Guillaume Bal

University of Michigan, Ann Arbor, Michigan, United States

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Publications (148)106.58 Total impact

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    Guillaume Bal, Chenxi Guo, François Monard
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    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.
    03/2014;
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    Guillaume Bal, John C Schotland
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    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. · 2.31 Impact Factor
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    Yu Gu, Guillaume Bal
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    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.
    01/2014;
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    William Martin, Brian Cairns, Guillaume Bal
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    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 01/2014; · 2.38 Impact Factor
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    Guillaume Bal, Wenjia Jing
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    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.
    ESAIM Mathematical Modelling and Numerical Analysis 01/2014; 48(2):387-409. · 1.03 Impact Factor
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    Yu Gu, Guillaume Bal
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    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 $d=3$.
    12/2013;
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    Guillaume Bal, Ningyao Zhang
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    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 $d\geq3$.
    11/2013;
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    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.
    10/2013;
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    Chenxi Guo, Guillaume Bal
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    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.
    08/2013;
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    Guillaume Bal, Ting Zhou
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    ABSTRACT: This paper concerns the quantitative step of the medical imaging modality Thermo-acoustic Tomography (TAT). We model the radiation propagation by a system of Maxwell's equations. We show that the index of refraction of light and the absorption coefficient (conductivity) can be uniquely and stably reconstructed from a sufficiently large number of TAT measurements. Our method is based on verifying that the linearization of the inverse problem forms a redundant elliptic system of equations. We also observe that the reconstructions are qualitatively quite different from the setting where radiation is modeled by a scalar Helmholtz equation as in [10].
    Inverse Problems 08/2013; 30(5). · 1.90 Impact Factor
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    Yu Gu, Guillaume Bal
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    ABSTRACT: An invariance principle is proved for Brownian motion in random scenery, which is chosen to be either a Gaussian or Poissonian random field.
    06/2013;
  • Yu Gu, Guillaume Bal
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    ABSTRACT: Solutions to partial differential equations with highly oscillatory, large random potential have been shown to converge either to homogenized, deterministic limits or to stochastic limits depending on the statistical properties of the potential. In this paper we consider a large class of piece-wise constant potentials and precisely describe how the limit depends on the the correlation properties of the potential and on spatial dimension $d\geq3$. The derivations are based on a Feynman-Kac probabilistic representation and on an invariance principle for Brownian motion in a random scenery.
    04/2013;
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    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, and one of the main sources of uncertainty 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 EarthCARE where capturing the complex, three-dimensional (3D) interactions between clouds and aerosols is a primary objective. 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). Later on, these potentially powerful inverse problem solutions will be fully integrated in a versatile satellite data analysis toolbox. At present, we can report substantial progress at the component level. Specifically, we will focus on the most elementary problems in atmospheric tomography with an emphasis on the vastly under-exploited class of multi-pixel techniques. One basic problem is to infer the outer shape and mean opacity of 3D clouds, along with a bulk measure of cloud particle size. Another is to separate high and low cloud layers based on their characteristically different spatial textures. Yet another is to reconstruct the 3D spatial distribution of aerosol density based on passive imaging. This suite of independent feasibility studies amounts to a compelling proofof- concept for the ambitious 3D-Tomographic Reconstruction of the Aerosol-Cloud Environment (3D-TRACE) project as a whole.
    04/2013;
  • Guillaume Bal, Chenxi Guo, Francois Monard
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    ABSTRACT: This paper concerns the reconstruction of an anisotropic conductivity tensor $\gamma$ from internal current densities of the form $J = \gamma\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(\gamma\nabla u) = 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum number of such functionals equal to $n + 2$, where $n$ is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that $\gamma$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $\gamma$ is scalar, it can be reconstructed with no loss of derivatives. We provide a precise statement of what components may be reconstructed with a loss of zero or one derivatives.
    03/2013; 30(2).
  • Guillaume Bal, Shari Moskow
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    ABSTRACT: Ultrasound modulated optical tomography, also called acousto-optics tomography, is a hybrid imaging modality that aims to combine the high contrast of optical waves with the high resolution of ultrasound. We follow the model of the influence of ultrasound modulation on the light intensity measurements developed in [Bal Schotland PRL 2010]. We present sufficient conditions ensuring that the absorption and diffusion coefficients modeling light propagation can locally be uniquely and stably reconstructed from the corresponding available information. We present an iterative procedure to solve such a problem based on the analysis of linear elliptic systems of redundant partial differential equations.
