and monochromatic high frequency waves pro...

We consider the reconstruction of a spatially-dependent scattering coefficient in a linear transport equation from diffusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for reconstruction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this error so that it is small when the high frequency content of the scattering kernel is small.

We demonstrate the effectiveness of a transport-based reconstruction method for imaging in highly scattering environments. Experimentally measured wave energy data in the micro-wave regime are used to reconstruct extended inclusions buried in scattering media or hidden behind non-penetrable obstacles. The performance of the imaging method is illustrated under various circumstances, via a set of electromagnetic experiments.

An experimental study is performed on electromagnetic time reversal in highly scattering environments, with a particular focus on performance when environmental conditions change. In particular, we consider the case for which there is a mismatch between the Green's function used on the forward measurement and that used for time-reversal inversion. We examine the degradation in the time-reversal image with increasing media mismatch, and consider techniques that mitigate such degradation. The experimental results are also compared with theoretical predictions for time reversal in changing media, with good agreement observed

We derive kinetic models for the correlations and the energy densities of wave fields propagating in random media. These models take the form of radiative transfer and diffusion equations. We use these macroscopic models to address the detection and imaging of small objects buried in highly heterogeneous media. More specifically, we quantify the influence of small objects on (i) the energy density measured at an array of detectors and (ii) the correlation between the wave field measured in the absence of the object and the wave field measured in the presence of the object. We analyze the advantages and disadvantages of such measurements as a function of the level of disorder in the random media. Numerical simulations verify the theoretical predictions.

This paper proposes an iterative technique to reconstruct the source term in transport equations, which account for scattering effects, from boundary measurements. In the two‐dimensional setting, the full outgoing distribution in the phase space (position and direction) needs to be measured. In three space dimensions, we show that measurements for angles that are orthogonal to a given direction are sufficient. In both cases, the derivation is based on a perturbation of the inversion of the two‐dimensional attenuated Radon transform and requires that (the anisotropic part of) scattering be sufficiently small. We present an explicit iterative procedure, which converges to the source term we want to reconstruct. Applications of the inversion procedure include optical molecular imaging, an increasingly popular medical imaging modality.

We consider the homogenization of the wave equation with high frequency initial conditions propagating in a medium with highly oscillatory random coefficients. By appropriate mixing assumptions on the random medium, we obtain an error estimate between the exact wave solution and the homogenized wave solution in the energy norm. This allows us to consider the limiting behavior of the energy density of high frequency waves propagating in highly heterogeneous media when the wavelength is much larger than the correlation length in the medium.

We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by the homogenized solution, the Green's function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations., 19 pages

We present explicit reconstruction algorithms for fully anisotropic unknown elasticity tensors from knowledge of a finite number of internal displacement fields, with applications to transient elastography. Under certain rank-maximality assumptions satified by the strain fields, explicit algebraic reconstruction formulas are provided. A discussion ensues on how to fulfill these assumptions, describing the range of validity of the approach. We also show how the general method can be applied to more specific cases such as the transversely isotropic one., 23 pages. Minor updates and additional references

International audience; We consider the reconstruction of internal elastic displacements from ultrasound measurements , which finds applications in the medical imaging modality called elastography. By appropriate interferometry and windowed Fourier transforms of the ultrasound measurements, we propose a reconstruction procedure of the vectorial structure of spatially varying elastic displacements in biological tissues. This provides a modeling and generalization of scalar reconstruction procedures routinely used in elastography. The proposed algorithm is justified using a single scattering approximation and local asymptotic analysis. Its validity is assessed by numerical simulations.

This paper addresses the validity of radiative transfer equations as a model for the energy density of waves propagating in highly heterogeneous media. Comparisons between acoustic wave simulations over domains of size comparable to 500 wavelengths in two space dimensions and Monte Carlo simulations of radiative transfer equations are performed. In the so-called weak coupling regime, the agreement between the energy densities obtained by solving the wave equations and those predicted by solving the radiative transfer equations is remarkable. The domain of validity of the radiative transfer equations is assessed by looking at the fluctuations in the energy density they predict in the presence of small-volume defects in the underlying media.

