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Guillaume Bal

Publications and source records attributed to Guillaume Bal.

At least 73 records · Page 4Linked to original sources
arXiv2015

Reconstruction of Cloud Geometry from High-Resolution Multi-Angle Images

We consider the reconstruction of the interface of compact, connected "clouds" from satellite or airborne light intensity measurements. In a two dimensional setting, the cloud is modeled by an interface, locally represented as a graph, and an outgoing radiation intensity that is consistent with a diffusion model for light propagation in the cloud. Light scattering inside the cloud and the optical internal parameters of the cloud are not modeled. The main objective is to understand what can or cannot be reconstructed in such a setting from intensity measurements in a finite (on the order of 10) number of directions along the path of a satellite. Numerical simulations illustrate the theoretical predictions.

math.CA↗
arXiv2015

Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields

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.

math.AP↗
arXiv2015

Imaging of complex-valued tensors for two-dimensional Maxwell's equations

This paper concerns the imaging of a complex-valued anisotropic tensor γ = σ+ιωε 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 γ 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 γ 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.

math.AP↗
arXiv2014

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

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.

math.AP↗
arXiv2014

Fluctuations of Parabolic Equations with Large Random Potentials

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 $d\geq 3$. The results depend highly on whether a stationary corrector exits.

math.PR↗
arXiv2014

Weak Convergence Approach for Parabolic Equations with Large, Highly Oscillatory, Random Potential

This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show that the random solution converges, as the correlation length of the medium tends to zero, to the deterministic solution of a homogenized equation in dimension $d\geq3$. Our derivation is based on a Feynman-Kac probabilistic representation and the Kipnis-Varadhan method applied to weak convergence of Brownian motions in random sceneries. For sufficiently mixing coefficients, we also provide an optimal rate of convergence to the homogenized limit using a quantitative martingale central limit theorem. As soon as the above integrability condition fails, the solution is expected to remain stochastic in the limit of a vanishing correlation length. For a large class of potentials given as functionals of Gaussian fields, we show the convergence of solutions to stochastic partial differential equations (SPDE) with multiplicative noise. The Feynman-Kac representation and the corresponding weak convergence of Brownian motions in random sceneries allows us to explain the transition from deterministic to stochastic limits as a function of the correlation function of the random potential.

math.PR↗
arXiv2014

Homogenization of Parabolic Equations with Large Time-dependent Random Potential

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.

math.PR↗
arXiv2014

Imaging of anisotropic conductivities from current densities in two dimensions

We consider the imaging of anisotropic conductivity tensors $γ=(γ_{ij})_{1\leq i,j\leq 2}$ from knowledge of several internal current densities $\mathcal{J}=γ\nabla u$ where $u$ satisfies a second order elliptic equation $\nabla\cdot(γ\nabla u)=0$ on a bounded domain $X\subset\mathbb{R}^2$ with prescribed boundary conditions on $\partial X$. We show that $γ$ 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 $γ(x)$ 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.

math.AP↗
arXiv2014

An invariance principle for Brownian motion in random scenery

We prove an invariance principle for Brownian motion in Gaussian or Poissonian random scenery by the method of characteristic functions. Annealed asymptotic limits are derived in all dimensions, with a focus on the case of dimension $d=2$, which is the main new contribution of the paper.

math.PR↗
arXiv2013

Hybrid inverse problems and redundant systems of partial differential equations

Hybrid inverse problems are mathematical descriptions of coupled-physics (also called multi-waves) imaging modalities that aim to combine high resolution with high contrast. The solution of a high-resolution inverse problem, a first step that is not considered in this paper, provides internal information combining unknown parameters and solutions of differential equations. In several settings, the internal information and the differential equations may be described as a redundant system of nonlinear partial differential equations. We propose a framework to analyze the uniqueness and stability properties of such systems. We consider the case when the linearization of the redundant system is elliptic and with boundary conditions satisfying the Lopatinskii conditions. General theories of elliptic systems then allow us to construct a parametrix for such systems and derive optimal stability estimates. The injectivity of the nonlinear problem or its linearization is not guaranteed by the ellipticity condition. We revisit unique continuation principles, such as the Holmgren theorem and the uniqueness theorem of Calderón, in the context of redundant elliptic systems of equations. The theory is applied to the case of power density measurements, which are internal functionals of the form $γ|\nabla u|^2$ where $γ$ is an unknown parameter and $u$ is the solution to the elliptic equation $\nabla\cdotγ\nabla u=0$ on a bounded domain with appropriate boundary conditions.

math.AP↗
arXiv2013

Homogenization of elliptic equations with large random potential

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 $d\geq3$.

math.AP↗
arXiv2013

Ultrasound Modulated Bioluminescence 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.

physics.optics↗
arXiv2013

Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential

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.

math.NA↗
arXiv2013

Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

This paper concerns the reconstruction of a complex-valued anisotropic tensor $γ=σ+ıω\varepsilon$ 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 $γ$ 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 $γ$. In the special case where $γ$ is close to a scalar tensor, boundary conditions are chosen by means of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $γ$, 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.

math.AP↗
arXiv2013

Hybrid Inverse Problems for a System of Maxwell's Equations

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].

math.AP↗
arXiv2013

Propagation of seismic waves through a spatio-temporally fluctuating medium: Homogenization

Measurements of seismic wave travel times at the photosphere of the Sun have enabled inferences of its interior structure and dynamics. In interpreting these measurements, the simplifying assumption that waves propagate through a temporally stationary medium is almost universally invoked. However, the Sun is in a constant state of evolution, on a broad range of spatio-temporal scales. At the zero wavelength limit, i.e., when the wavelength is much shorter than the scale over which the medium varies, the WKBJ (ray) approximation may be applied. Here, we address the other asymptotic end of the spectrum, the infinite wavelength limit, using the technique of homogenization. We apply homogenization to scenarios where waves are propagating through rapidly varying media (spatially and temporally), and derive effective models for the media. One consequence is that a scalar sound speed becomes a tensorial wavespeed in the effective model and anisotropies can be induced depending on the nature of the perturbation. The second term in this asymptotic two-scale expansion, the so-called corrector, contains contributions due to higher-order scattering, leading to the decoherence of the wavefield. This decoherence may be causally linked to the observed wave attenuation in the Sun. Although the examples we consider here consist of periodic arrays of perturbations to the background, homogenization may be extended to ergodic and stationary random media. This method may have broad implications for the manner in which we interpret seismic measurements in the Sun and for modeling the effects of granulation on the scattering of waves and distortion of normal-mode eigenfunctions.

astro-ph.EP↗
arXiv2013

Inverse anisotropic conductivity from internal current densities

This paper concerns the reconstruction of an anisotropic conductivity tensor $γ$ from internal current densities of the form $J = γ\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(γ\nabla u) = 0$ 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 $γ$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $γ$ 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.

math.AP↗