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Josselin Garnier

Publications and source records attributed to Josselin Garnier.

At least 109 records · Page 6Linked to original sources
arXiv2017

Global acoustic daylight imaging in a stratified Earth-like model

We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation and that the measurements are of the wave field at the surface. Then, we obtain a relation between the autocorrelation function of the measurements and the trace of the scattered field generated by an impulsive source localized just below the surface. From this relation it is, for example, clear that the eigenfrequencies can be recovered from the autocorrelation. Moreover, the complete scattering operator can be extracted under the additional assumption that the annulus is close to the surface and has a thickness smaller than the typical wavelength.

math.AP↗
arXiv2017

Laser beam imaging from the speckle pattern of the off-axis scattered intensity

We study the inverse problem of localization (imaging) of a laser beam from measurements of the intensity of light scattered off-axis by a Poisson cloud of small particles. Starting from the wave equation, we analyze the microscopic coherence of the scattered intensity and show that it is possible to determine the laser beam from the speckle pattern captured by a group of CCD cameras. Two groups of cameras are sufficient when the particles are either small or large with respect to the wavelength. For general particle sizes the accuracy of the laser localization with two groups of cameras is subject to knowing the scattering properties of the cloud. However, three or more groups of cameras allow accurate localization that is robust to uncertainty of the type, size, shape and concentration of the particles in the cloud. We introduce a novel laser beam localization algorithm and give some numerical illustrations in a regime relevant to the application of imaging high energy lasers in a maritime atmosphere.

physics.data-an↗
arXiv2017

On the origins of spectral broadening of incoherent waves:Catastrophic process of coherence degradation

We revisit the mechanisms underlying the process of spectral broadening of incoherent optical waves propagating in nonlinear media on the basis of nonequilibrium thermodynamic considerations. A simple analysis reveals that a prerequisite for the existence of a significant spectral broadening of the waves is that the linear part of the energy (Hamiltonian) has different contributions of opposite signs. It turns out that, at variance with the expected soliton turbulence scenario, an increase of the amount of disorder (incoherence) in the system does not require the generation of a coherent soliton structure. We illustrate the idea by considering the propagation of two wave components in an optical fiber with opposite dispersion coefficients. A wave turbulence approach of the problem reveals that the increase of kinetic energy in one component is offset by the negative reduction in the other component, so that the waves exhibit, as a general rule, a virtually unlimited spectral broadening. More precisely, a self-similar solution of the kinetic equations reveals that the spectra of the incoherent waves tend to relax toward a homogeneous distribution in the wake of a front that propagates in frequency space with a decelerating velocity. We discuss this catastrophic process of spectral broadening in the light of different important phenomena, in particular supercontinuum generation, soliton turbulence, wave condensation, and the runaway motion of mechanical systems composed of positive and negative masses.

nlin.PS↗
arXiv2017

Correction to Black-Scholes formula due to fractional stochastic volatility

Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a function of maturity to a fractional power.

q-fin.MF↗
arXiv2017

Transport of power in random waveguides with turning points

We present a mathematical theory of time-harmonic wave propagation and reflection in a two-dimensional random acoustic waveguide with sound soft boundary and turning points. The boundary has small fluctuations on the scale of the wavelength, modeled as random. The waveguide supports multiple propagating modes. The number of these modes changes due to slow variations of the waveguide cross-section. The changes occur at turning points, where waves transition from propagating to evanescent or the other way around. We consider a regime where scattering at the random boundary has significant effect on the wave traveling from one turning point to another. This effect is described by the coupling of its components, the modes. We derive the mode coupling theory from first principles, and quantify the randomization of the wave and the transport and reflection of power in the waveguide. We show in particular that scattering at the random boundary may increase or decrease the net power transmitted through the waveguide depending on the source.

math.AP↗
arXiv2017

Ambient noise correlation-based imaging with moving sensors

Waves can be used to probe and image an unknown medium. Passive imaging uses ambient noise sources to illuminate the medium. This paper considers passive imaging with moving sensors. The motivation is to generate large synthetic apertures, which should result in enhanced resolution. However Doppler effects and lack of reciprocity significantly affect the imaging process. This paper discusses the consequences in terms of resolution and it shows how to design appropriate imaging functions depending on the sensor trajectory and velocity.

