[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We propose a method to reconstruct the density of an optical 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
radiative transport equation (RTE). A controllability result for the RTE plays
an essential role in the analysis.
[Show abstract][Hide abstract] ABSTRACT: Polynomial chaos expansions (PCE) allow us to propagate uncertainties in the
coefficients of differential equations to the statistics of their solutions.
Their main advantage is that they replace stochastic equations by systems of
deterministic equations. Their main challenge is that the computational cost
becomes prohibitive when the dimension of the parameters modeling the
stochasticity is even moderately large. We propose a generalization of the PCE
framework that allows us to keep this dimension as small as possible in
favorable situations. For instance, in the setting of stochastic differential
equations (SDE) with Markov random forcing, we expect the future evolution to
depend on the present solution and the future stochastic variables. We present
a restart procedure that precisely allows PCE to depend only on that
information. The computational difficulty then becomes the construction of
orthogonal polynomials for dynamically evolving measures. We present
theoretical results of convergence for our Dynamically Orthogonal generalized
Polynomial Chaos (DO-gPC) method. Numerical simulations for linear and
nonlinear SDEs show that it adequately captures the long-time behavior of their
solutions as well as their invariant measures when the latter exist.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: This paper concerns the imaging of a complex-valued anisotropic tensor
{\gamma} = {\sigma}+{\iota}{\omega}{\epsilon} from knowledge of several inter
magnetic fields H where H satisfies the anisotropic Maxwell system on a bounded
domain with prescribed boundary conditions. We show that {\gamma} can be
uniquely reconstructed with a loss of two derivatives from errors in the
acquisition H. A minimum number of five well-chosen functionals guaranties a
local reconstruction of {\gamma} in dimension two. 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
modalities Current Density Imaging and Magnetic Resonance Electrical Impedance
Tomography.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: This paper reviews several results obtained recently in the convergence of solutions to elliptic or parabolic equations with large highly oscillatory random potentials. Depending on the correlation properties of the potential, the resulting limit may be either deterministic and solution of a homogenized equation or random and solution of a stochastic PDE. In the former case, the residual random fluctuations of the heterogeneous solution may also be characterized, or at least the rate of convergence to the deterministic limit established. We present several results that can be obtained by the methods of asymptotic perturbations, diagrammatic expansions, probabilistic representations, and the multiscale method.
[Show abstract][Hide abstract] ABSTRACT: This work is devoted to the stability/resolution analysis of several imaging
functionals in complex environments. We consider both linear functionals in the
wavefield as well as quadratic functionals based on wavefield correlations.
Using simplified measurement settings and reduced functionals that retain the
main features of functionals used in practice, we obtain optimal asymptotic
estimates of the signal-to-noise ratios depending on the main physical
parameters of the problem. We consider random media with possibly long-range
dependence and with a correlation length that is less than or equal to the
central wavelength of the source we aim to reconstruct. This corresponds to the
wave propagation regimes of radiative transfer or homogenization.
[Show abstract][Hide abstract] ABSTRACT: This paper derives an efficient procedure for using the three-dimensional (3D) vector radiative transfer equation (VRTE) to adjust atmosphere and surface properties and improve their fit with multi-angle/multi-pixel radiometric and polarimetric measurements of scattered sunlight. The proposed adjoint method uses the 3D VRTE to compute the measurement misfit function and the adjoint 3D VRTE to compute its gradient with respect to all unknown parameters. In the remote sensing problems of interest, the scalar-valued misfit function quantifies agreement with data as a function of atmosphere and surface properties, and its gradient guides the search through this parameter space. Remote sensing of the atmosphere and surface in a three-dimensional region may require thousands of unknown parameters and millions of data points. Many approaches would require calls to the 3D VRTE solver in proportion to the number of unknown parameters or measurements. To avoid this issue of scale, we focus on computing the gradient of the misfit function as an alternative to the Jacobian of the measurement operator. The resulting adjoint method provides a way to adjust 3D atmosphere and surface properties with only two calls to the 3D VRTE solver for each spectral channel, regardless of the number of retrieval parameters, measurement view angles or pixels. This gives a procedure for adjusting atmosphere and surface parameters that will scale to the large problems of 3D remote sensing. For certain types of multi-angle/multi-pixel polarimetric measurements, this encourages the development of a new class of three-dimensional retrieval algorithms with more flexible parameterizations of spatial heterogeneity, less reliance on data screening procedures, and improved coverage in terms of the resolved physical processes in the Earth's atmosphere.