We consider the reconstruction of singular surfaces from the over-determined boundary conditions of an elliptic problem. The problem arises in optical and impedance tomography, where void-like structure or cracks may be modeled as diffusion processes supported on co-dimension one surfaces. The reconstruction of such surfaces is obtained theoretically and numerically by combining a shape sensitivity analysis with a level set method. The shape sensitivity analysis is used to define a velocity field, which allows us to update the surface while decreasing a given cost function, which quantifies the error between the prediction of the forward model and the measured data. The velocity field depends on the geometry of the surface and the tangential diffusion process supported on it. The latter process is assumed to be known in this paper. The level set method is next applied to evolve the surface in the direction of the velocity field. Numerical simulations show how the surface may be reconstructed from noisy estimates of the full, or local, Neumann-to-Dirichlet map.

This paper concerns the derivation of kinetic models for high frequency scalar wave fields propagating in random media. The kinetic equations model the propagation in the phase space of the energy density of a wave field or the correlation function of two wave fields propagating in two possibly different media. Dispersive effects due to, e.g. spatial and temporal discretizations, which are modeled as non-local pseudo-differential operators, are taken into account. The derivation of the models is based on a multiple-scale asymptotic expansion of the spatio-temporal Wigner transform of two scalar wave fields.

We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann–Hilbert problem. The inversion formulae have a very similar structure to their counterparts in Euclidean geometry and may therefore be amenable to efficient discretizations and numerical inversions. An important field of application is geophysical imaging when absorption effects are accounted for.

We consider the detection and imaging of inclusions buried in highly heterogeneous media. We assume that only the statistical properties of the heterogeneous media can be observed and that the wave energy density may be modelled by macroscopic equations. The detection and imaging capabilities hinge on ensuring that the measured data are statistically stable, which means that they depend only on the macroscopic statistical parameters of the random media and not on the microscopic statistical realization. In this paper, the macroscopic model is a diffusion equation. In this context, we construct statistical tests to detect inclusions based on macroscopic diffusion measurements and perform asymptotic expansions to image their location and volume. We show that time-reversal measurements enjoy a much larger signal-to-noise ratio in the presence of background noise than do direct wave energy measurements. This is a direct consequence of the enhanced refocusing properties that characterize time reversed waves propagating in heterogeneous media. Finally, we present numerical simulations of acoustic waves propagating in heterogeneous two-dimensional media. The numerical simulations illustrate which factors contribute to 'noise' in the measured data and how they affect the detection and imaging capabilities.

We consider the reconstruction of vertical concentration profiles of atmospheric gases from a spectral distribution of radiation measured from a space-borne infrared spectrometer. Under some separability assumptions of the gases' spectral absorption coefficients, we obtain uniqueness results on the reconstruction of concentration profiles from (multiple-wavenumber) radiance measurements and provide an explicit reconstruction procedure. We show that the reconstruction is a severely ill-posed problem. To address the reconstruction of localized layers, such as ozone or dust layers, we model the reconstruction of strong localized variations in the concentration profiles by using asymptotic expansions in the layer thickness. Assuming the background is known, we obtain that the location as well as the product of the concentration variability within the layer multiplied and the thickness of the layer may be reconstructed from moderately noisy data. The reconstructions of both the concentration and the thickness of the layer require more accurate data.

Singular layers modelled by a tangential diffusion process supported on an embedded closed surface (of co-dimension 1) have found applications in tomography problems. In optical tomography they may model the propagation of photons in thin clear layers, which are known to hamper the use of classical diffusion approximations. In impedance tomography they may be used to model thin regions of very high conductivity profile. In this paper we show that such surfaces can be reconstructed from boundary measurements (more precisely, from a local Neumann-to-Dirichlet operator) provided that the material properties between the measurement surface and the embedded surface are known. The method is based on the factorization technique introduced by Kirsch. Once the location of the surface is reconstructed, we show under appropriate assumptions that the full tangential diffusion process and the material properties in the region enclosed by the surface can also uniquely be determined.

Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified regime of the parabolic wave equation in a random medium. The mathematical theory of the convergence and statistical properties of the algorithm is based on the analysis of the Wigner transforms in random media. Our results provide a step toward understanding time and space discretizations that are needed in order for the numerical algorithm to capture the correct macroscopic statistics of the wave energy density in a random medium.