physics.class-ph↗
arXiv2016

Mathematical modelling of the electric sense of fish: the role of multi-frequency measurements and movement

Understanding active electrolocation in weakly electric fish remains a challenging issue. In this article we propose a mathematical formulation of this problem, in terms of partial differential equations. This allows us to detail two algorithms: one for localizing a target using the multi-frequency aspect of the signal, and antoher one for identifying the shape of this target. Shape recognition is designed in a machine learning point of view, and take advantage of both the multi-frequency setup and the movement of the fish around its prey. Numerical simulations are shown for the computation of the electric field emitted and sensed by the fish; they are then used as an input for the two algorithms.

q-bio.TO↗
arXiv2016

Pulse reflection in a random waveguide with a turning point

We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected. To focus attention on this reflection, we assume that the waveguide supports a single propagating mode from the source to the turning point, beyond which all the waves are evanescent. We consider a scaling regime where scattering at the random boundary has a significant effect on the reflected pulse. In this regime scattering from the random boundary away from the turning point is negligible, while scattering from the random boundary around the turning point results in a strong, deterministic pulse deformation. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. The reflected pulse also carries a random phase.

math.AP↗
arXiv2016

Focusing Waves Through a Randomly Scattering Medium in the White-Noise Paraxial Regime

When waves propagate through a complex or heterogeneous medium the wave field is corrupted by the heterogeneities. Such corruption limits the performance of imaging or communication schemes. One may then ask the question: is there an optimal way of encoding a signal so as to counteract the corruption by the medium? In the ideal situation the answer is given by time reversal: for a given target or focusing point, in a first step let the target emit a signal and then record the signal transmitted to the source antenna, time reverse this and use it as the source trace at the source antenna in a second step. This source will give a sharply focused wave at the target location if the source aperture is large enough. Here we address this scheme in the more practical situation with a limited aperture, time-harmonic signal, and finite-sized elements in the source array. Central questions are then the focusing resolution and signal-to-noise ratio at the target, their dependence on the physical parameters, and the capacity to focus selectively in the neighborhood of the target point and therefore to transmit images. Sharp results are presented for these questions.

math.AP↗
arXiv2016

Consensus Convergence with Stochastic Effects

We consider a stochastic, continuous state and time opinion model where each agent's opinion locally interacts with other agents' opinions in the system, and there is also exogenous randomness. The interaction tends to create clusters of common opinion. By using linear stability analysis of the associated nonlinear Fokker-Planck equation that governs the empirical density of opinions in the limit of infinitely many agents, we can estimate the number of clusters, the time to cluster formation and the critical strength of randomness so as to have cluster formation. We also discuss the cluster dynamics after their formation, the width and the effective diffusivity of the clusters. Finally, the long term behavior of clusters is explored numerically. Extensive numerical simulations confirm our analytical findings.

cs.SI↗
arXiv2016

Robust imaging with electromagnetic waves in noisy environments

We study imaging with an array of sensors that probes a medium with single frequency electromagnetic waves and records the scattered electric field. The medium is known and homogenous except for some small and penetrable inclusions. The goal of inversion is to locate and characterize these inclusions from the data collected by the array, which are corrupted by additive noise. We use results from random matrix theory to obtain a robust inversion method. We assess its performance with numerical simulations and quantify the benefit of measuring more than one component of the scattered electric field.

math.AP↗
arXiv2016

Polarization effects for electromagnetic wave propagation in random media

We study Maxwell's equations in random media with small fluctuations of the electric permittivity. We consider a setup where the waves propagate toward a preferred direction, called range. We decompose the electromagnetic wave field in transverse electric and transverse magnetic plane waves, called modes, with random amplitudes that model cumulative scattering effects in the medium. Their evolution in range is described by a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the Markov limit theorem and obtain a detailed asymptotic characterization of the electromagnetic wave field in the long range limit. In particular, we quantify the loss of coherence of the waves due to scattering by calculating the range scales (scattering mean free paths) on which the mean amplitudes of the modes decay. We also quantify the energy exchange between the modes, and consequently the loss of polarization induced by scattering, by analyzing the Wigner transform (energy density) of the electromagnetic wave field. This analysis involves the derivation of transport equations with polarization. We study in detail these equations and connect the results with the existing literature in radiative transport and paraxial wave propagation.