    03/2013; 30(2).
  • Guillaume Bal, Chenxi Guo, Francois Monard
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    ABSTRACT: This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupled-physics medical imaging modalities such as ultrasound modulated electrical impedance tomography and impedance-acoustic tomography. We consider the linearization of the nonlinear hybrid inverse problem. We find sufficient conditions for the linearized problem, a system of partial differential equations, to be elliptic and for the system to be injective. Such conditions are found to hold for a lesser number of measurements than those required in recently established explicit reconstruction procedures for the nonlinear problem.
    02/2013;
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    Guillaume Bal, Olivier Pinaud
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    ABSTRACT: High frequency waves propagating in highly oscillatory media are often modeled by radiative transfer equations that describes the propagation of the energy density of the waves. When the medium is statistically homogeneous, averaging effects occur in such a way that in the limit of vanishing wavelength, the wave energy density solves a deterministic radiative transfer equation. In this paper, we are interested in the remaining stochasticity of the energy density. More precisely, we wish to understand how such stochasticity depends on the statistics of the random medium and on the initial phase-space structure of the propagating wave packets. The analysis of stochasticity is a formidable task involving complicated ana-lytical calculations. In this paper, we consider the propagation of waves modeled by a scalar Schrödinger equation and limit the interaction of the waves with the underlying structure to second order. We calculate the scintillation function (second statistical moment) for such signals, which thus involve fourth-order moments of the random fluctuations, which we assume have Gaussian statistics. Our main result is a detailed analysis of the scintillation function in that setting. This requires the analysis of non-trivial oscillatory integrals, which is carried out in detail.
    Communications in Partial Differential Equations 01/2013; 38(6). · 1.03 Impact Factor
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    ABSTRACT: Physics-based retrievals of atmosphere and/or surface properties are generally multi- or hyperspectral in nature; some use multi-angle information as well. Recently, polarization has been added to the available input from sensors and accordingly modeled with vector radiative transfer (RT). At any rate, a single pixel is processed at a time using a forward RT model predicated on 1-D transport theory. Neighboring pixels are sometimes considered but, generally, just to formulate statistical constraints on the inversion based on spatial context. Herein, we demonstrate the power to be harnessed by adding bona fide multipixel techniques to the mix. We use a forward RT model in 2-D, sufficient for this demonstration and easily extended to 3-D, for the response of a single-wavelength imaging sensor. The data, an image, is used to infer position, size, and opacity of an absorbing atmospheric plume somewhere in a deep valley in the presence of partially known/partially unknown aerosol. We first describe the necessary innovation to speed-up forward multidimensional RT. In spite of its reputation for inefficiency, we use a Monte Carlo (MC) technique. However, the adopted scheme is highly accelerated without loss of accuracy by using “recycled” MC paths. This forward model is then put to work in a novel Bayesian inversion adapted to this kind of RT model where it is straightforward to trade precision and efficiency. Retrievals target the plume properties and the specific amount of aerosol. In spite of the limited number of pixels and low signal-to-noise ratio, there is added value for certain nuclear treaty verification applications.
    IEEE Transactions on Geoscience and Remote Sensing 01/2013; 51(5):2903-2919. · 3.47 Impact Factor
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    ABSTRACT: In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg-Marquardt iteration for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity $\sigma$ from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg-Marquardt algorithm and tested it on simulated data.
    11/2012;
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    Ningyao Zhang, Guillaume Bal
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    ABSTRACT: We study the asymptotic behavior of solutions to the Schr{\"o}dinger equation with large-amplitude, highly oscillatory, random potential. In dimension $d<\mathfrak{m}$, where $\mathfrak{m}$ is the order of the leading operator in the Schr\"odinger equation, we construct the heterogeneous solution by using a Duhamel expansion and prove that it converges in distribution, as the correlation length $\varepsilon$ goes to 0, to the solution of a stochastic differential equation, whose solution is represented as a sum of iterated Stratonovich integral, over the space $C([0,+\infty),\mathcal{S}')$. The uniqueness of the limiting solution in a dense space of $L^2(\Omega\times\mathbb{R}^d)$ is shown by verifying the property of conservation of mass for the Schr\"odinger equation. In dimension $d>\mathfrak{m}$, the solution to the Schr{\"o}dinger equation is shown to converge in $L^2(\Omega\times\mathbb{R}^d)$ to a deterministic Schr{\"o}dinger solution in \cite{ZB-12}.
    11/2012;

Publication Stats

1k Citations
106.58 Total Impact Points

Institutions

  • 2014
    • University of Michigan
      • Department of Mathematics
      Ann Arbor, Michigan, United States
  • 2001–2014
    • Columbia University
      • Department of Applied Physics and Applied Mathematics
      New York City, New York, United States
  • 2009–2011
    • University of Texas at Austin
      • Department of Mathematics
      Austin, Texas, United States
  • 2000–2002
    • University of Chicago
      • Department of Mathematics
      Chicago, Illinois, United States
  • 1998–2001
    • Stanford University
      • Department of Mathematics
      Stanford, CA, United States