Journal of Quantitative Spectroscopy and Radiative Transfer 09/2014; 144. DOI:10.1016/j.jqsrt.2014.03.030 · 2.65 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Remote sensing is a key tool for sorting cloud ensembles by dynamical state, aerosol environments by source region, and establishing causal relationships between aerosol amounts, type, and cloud microphysics---the so-called indirect aerosol climate impacts, which is identified as one of the main sources of error in current climate models. Current satellite imagers use data processing approaches that invariably start with cloud detection/masking to isolate aerosol air-masses from clouds, and then rely on one-dimensional (1D) radiative transfer (RT) to interpret the aerosol and cloud measurements in isolation. Not only does this lead to well-documented biases for the estimates of aerosol radiative forcing and cloud optical depths in current missions, but it is fundamentally inadequate for future missions such as ESA's Earth Cloud, Aerosol and Radiation Experiment (EarthCARE), NASA's Pre-ACE (PACE) and Aerosol-Cloud-Ecosystem mission (ACE, a NASA Tier-2 Decadal Survey mission). All of these missions are intent on capturing the complex, three-dimensional (3D) interactions between clouds and aerosols.
In order to advance the state-of-the-art, the next generation of satellite information processing systems must incorporate technologies that will enable the treatment of the atmosphere as a fully 3D environment, represented more realistically as a continuum. At one end, there is an optically thin background dominated by aerosols and molecular scattering that is strongly stratified and relatively homogeneous in the horizontal. At the other end, there are optically thick embedded elements, clouds and aerosol plumes, which can be more or less uniform and quasi-planar or else highly 3D with boundaries in all directions; in both cases, strong internal variability may be present.
To make this paradigm shift possible, we propose to combine the standard models for satellite signal prediction physically grounded in 1D and 3D RT, both scalar and vector, with technologies adapted from biomedical imaging, digital image processing, and computer vision. This will enable us to demonstrate how the 3D distribution of atmospheric constituents, and their associated microphysical properties, can be reconstructed from multi-angle/multi-spectral imaging radiometry and, more and more, polarimetry. Specific technologies of interest are computed tomography (reconstruction from projections), optical tomography (using cross-pixel radiation transport in the diffusion limit), stereoscopy (depth/height retrievals), blind source and scale separation (signal unmixing), and disocclusion (information recovery in the presence of obstructions).
In time, these potentially powerful inverse problem solutions will be fully integrated in a versatile satellite data analysis toolbox. At present, we will report on substantial progress at the component level achieved in the course of a one-year pilot study sponsored by NASA's Earth Science and Technology office (ESTO). We focused specifically on the most elementary problems in atmospheric tomography:
* One basic problem is to infer the outer shape and mean extinction of optically thick cumulus-type 3D clouds, along with a bulk measure of cloud particle size. Two independent approaches were tested at JPL and Columbia University.
* Another is to reconstruct the 3D spatial distribution of aerosol particle density in a plume, or crystal density in a cirrus layer, using only passive imaging. Here again, two independent approaches were tested at JPL and Technion - IIT. See Aides et al. [2013] for a description and demonstration of the latter tomographic reconstruction method.
* Yet another is to separate high (cirrus) and low (broken cumulus) cloud layers based on their characteristically different spatial textures. See Yanovsky et al. [2014] for a description and demonstration of the image processing methodology that was used to solve this basic problem.
Across all of these efforts, the wide-open frontier of multi-angle/multi-pixel algorithms was explored. The suite of five independent feasibility studies will amount to a compelling proof-of-concept for the ambitious 3D-Tomographic Reconstruction of the Aerosol-Cloud Environment (3D-TRACE) project as a whole.
Finally, a notable spin-off of the 3D-TRACE project is the development of a high-performance computing framework for generating high-fidelity synthetic multi-angle imagery to test new algorithms in a setting where the truth is known at every level of detail. First, the JPL Large-Eddy Simulation (LES) code [Matheou and Chung, 2014] is used to obtain very realistic clouds or aerosol plumes (5 to 20 m grid-cells over 5 to 20 km domains). Then Lorentz-Mie code is used to convert the LES's bulk or bin microphysical quantities into optical ones. Finally, the state-of-the-art MYSTIC 3D vector radiative transfer code [Emde et al., 2010] is applied to this large gridded scene using highly optimized backward Monte Carlo methods [Buras and Mayer, 2011] to deliver the imagery just as a remote sensing instrument would record it. This stand-alone capability at JPL will also be used to test operational algorithms in new ways.
References:
Aides, A., Y. Y. Schechner, V. Holodovsky, M. J. Garay, and A. B. Davis (2013). Multi Sky-View 3D Aerosol Distribution Recovery, Opt. Express, 21, 25820-25833.
Emde, C., R. Buras, B. Mayer, and M. Blumthaler (2010). The impact of aerosols on polarized sky radiance: Model development, validation, and applications. Atmos. Chem. Phys., 10, 383-396.
Buras, R., and B. Mayer (2011). Efficient unbiased variance reduction techniques for Monte Carlo simulations of radiative transfer in cloudy atmospheres: The solution. J. Quant. Spectrosc. Radiat. Transfer, 112, 434-447.
Matheou, G., and D. Chung (2014). Large-eddy simulation of stratified turbulence. Part II: Application of the stretched-vortex model to the atmospheric boundary layer. J. Atmos. Sci. (under revision).