We propose a numerical method to simulate and invert the two-dimensional attenuated Radon transform (AtRT) from full (360 ◦ ) or partial (180 ◦ ) measurements. The method is based on an extension of the fast slant stack algorithm developed for the Radon transform. We show that the algorithm offers robust and fast inversion of the AtRT for a wide class of synthetic sources and absorptions. The complexity of the fast algorithm to compute the AtRT of a n × n image and perform the reconstruction from the AtRT data is O(Nn 2 log n) operations, with N the number of Fourier modes necessary to accurately represent the absorption map. The algorithm is applied to the reconstruction of the exponential Radon transform, where the absorption coefficient is constant, and of the AtRT when only 180 ◦ measurements are available. The reconstruction from partial measurements is based on an iterative scheme introduced recently in Bal (2004 Inverse Problems 20 399– 419). Single-photon emission computed tomography is an important medical imaging technique based on the inversion of the AtRT.

We present the asymptotic expansion of the solution to a diffusion equation with a finite number of absorbing inclusions of small volume. We use the first few terms in this expansion measured at the domain boundary to reconstruct the absorption parameters of the inclusions and certain geometrical characteristics. We demonstrate theoretically and numerically that the number of inclusions, their location and their capacity can be reconstructed in a stable way even from moderately noisy data. The reconstruction of the absorption parameter, which is important in optical tomography to discriminate between healthy and unhealthy tissues, requires us however to have far less noisy data. Since the reconstruction of absorption maps from boundary measurements is an extremely ill posed problem, the method of asymptotic expansions of small volume inclusions provides a useful framework to decide which information can be reconstructed from boundary measurements with a given noise level.

In time reversal acoustics experiments, a signal is emitted from a localized source, recorded at an array of receivers, time reversed, and finally reemitted into the medium. A celebrated feature of time reversal experiments is that the refocusing of the reemitted signals at the location of the initial source is improved when the medium is heterogeneous. Contrary to intuition, multiple scattering enhances the spatial resolution of the refocused signal and allows one to beat the diffraction limit obtained in homogeneous media. This paper presents a quantitative explanation of time reversal and other more general refocusing phenomena for general classical waves in heterogeneous media. The theory is based on the asymptotic analysis of the Wigner transform of wave fields in the high frequency limit. Numerical experiments complement the theory.

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.

This paper introduces generalized diffusion models for the transport of particles in scattering media with nonscattering inclusions. Classical diffusion is known as a good approximation of transport only in scattering media. Based on asymptotic expansions and the coupling of transport and diffusion models, generalized diffusion equations with nonlocal interface conditions are proposed which offer a computationally cheap, yet accurate, alternative to solving the full phase-space transport equations. The paper shows which computational model should be used depending on the size and shape of the nonscattering inclusions in the simplified setting of two space dimensions. An important application is the treatment of clear layers in near-infrared (NIR) spectroscopy, an imaging technique based on the propagation of NIR photons in human tissues.

We study transport for a scalar model of Love waves. These waves arise in the propagation of seismic waves whose energy is concentrated in the vicinity of the earth surface. We derive radiative transfer equations from first principles for the angularly resolved energy density of the Love waves in a simplified acoustic model. We consider a rough top surface with weak fluctuations at the scale of the wavelength. The transport equation accounts for the multiple scattering of the Love waves and their scattering into volume waves. We also analyze a diffusive regime when energy is universally distributed over various modes of the Love waves.

We give a detailed mathematical analysis of the radiative transport limit for the average phase space density of solutions of the Schroedinger equation with time dependent random potential. Our derivation is based on the construction of an approximate martingale for the random Wigner distribution.

We derive transport equations for the propagation of water wave action in the presence of subsurface random flows. Using the Wigner distribution W(x, k ,t )to represent the envelope of the wave amplitude at position x, time t contained in high frequency waves with wave vector k/e (where e is a small parameter compared to a characteristic distance of propagation), we describe surface wave transport over flows consisting of two length scales; one varying slowly on the wavelength scale, the other varying on a scale comparable to the wavelength. Both static underlying flows and time-varying underlying flows are considered. The spatially rapidly varying but weak surface flows augment the characteristic equations with scattering terms that are explicit functions of the correlations of the random surface currents. These scattering terms depend parametrically on the magnitudes and directions of the smoothly varying drift and are shown to give rise to a Doppler-coupled scattering mechanism. Conservation of wave action (CWA), typically derived for drift varying over long distances, is extended to systems with flow that varies on small length scales of order the surface wavelength. Our results provide a formal set of equations to analyze transport of surface wave action, intensity, energy, and wave scattering as a function of the smoothly varying drifts and the correlation functions of the random, highly oscillating surface flows. © 2002 Elsevier Science B.V. All rights reserved.

This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues.

