arXiv subjects

Guillaume Bal

Publications and source records attributed to Guillaume Bal.

At least 19 recordsLinked to original sources

arXiv2026

A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations

Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost.

math.NA↗

arXiv2026

Effects of interface regularity on the bulk-edge correspondence in continuum photonic systems

In this study we analyze the topological invariants and edge states of transverse magnetic wave propagation in continuum photonic systems at a finite-width interface between two gyrotropic materials with different magnetic biases. Where previous studies have almost exclusively considered sharp transitions between two different electromagnetic media, we consider the more general geometry where the magnetic field bias is allowed to vary arbitrarily in a finite-width interface between two bulk regions. We find that when the magnetic field bias varies continuously between the two bulk regions, the Bulk Edge Correspondence (BEC) holds robustly with respect to well-defined Chern invariants. However, discontinuities in the magnetic field bias introduce new edge modes and anomalous high-wavenumber spectral effects which alter the BEC. We analyze these spectral alterations and define a new anomalous BEC in continuum photonic systems which includes contributions from topological invariants and discontinuities in magnetic field bias.

physics.optics↗

arXiv2026

ML-based approach to classification and generation of structured light propagation in turbulent media

We study the classification task of structured-light beams after propagation through a random turbulent medium. The received speckle patterns are generated by numerical simulation of a stochastic paraxial propagation model, and the classification task is formulated over a finite alphabet of 15 OAM source classes. We benchmark intensity and autocorrelation inputs using SimpleCNN and ResNet-18 as classifiers. We also quantify the effect of training-set size and receiver-window misalignment. Since additional propagated samples may be costly to obtain, we develop a class-conditioned diffusion model for generative augmentation of turbulence-degraded intensity images. The main contribution is a spectrum-aware diffusion objective: a pixel-domain loss combined with a Fourier-domain Bregman regularizer designed to preserve high-frequency speckle statistics. We prove that this hybrid objective is consistent with the posterior-mean regression target of the diffusion model and show that generated samples substantially improve low-data classification.

physics.optics↗

arXiv2026

Optimal error bounds on the exponential integrator for dispersive equations with highly concentrated potential

We study a one-dimensional linear dispersive equation of differential order $\kappa \geq 2$ with concentrated potential of extension $\varepsilon$ with $0 < \varepsilon \ll 1$, featuring a competition between weak dispersion of strength $\varepsilon^\alpha \ (0 \leq \alpha \leq \kappa)$ and localization induced by the concentrated potential. We first obtain precise regularity estimates of the exact solution in terms of $\varepsilon$. We then apply a natural first-order exponential integrator with step size $\tau$ to discretize the equation, and establish an optimal error bound of the form $O_{L^\infty}(\tau \varepsilon^\beta)$ (up to logarithmic factors in $\tau$ and $\varepsilon$). Salient features of the result are: (i) error bounds are not only uniform in $\varepsilon$ but improve as $\varepsilon \rightarrow 0$; and (ii) no restriction on $\tau$ in terms of $\varepsilon$. The analysis combines iterated Duhamel's expansions and a transformation that exploits cancellations in oscillatory phases that cannot be obtained directly from regularity estimates of the exact solution. We also show that other classical numerical schemes, such as Lie or centered splitting schemes and low regularity integrators, fail to display optimal rates of convergence. Extensive numerical results are presented and confirm the theoretical error estimates.

math.NA↗

arXiv2026

Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points $(K,E)$ in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.

math-ph↗

arXiv2025

Inverse scattering for waveguides in topological insulators

This paper concerns the inverse scattering problem of a topologically non-trivial waveguide separating two-dimensional topological insulators. We consider the specific model of a Dirac system. We show that a short-range perturbation can be fully reconstructed from scattering data in a linearized setting and in a finite-dimensional setting under a smallness constraint. We also provide a stability result in appropriate topologies. We then solve the problem numerically by means of a standard adjoint method and illustrate our theoretical findings with several numerical simulations.

