arXiv subjects

Guillaume Bal

Publications and source records attributed to Guillaume Bal.

At least 55 records · Page 3Linked to original sources

arXiv2021

Boundary Control for Transport Equations

This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain X can be controlled exactly from incoming boundary conditions for X under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.

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arXiv2021

Multiscale Invariants of Floquet Topological Insulators

This paper analyzes Floquet topological insulators resulting from the time-harmonic irradiation of electromagnetic waves on two dimensional materials such as graphene. We analyze the bulk and edge topologies of approximations to the evolution of the light-matter interaction. Topologically protected interface states are created by spatial modulations of the drive polarization across an interface. In the high-frequency modulation regime, we obtain a sequence of topologies that apply to different time scales. Bulk-difference invariants are computed in detail and a bulk-interface correspondence is shown to apply. We also analyze a high-frequency high-amplitude modulation resulting in a large-gap effective topology topologically that remains valid only for moderately long times.

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arXiv2021

Topological invariants for interface modes

We consider topologically non-trivial interface Hamiltonians, which find several applications in materials science and geophysical fluid flows. The non-trivial topology manifests itself in the existence of topologically protected, asymmetric edge states at the interface between two two-dimensional half spaces in different topological phases. It is characterized by a quantized interface conductivity. The objective of this paper is to compute such a conductivity and show its stability under perturbations. We present two methods. The first one computes the conductivity using the winding number of branches of absolutely continuous spectrum of the interface Hamiltonian. This calculation is independent of any bulk properties but requires a sufficient understanding of the spectral decomposition of the Hamiltonian. In the fluid flow setting, it also applies in cases where the so-called bulk-interface correspondence fails. The second method establishes a bulk-interface correspondence between the interface conductivity and a so-called bulk-difference invariant. We introduce the bulk-difference invariants characterizing pairs of half spaces. We then relate the interface conductivity to the bulk-difference invariant by means of a Fedosov-Hörmander formula, which computes the index of a related Fredholm operator. The two methods are used to compute invariants for representative $2\times2$ and $3\times3$ systems of equations that appear in the applications.

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arXiv2020

Pencil-beam approximation of fractional Fokker-Planck

We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil-beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.

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arXiv2020

Pencil-beam approximation of stationary Fokker-Planck

Solutions of stationary Fokker-Planck equations in the narrow beam regime are commonly approximated by either ballistic linear transport or by a Fermi pencil-beam equation. We present a rigorous approximation analysis of these three models in a half-space geometry. Error estimates are obtained in a 1-Wasserstein sense, which is an adapted metric to quantify beam spreading. The required well-posedness and regularity results for the stationary Fokker-Planck equation with singular internal and boundary sources are also presented in detail.

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arXiv2020

Homogenization of hydrodynamic transport in Dirac fluids

Large-scale electrical and thermal currents in ordinary metals are well approximated by effective medium theory: global transport properties are governed by the solution to homogenized coupled diffusion equations. In some metals, including the Dirac fluid of nearly charge neutral graphene, microscopic transport is not governed by diffusion, but by a more complicated set of linearized hydrodynamic equations, which form a system of degenerate elliptic equations coupled with the Stokes equation for fluid velocity. In sufficiently inhomogeneous media, these hydrodynamic equations reduce to homogenized diffusion equations. We re-cast the hydrodynamic transport equations as the infimum of a functional over conserved currents, and present a functional framework to model and compute the homogenized diffusion tensor relating electrical and thermal currents to charge and temperature gradients. We generalize to this system two well-known results in homogenization theory: Tartar's proof of local convergence to the homogenized theory in periodic and highly oscillatory media, and sub-additivity of the above functional in random media with highly oscillatory, stationary and ergodic coefficients.

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arXiv2020

Uniqueness criteria in multi-energy CT

Multi-Energy Computed Tomography (ME-CT) is a medical imaging modality aiming to reconstruct the spatial density of materials from the attenuation properties of probing x-rays. For each line in two- or three-dimensional space, ME-CT measurements may be written as a nonlinear mapping from the integrals of the unknown densities of a finite number of materials along said line to an equal or larger number of energy-weighted integrals corresponding to different x-ray source energy spectra. ME-CT reconstructions may thus be decomposed as a two-step process: (i) reconstruct line integrals of the material densities from the available energy measurements; and (ii) reconstruct densities from their line integrals. Step (ii) is the standard linear x-ray CT problem whose invertibility is well-known, so this paper focuses on step (i). We show that ME-CT admits stable, global inversion provided that (a well-chosen linear transform of) the differential of the transform in step (i) satisfies appropriate orientation constraints that makes it a P-matrix. We introduce a notion of quantitative P-function that allows us to derive global stability results for ME-CT in the determined as well as over-determined (with more source energy spectra than the number of materials) cases. Numerical simulations based on standard material properties in imaging applications (of bone, water, contrast agents) and well accepted models of source energy spectra show that ME-CT is often (always in our simulations) either (i) non-globally injective because it is non-injective locally (differential not of full rank), or (ii) globally injective as soon as it is locally injective (differentials satisfy our proposed constraints).

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arXiv2019

Inverse source problems in transport via attenuated tensor tomography

We establish results for the injectivity and injectivity modulo gauge of certain inverse source problems in transport on a simply connected domain with variable index of refraction inducing a 'simple geometry'. The model given by radiative transfer involves a scattering kernel with finite harmonic content in the deviation angle. The results on injectivity are constructive, and they are connected to the explicit inversion (modulo kernel) of the attenuated X-ray transform on tensor fields on simple Riemannian surfaces.