We consider the imaging of anisotropic conductivity tensors $\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}$ from knowledge of several internal current densities $\mathcal{J}=\gamma\nabla u$ where $u$ satisfies a second order elliptic equation $\nabla\cdot(\gamma\nabla u)=0$ on a bounded domain $X\subset\mathbb{R}^2$ with prescribed boundary conditions on $\partial X$. We show that $\gamma$ can be uniquely reconstructed from four {\em well-chosen} functionals $\mathcal{J}$ and that noise in the data is differentiated once during the reconstruction. The inversion procedure is local in the sense that (most of) the tensor $\gamma(x)$ can be reconstructed from knowledge of the functionals $\mathcal{J}$ in the vicinity of $x$. We obtain the existence of an open set of boundary conditions on $\partial X$ that guaranty stable reconstructions by using the technique of complex geometric optics (CGO) solutions. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modality called Current Density Imaging or Magnetic Resonance Electrical Impedance Tomography.

We propose a method to reconstruct the density of a luminescent source in a highly-scattering medium from ultrasound modulated optical measurements. Our approach is based on the solution to a hybrid inverse source problem for the diffusion equation., Comment: Supplementary information included

This paper concerns the homogenization problem of a parabolic equation with large, time-dependent, random potentials in high dimensions $d\geq 3$. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE., 23 pages, to appear in SPA

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach., Comment: 23 pages, 7 figures

This paper analyzes, theoretically and numerically, the diamond discretization (DD) of free particle transport equations. DD is one of the most commonly used discretizations used in particle transport theory, with applications ranging from the propagation of neutrons in nuclear physics to that of near-infra-red photons in medical imaging. Based on the theory of scalar hyperbolic equations in the Fourier domain, we show that DD performs very well when the boundary data are smooth and the absorption coefficient small. However, because high frequency modes are not damped at all by DD even in the dissipative case, several unphysical modes appear when the boundary conditions are not so smooth. These spurious oscillations are analyzed carefully and displayed numerically.

We consider the matrix-valued radiative transfer equations for the Stokes parameters for the propagation of light through turbulent atmospheres. A Monte Carlo method is introduced to solve the time dependent matrix-valued radiative transfer equations in 3D geometry. The Monte Carlo method is based on a probabilistic representation of the radiative transfer equations involving an augmented scalar transport equation where the polarization parameters are independent variables. The linear moments of the augmented transport equation with respect to the polarization parameters solve the matrix-valued radiative transfer equations. We show how polarization and depolarization effects develop in time for isotropic and unpolarized point sources, considered for concreteness in spherical and half-space geometries. We analyze in detail the creation of polarization by single- and multiple-scattering effects.

We address the propagation of elastic waves generated by an earthquake in the earth crust modeled by a channel separated from the atmosphere and the mantel by two horizontal interfaces. Geophysical studies have shown the validity of radiative transfer in this frequency regime to describe the phase space energy density of seismic waves. For long times and large distances, radiative transfer in weakly absorbing media can be approximated by a diffusion equation. However, the thickness of the crust is of the order of the transport mean free path, the average distance it takes for waves with a wavenumber v to be scattered into another wavenumber v′ by interaction with the inhomogeneous underlying medium. Hence there cannot be diffusion in the vertical direction. This paper shows that diffusion is still valid in the following sense. The radiative transfer solution factors asymptotically in the limit of vanishing mean free paths as the product of a two-dimensional diffusion term in the horizontal directions and ...

This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.

We consider the coupling of angular discretizations of the two-dimensional linear transport equation. We show the well-posedness of the coupled problem and give an error estimate. The angular discretization is based on the discrete ordinates method. It involves a quadrature rule specifically designed for our coupling. The theory uses the integral formulation of transport and will be demonstrated on simplified geometries.

We study transport and diffusion of classical waves in two-dimensional disordered systems and in particular surface waves on a flat surface with randomly fluctuating impedance. We derive from first principles a radiative transport equation for the angularly resolved energy density of the surface waves. This equation accounts for multiple scattering of surface waves as well as for their decay because of leakage into volume waves. We analyze the dependence of the scattering mean free path and of the decay rate on the power spectrum of fluctuations. We also consider the diffusion approximation of the surface radiative transport equation and calculate the angular distribution of the energy transmitted by a strip of random surface impedance.