math-ph↗

arXiv2025

Complex Gaussianity and spatio-frequential memory effect of random wave processes

Wavefield speckle patterns are generated by interference of randomly scattered coherent light. In the weak-coupling regime of the It\^o-Schr\"odinger paraxial model for long-distance wave propagation, we show the following multiscale character: a macroscopic envelope solves a deterministic diffusion equation while the local wavefield (the speckle) is described by a complex Gaussian process both in terms of spatial axial and lateral displacements as well as frequency and angular variations of the incident wavebeam. These results describe speckle patterns and corroborate chromato-spatial memory effects observed in laser light propagation through heterogeneous media.

math.AP↗

arXiv2025

Quantum anomalous Hall phases in gated rhombohedral graphene

We consider a coupled system of Dirac operators that models spin- and valley-polarized gated rhombohedral graphene (RHG) with an arbitrary number of layers. We classify all quantum anomalous Hall phases that are compatible with the model and show that a bulk-edge correspondence exists between bulk phases and chiral edge states carrying a quantized anomalous Hall current. When the displacement field is sufficiently small compared to the interlayer coupling in the RHG application, we retrieve the known phases where the charge is given by the number of graphene layers. When the displacement field increases, we identify all possible topological phase transitions and corresponding quantized chiral edge charges. Numerical simulations confirm the theoretical findings.

cond-mat.mes-hall↗

arXiv2025

Topological edge states of continuous Hamiltonians

This paper concerns the topological classification of continuous Hamiltonians that find applications in biased cold plasmas and photonics. Besides a magnetic bias, the Hamiltonians are parametrized by a plasma frequency and a fixed vertical wavenumber. Eight distinct phases of matter are identified as these parameters vary. When insulating gaps are shared by two such phases, asymmetric edge modes propagate along interfaces separating the two phases. Here we apply the notion of a bulk difference invariant (BDI) to this Hamiltonian, and show by numerical diagonalizations of interface Hamiltonians that after an appropriate regularization our BDI correctly predicts edge transport as described by a bulk edge correspondence. We also derive theoretical tools to compute the BDI and show the limitations of the bulk edge correspondence (BEC) when the phase transition is too singular.

math-ph↗

arXiv2025

Splitting algorithms for paraxial and It\^o-Schr\"odinger models of wave propagation in random media

This paper introduces a full discretization procedure to solve wave beam propagation in random media modeled by a paraxial wave equation or an It\^o-Schr\"odinger stochastic partial differential equation. This method bears similarities with the phase screen method used routinely to solve such problems. The main axis of propagation is discretized by a centered splitting scheme with step $\Delta z$ while the transverse variables are treated by a spectral method after appropriate spatial truncation. The originality of our approach is its theoretical validity even when the typical wavelength $\theta$ of the propagating signal satisfies $\theta\ll\Delta z$. More precisely, we obtain a convergence of order $\Delta z$ in mean-square sense while the errors on statistical moments are of order $(\Delta z)^2$ as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when $\Delta z$ is not the smallest scale of the problem. The analysis is based on equations satisfied by statistical moments in the It\^o-Schr\"odinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings.

math.NA↗

arXiv2024

Continuous Topological Insulators Classification and Bulk Edge Correspondence

This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface-current-observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-H\"ormander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators.

math-ph↗

arXiv2024

Non-unique water and contrast agent solutions in dual-energy CT

The goal of this work is to study occurrences of non-unique solutions in dual-energy CT (DECT) for objects containing water and a contrast agent. Previous studies of the Jacobian of nonlinear systems identified that a vanishing Jacobian determinant indicates the existence of multiple solutions to the system. Vanishing Jacobian determinants are identified for DECT setups by simulating intensity data for practical thickness ranges of water and contrast agent. Once existence is identified, non-unique solutions are found by simulating scan data and finding intensity contours with that intersect multiple times. With this process non-unique solutions are found for DECT setups scanning iodine and gadolinium, including setups using tube potentials in practical ranges. Non-unique solutions demonstrate a large range of differences and can result in significant discrepancies between recovered and true material mapping.