math.AP↗
arXiv2016

Derivation of a one-way radiative transfer equation in random media

We derive from first principles a one-way radiative transfer equation for the wave intensity resolved over directions (Wigner transform of the wave field) in random media. It is an initial value problem with excitation from a source which emits waves in a preferred, forward direction. The equation is derived in a regime with small random fluctuations of the wave speed but long distances of propagation with respect to the wavelength, so that cumulative scattering is significant. The correlation length of the medium and the scale of the support of the source are slightly larger than the wavelength, and the waves propagate in a wide cone with opening angle less than $180^o$, so that the backward and evanescent waves are negligible. The scattering regime is a bridge between that of radiative transfer, where the waves propagate in all directions and the paraxial regime, where the waves propagate in a narrow angular cone. We connect the one-way radiative transport equation with the equations satisfied by the Wigner transform of the wave field in these regimes.

math.AP↗
arXiv2015

Fourth-moment Analysis for Wave Propagation in the White-Noise Paraxial Regime

In this paper we consider the Ito-Schrodinger model for wave propagation in random media in the paraxial regime. We solve the equation for the fourth-order moment of the field in the regime where the correlation length of the medium is smaller than the initial beam width. As applications we prove that the centered fourth-order moments of the field satisfy the Gaussian summation rule, we derive the covariance function of the intensity of the transmitted beam, and the variance of the smoothed Wigner transform of the transmitted field. The second application is used to explicitly quantify the scintillation of the transmitted beam and the third application to quantify the statistical stability of the Wigner transform.

math.AP↗
arXiv2015

A risk analysis for a system stabilized by a central agent

We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force in the system. The local agents derive their stability only from the central agent. In the mean field limit of a large number of local agents we show that the systemic risk decreases when the strength of the interaction of the local agents with the central agent increases. This means that the probability of transition from one of the two stable quasi-equilibria to the other one decreases. We also show that the systemic risk increases when the strength of the interaction of the central agent with the mean field of the local agents increases. Following the financial interpretation of such models and their behavior given in our previous paper (Garnier, Papanicolaou and Yang, SIAM J. Fin. Math. 4, 2013, 151-184), we may interpret the results of this paper in the following way. From the point of view of systemic risk, and while keeping the perceived risk of the local agents approximately constant, it is better to strengthen the interaction of the local agents with the central agent than the other way around.

q-fin.MF↗
arXiv2013

Role of scattering in virtual source array imaging

We consider imaging in a scattering medium where the illumination goes through this medium but there is also an auxiliary, passive receiver array that is near the object to be imaged. Instead of imaging with the source-receiver array on the far side of the object we image with the data of the passive array on the near side of the object. The imaging is done with travel time migration using the cross correlations of the passive array data. We showed in [J. Garnier and G. Papanicolaou, Inverse Problems {28} (2012), 075002] that if (i) the source array is infinite, (ii) the scattering medium is modeled by either an isotropic random medium in the paraxial regime or a randomly layered medium, and (iii) the medium between the auxiliary array and the object to be imaged is homogeneous, then imaging with cross correlations completely eliminates the effects of the random medium. It is as if we imaged with an active array, instead of a passive one, near the object. The purpose of this paper is to analyze the resolution of the image when both the source array and the passive receiver array are finite. We show with a detailed analysis that for isotropic random media in the paraxial regime, imaging not only is not affected by the inhomogeneities but the resolution can in fact be enhanced. This is because the random medium can increase the diversity of the illumination. We also show analytically that this will not happen in a randomly layered medium, and there may be some loss of resolution in this case.

math.AP↗
arXiv2013

Spectroscopic imaging of a dilute cell suspension

A rigorous homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with respect to the membrane polarization. Different microstructures are shown to be distinguishable via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell-Wagner-Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells.

math.AP↗