Summary Radiative transfer theory provides a good framework for the study of multiple scattering in the randomly inhomogeneous lithosphere. Envelopes of high-frequency seismograms (mainly S coda waves) of local earthquakes have been synthesized on the basis of this theory, and inversions for some Earth parameters such as intrinsic attenuation, scattering attenuation and degree of non-isotropic scattering have been carried out. However, a scalar model has often been assumed because of its mathematical relative simplicity. The simplification amounts to neglecting the polarized nature of the underlying motion. This approach is only valid for long lapse times when S waves become unpolarized because of high-order scattering, and cannot be justified by only assuming that the source is unpolarized. We show that incoming unpolarized S waves can be up to 80 per cent polarized after single scattering. Depolarization of S waves after multiple scattering is studied by a Monte Carlo method. We show that the scattering of S waves off different kinds of inhomogeneities gives rise to different polarization and depolarization patterns. Consequently, polarization should provide valuable information for the understanding of the physics of wave motion and the properties of the Earth’s lithosphere.

A companion paper by Bal (Bal G 2000 Inverse Problems 16 997) and this paper are parts I and II of a series dealing with the reconstruction from boundary measurements of the scattering operator of homogeneous linear transport equations. This part II deals with the case of convex bounded domains in dimensions higher than one. We distinguish the analysis of smooth boundaries from that of boundaries with discontinuities such as corners. We propose a reconstruction in the case of degenerate symmetric scattering operators and show the well-posedness of the inverse problem. The proof of well-posedness is based on a decomposition of angular moments of the transport solution into unbounded and bounded components. This decomposition allows us to show the linear independence of a sufficiently large number of angular moments of the transport solution that are used to construct an invertible system for the scattering coefficients to be reconstructed.

We derive a probabilistic representation for solutions of matrix-valued transport equations that account for polarization effects. Such equations arise in radiative transport for the Stokes parameters that model the propagation of light through turbulent atmospheres. They also arise in radiative transport for seismic wave propagation in the earth's crust. The probabilistic representation involves an augmented scalar transport equation in which the polarization parameters become independent variables. Our main result is that the linear moments of the augmented transport equation with respect to the polarization variables are the solution of the matrix-valued transport equation. The augmented scalar transport equation is well suited to analyzing the hydrodynamic regime of small mean free paths. It is also well suited to getting approximate solutions by Monte Carlo simulation.

We derive the diffusion approximation of transport equations with discontinuities at interfaces. The transport equations model the energy density of acoustic waves. The waves are reflected and transmitted at the interface between different media,which leads to discontinuities of the energy density across the interface. The diffusion approximation, which is valid inside each region is not correct at the vicinity of the interface. However, using interface layer analysis, we prove that the transport solution can be approximated by a diffusion term plus an interface layer which decays exponentially fast. We derive systematically the correct form of the interface conditions for this diffusion term.

Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission at a rough interface with small random fluctuations. The Wigner distribution is used to go from waves to energy transport in the high frequency limit, and the Born expansion is used to calculate the effect of the random rough surface. The smoothing method is also used to remove the grazing angle singularity due to the Born approximation. The results are presented in a form that is convenient both for theoretical analysis and for numerical computations.

We derive transport theoretic boundary conditions for acoustic wave reflection at a weakly rough boundary in an inhomogeneous half space. We use the Wigner distribution to go from waves to energy transport in the high frequency limit. We generalize known results on the reflection of acoustic plane waves in a homogeneous medium. We analyze higher order corrections, which include a enhanced backscattering effect in the back direction.

We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. One term is the first eigenvector of the transport equation in the periodicity cell. The other term is the first eigenvector of a diffusion equation in the homogenized domain. Furthermore, the corresponding eigenvalue gives a second order corrector for the eigenvalue of the heterogeneous transport problem. This result justifies and improves the engineering procedure used in practice for nuclear reactor cores computations.

We consider the homogenization of the criticality eigenvalue problem for the even parity flux of neutron transport in a domain with isotropic and periodically oscillating coefficients. We prove that the neutron density is factored in the product of two terms. The first one describes local behavior of the density at the cell level. It is a solution of a heterogeneous transport problem with periodic boundary conditions. The second term gives global behavior on the whole domain. It satisfies a homogeneous diffusion equation posed on the whole domain with Dirichlet boundary conditions. We also give the asymptotic analysis of the corresponding eigenvalues. This expansion gives rise to errors of the order of the cell size. It does not account for neutron leakage at the boundary of the core and yields unacceptable errors in practice. We derive a more accurate expansion of the eigenelements in the case of a symmetric and cubic domain. The analysis of a boundary layer allows us to derive modified boundary conditio...