Yanovsky, I., A. B. Davis, and V. M. Jovanovic (2014). Separation of radiances from a cirrus layer and broken cumulus clouds in multispectral images. IEEE Trans. Geosc. and Remote Sens. (submitted).
14th Conference on Cloud Physics/14th Conference on Atmospheric Radiation/Anthony Slingo Symposium 2014 American Meteorological Society; 07/2014
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: This paper analyzes the random fluctuations obtained by a heterogeneous
multi-scale first-order finite element method applied to solve elliptic
equations with a random potential. We show that the random fluctuations of such
solutions are correctly estimated by the heterogeneous multi-scale algorithm
when appropriate fine-scale problems are solved on subsets that cover the whole
computational domain. However, when the fine-scale problems are solved over
patches that do not cover the entire domain, the random fluctuations may or may
not be estimated accurately. In the case of random potentials with short-range
interactions, the variance of the random fluctuations is amplified as the
inverse of the fraction of the medium covered by the patches. In the case of
random potentials with long-range interactions, however, such an amplification
does not occur and random fluctuations are correctly captured independent of
the (macroscopic) size of the patches.
These results are consistent with those obtained by the authors for more
general equations in the one-dimensional setting and provide indications on the
loss in accuracy that results from using coarser, and hence less
computationally intensive, algorithms.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we present a fluctuation analysis of a type of parabolic equations with large, highly oscillatory, random potentials around the homogenization limit. With a Feynman-Kac representation, the Kipnis-Varadhan’s method, and a quantitative martingale central limit theorem, we derive the asymptotic distribution of the rescaled error between heterogeneous and homogenized solutions under different assumptions in dimension \(d\ge 3\) . The results depend highly on whether a stationary corrector exits.
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review E 03/2014; 89(3-1):031201. DOI:10.1103/PhysRevE.89.031201 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper concerns the homogenization problem of heat 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.
Stochastic Processes and their Applications 01/2014; 125(1). DOI:10.1016/j.spa.2014.07.024 · 1.06 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this note, we prove a central limit result for heat equation with large,
highly oscillatory, random potential. With a Feynman-Kac representation, the
Kipnis-Varadhan's method and a refined quantitative martingale central limit
theorem, we derive the asymptotic Gaussian distribution of the rescaled
corrector when the random potential is Gaussian or Poissonian and the dimension
$d=3$.
[Show abstract][Hide abstract] ABSTRACT: We consider an elliptic equation with purely imaginary, highly heterogeneous,
and large random potential with a sufficiently rapidly decaying correlation
function. We show that its solution is well approximated by the solution to a
homogeneous equation with a real-valued homogenized potential as the
correlation length of the random medium $\varepsilon\rightarrow 0$ and estimate
the size of the random fluctuations in the setting $d\geq3$.
[Show abstract][Hide abstract] ABSTRACT: Assuming the availability of internal full-field measurements of the
continuum deformations associated with a non-homogeneous isotropic linear
elastic solid, this article focuses on the quantitative reconstruction of its
constitutive parameters. Starting from the governing momentum equation,
algebraic manipulations are employed to construct a simple gradient system for
the quantities of interest in which the featured coefficients are expressed in
terms of the measured displacement fields and their spatial derivatives. A
direct integration of this system is discussed to finally demonstrate the
inexpediency of such an approach when dealing with polluted measurements. Upon
using noisy data, an alternative variational formulation is deployed to invert
for the unknown 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 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.
[Show abstract][Hide abstract] ABSTRACT: This paper concerns the reconstruction of possibly complex-valued
coefficients in a second-order scalar elliptic equation posed on a bounded
domain from knowledge of several solutions of that equation. We show that for a
sufficiently large number of solutions and for an open set of corresponding
boundary conditions, all coefficients can be uniquely and stably reconstructed
up to a well characterized gauge transformation. We also show that in some
specific situations, a minimum number of such available solutions equal to
$I_n=\frac12n(n+3)$ is sufficient to uniquely and globally reconstruct the
unknown coefficients. This theory finds applications in several coupled-physics
medical imaging modalities including photo-acoustic tomography, transient
elastography, and magnetic resonance elastography.
Communications on Pure and Applied Mathematics 10/2013; 66(10). DOI:10.1002/cpa.21453 · 3.13 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper concerns the reconstruction of a complex-valued anisotropic tensor
$\gamma=\sigma+\i\omega\varepsilon$ from knowledge of several internal magnetic
fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded
domain with prescribed boundary conditions. We show that $\gamma$ can be
uniquely reconstructed with a loss of two derivatives from errors in the
acquisition of $H$. A minimum number of 6 such functionals is sufficient to
obtain a local reconstruction of $\gamma$. In the special case where $\gamma$
is close to a scalar tensor, boundary conditions are chosen by means of complex
geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a
Runge approximation property is used to obtain partial results. This problem
finds applications in the medical imaging modalities Current Density Imaging
and Magnetic Resonance Electrical Impedance Tomography.