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arXiv2019

Photo-acoustic tomography in a rotating setting

Photo-acoustic tomography is a coupled-physics (hybrid) medical imaging modality that aims to reconstruct optical parameters in biological tissues from ultrasound measurements. As propagating light gets partially absorbed, the resulting thermal expansion generates minute ultrasonic signals (the photo-acoustic effect) that are measured at the boundary of a domain of interest. Standard inversion procedures first reconstruct the source of radiation by an inverse ultrasound (boundary) problem and second describe the optical parameters from internal information obtained in the first step. This paper considers the rotating experimental setting. Light emission and ultrasound measurements are fixed on a rotating gantry, resulting in a rotation-dependent source of ultrasound. The two-step procedure we just mentioned does not apply. Instead, we propose an inversion that directly aims to reconstruct the optical parameters quantitatively. The mapping from the unknown (absorption and diffusion) coefficients to the ultrasound measurement via the unknown ultrasound source is modeled as a composition of a pseudo-differential operator and a Fourier integral operator. We show that for appropriate choices of optical illuminations, the above composition is an elliptic Fourier integral operator. Under the assumption that the coefficients are unknown on a sufficiently small domain, we derive from this a (global) injectivity result (measurements uniquely characterize our coefficients) combined with an optimal stability estimate. The latter is the same as that obtained in the standard (non-rotating experimental) setting.

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arXiv2018

Topological protection of perturbed edge states

This paper proposes a quantitative description of the low energy edge states at the interface between two-dimensional topological insulators. They are modeled by continuous Hamiltonians as systems of Dirac equations that are amenable to a large class of random perturbations. We consider general as well as fermionic time reversal symmetric models. In the former case, Hamiltonians are classified using the index of a Fredholm operator. In the latter case, the classification involves a ${\mathbb Z}_2$ index. These indices dictate the number of topologically protected edge states. A remarkable feature of topological insulators is the asymmetry (chirality) of the edge states, with more modes propagating, say, up than down. In some cases, backscattering off imperfections is prevented when no mode can carry signals backwards. This is a desirable feature from an engineering perspective, which raises the question of how back-scattering is protected topologically. A major motivation for the derivation of continuous models is to answer such a question. We quantify how backscattering is affected but not suppressed by the non-trivial topology by introducing a scattering problem along the edge and describing the effects of topology and randomness on the scattering matrix. Explicit macroscopic models are then obtained within the diffusion approximation of field propagation to show the following: the combination of topology and randomness results in un-hindered transport of the protected modes while all other modes (Anderson) localize.

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arXiv2017

On the propagation of singularities for a class of linearised hybrid inverse problems

For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure based of the least squares finite element method is derived, and a series of numerical experiments visualise exactly how the loss of ellipticity manifests itself as propagating singularities.

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arXiv2017

A Dynamical Sparse Grid Collocation Method for Differential Equations Driven by White Noise

We propose a sparse grid stochastic collocation method for long-time simulations of stochastic differential equations (SDEs) driven by white noise. The method uses pre-determined sparse quadrature rules for the forcing term and constructs evolving set of sparse quadrature rules for the solution variables in time. We carry out a restarting scheme to keep the dimension of random variables for the forcing term, therefore also the number of quadrature points, independent of time. At each restart, a sparse quadrature rule for the current solution variables is constructed based on the knowledge of moments and the previous quadrature rules via a minimization procedure. In this way, the method allows us to capture the long-time solutions accurately using small degrees of freedom. We apply the algorithm to low-dimensional nonlinear SDEs and demonstrate its capability in long-time simulations numerically.

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arXiv2017

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot σ(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension $n=2$, it is known that the number of critical points (where $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $σ$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $σ(x)$ such that $\nabla u$ vanishes at least at one point in $X$. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $σ(x)$. These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $σ(x)$ for which the stability of the reconstructions will inevitably degrade.

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arXiv2017

Generalized stability estimates in inverse transport theory

Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ error predictions on the reconstruction of the coefficients, which are not useful. This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the $1-$Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting. We also consider the effect of errors, still measured in the $1-$Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allow us to consider a discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.

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arXiv2017

Fluctuations in the Homogenization of Semilinear Equations with Random Potentials

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.

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arXiv2016

A dynamical polynomial chaos approach for long-time evolution of SPDEs

We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen--Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier--Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

math.NA↗

arXiv2016

Photo-acoustic tomography in a rotating measurement setting

Photo-acoustic tomography (PAT) aims to leverage the photo-acoustic coupling between optical absorption of light sources and ultrasound (US) emission to obtain high contrast reconstructions of optical parameters with the high resolution of sonic waves. Quantitative PAT often involves a two-step procedure: first the map of sonic emission is reconstructed from US boundary measurements; and second optical properties of biological tissues are evaluated. We consider here a practical measurement setting in which such a separation does not apply. We assume that the optical source and an array of ultrasonic transducers are mounted on a rotating frame (in two or three dimensions) so that the light source rotates at the same time as the US measurements are acquired. As a consequence, we no longer have the option to reconstruct a map of sonic emission corresponding to a given optical illumination. We propose here a framework where the two steps are combined into one and an absorption map is directly reconstructed from the available ultrasound measurements.

math.AP↗

arXiv2016

Dynamical polynomial chaos expansions and long time evolution of differential equations with random forcing

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 (SDEs) 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 Dynamical generalized Polynomial Chaos (DgPC) 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.

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