physics.med-ph↗

arXiv2024

Long distance propagation of wave beams in paraxial regime

This paper concerns the propagation of high frequency wave-beams in highly turbulent atmospheres. Using a paraxial model of wave propagation, we show in the long-distance weak-coupling regime that the wavefields are approximately described by a complex Gaussian field whose scintillation index is unity. This provides a model of the speckle formation observed in many practical settings. The main step of the derivation consists in showing that closed-form moment equations in the It\^o-Schr\"odinger regime are still approximately satisfied in the paraxial regime. The rest of the proof is then an extension of results derived in [Bal, G. and Nair, A., arXiv:2402.17107.]

math.AP↗

arXiv2024

Long distance propagation of light in random media with partially coherent sources

Optical beam propagation in random media is characterized by familiar speckle patterns generated by intricate interference effects. Such patterns may be modified and possibly attenuated for partially coherent incident beam profiles. In the weak-coupling regime of the It\^o-Schr\"odinger paraxial model of wave propagation, we show how the spatio-temporal statistics of the partially coherent beams interact with the statistics of the random medium to enhance or suppress scintillation effects.

math.AP↗

arXiv2024

Topological Equatorial Waves and Violation (or not) of the Bulk Edge Correspondence

Atmospheric and oceanic mass transport near the equator display a well-studied asymmetry characterized by two modes moving eastward. This asymmetric edge transport is characteristic of interfaces separating two-dimensional topological insulators. The northern and southern hemispheres are insulating because of the presence of a Coriolis force parameter that vanishes only in the vicinity of the equator. A central tenet of topological insulators, the bulk edge correspondence, relates the quantized edge asymmetry to bulk properties of the insulating phases, which makes it independent of the Coriolis force profile near the equator. We show that for a natural differential Hamiltonian model of the atmospheric and oceanic transport, the bulk-edge correspondence does not always apply. In fact, an arbitrary quantized asymmetry can be obtained for specific, discontinuous, such profiles. The results are based on a careful analysis of the spectral flow of the branches of absolutely continuous spectrum of a shallow-water Hamiltonian. Numerical simulations validate our theoretical findings.

math.AP↗

arXiv2024

Complex Gaussianity of long-distance random wave processes

Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. In the It\^o-Schr\"odinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime.

math.AP↗

arXiv2023

Integral formulation of Dirac singular waveguides

This paper concerns a boundary integral formulation for the two-dimensional massive Dirac equation. The mass term is assumed to jump across a one-dimensional interface, which models a transition between two insulating materials. This jump induces surface waves that propagate outward along the interface but decay exponentially in the transverse direction. After providing a derivation of our integral equation, we prove that it has a unique solution for almost all choices of parameters using holomorphic perturbation theory. We then extend these results to a Dirac equation with two interfaces. Finally, we implement a fast numerical method for solving our boundary integral equations and present several numerical examples of solutions and scattering effects.

math-ph↗

arXiv2023

Topological Anderson Insulators by homogenization theory

A central property of (Chern) topological insulators is the presence of robust asymmetric transport along interfaces separating two-dimensional insulating materials in different topological phases. A Topological Anderson Insulator is an insulator whose topological phase is induced by spatial fluctuations. This paper proposes a mathematical model of perturbed Dirac equations and shows that for sufficiently large and highly oscillatory perturbations, the systems is in a different topological phase than the unperturbed model. In particular, a robust asymmetric transport indeed appears at an interface separating perturbed and unperturbed phases. The theoretical results are based on careful estimates of resolvent operators in the homogenization theory of Dirac equations and on the characterization of topological phases by the index of an appropriate Fredholm operator.

math.AP↗