Vortex spectrum in superfluid turbulence: interpretation of a recent experiment

We discuss a recent experiment in which the spectrum of the vortex line density fluctuations has been measured in superfluid turbulence. The observed frequency dependence of the spectrum, $f^{-5/3}$, disagrees with classical vorticity spectra if, following the literature, the vortex line density is interpreted as a measure of the vorticity or enstrophy. We argue that the disagrement is solved if the vortex line density field is decomposed into a polarised field (which carries most of the energy) and an isotropic field (which is responsible for the spectrum).Comment: Submitted for publication http://crtbt.grenoble.cnrs.fr/helio/GROUP/infa.html http://www.mas.ncl.ac.uk/~ncfb

Bayesian Image Quality Transfer with CNNs: Exploring Uncertainty in dMRI Super-Resolution

In this work, we investigate the value of uncertainty modeling in 3D super-resolution with convolutional neural networks (CNNs). Deep learning has shown success in a plethora of medical image transformation problems, such as super-resolution (SR) and image synthesis. However, the highly ill-posed nature of such problems results in inevitable ambiguity in the learning of networks. We propose to account for intrinsic uncertainty through a per-patch heteroscedastic noise model and for parameter uncertainty through approximate Bayesian inference in the form of variational dropout. We show that the combined benefits of both lead to the state-of-the-art performance SR of diffusion MR brain images in terms of errors compared to ground truth. We further show that the reduced error scores produce tangible benefits in downstream tractography. In addition, the probabilistic nature of the methods naturally confers a mechanism to quantify uncertainty over the super-resolved output. We demonstrate through experiments on both healthy and pathological brains the potential utility of such an uncertainty measure in the risk assessment of the super-resolved images for subsequent clinical use.Comment: Accepted paper at MICCAI 201

Renormalization group and perfect operators for stochastic differential equations

We develop renormalization group methods for solving partial and stochastic differential equations on coarse meshes. Renormalization group transformations are used to calculate the precise effect of small scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator: an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.Comment: 35 pages, 16 figure

Fluid--Gravity Correspondence under the presence of viscosity

The present work addresses the analogy between the speed of sound of a viscous, barotropic, and irrotational fluid and the equation of motion for a non--massive field in a curved manifold. It will be shown that the presence of viscosity implies the introduction, into the equation of motion of the gravitational analogue, of a source term which entails the flow of energy from the non--massive field to the curvature of the spacetime manifold. The stress-energy tensor is also computed and it is found not to be constant, which is consistent with such energy interchange

Heterogeneous dynamics of the three dimensional Coulomb glass out of equilibrium

The non-equilibrium relaxational properties of a three dimensional Coulomb glass model are investigated by kinetic Monte Carlo simulations. Our results suggest a transition from stationary to non-stationary dynamics at the equilibrium glass transition temperature of the system. Below the transition the dynamic correlation functions loose time translation invariance and electron diffusion is anomalous. Two groups of carriers can be identified at each time scale, electrons whose motion is diffusive within a selected time window and electrons that during the same time interval remain confined in small regions in space. During the relaxation that follows a temperature quench an exchange of electrons between these two groups takes place and the non-equilibrium excess of diffusive electrons initially present decreases logarithmically with time as the system relaxes. This bimodal dynamical heterogeneity persists at higher temperatures when time translation invariance is restored and electron diffusion is normal. The occupancy of the two dynamical modes is then stationary and its temperature dependence reflects a crossover between a low-temperature regime with a high concentration of electrons forming fluctuating dipoles and a high-temperature regime in which the concentration of diffusive electrons is high.Comment: 10 pages, 9 figure

Variational methods, multisymplectic geometry and continuum mechanics

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper

A stochastic perturbation of inviscid flows

We prove existence and regularity of the stochastic flows used in the stochastic Lagrangian formulation of the incompressible Navier-Stokes equations (with periodic boundary conditions), and consequently obtain a \holderspace{k}{\alpha} local existence result for the Navier-Stokes equations. Our estimates are independent of viscosity, allowing us to consider the inviscid limit. We show that as $\nu \to 0$, solutions of the stochastic Lagrangian formulation (with periodic boundary conditions) converge to solutions of the Euler equations at the rate of $O(\sqrt{\nu t})$.Comment: 13 pages, no figures

Formation and evolution of density singularities in hydrodynamics of inelastic gases

We use ideal hydrodynamics to investigate clustering in a gas of inelastically colliding spheres. The hydrodynamic equations exhibit a new type of finite-time density blowup, where the gas pressure remains finite. The density blowups signal formation of close-packed clusters. The blowup dynamics are universal and describable by exact analytic solutions continuable beyond the blowup time. These solutions show that dilute hydrodynamic equations yield a powerful effective description of a granular gas flow with close-packed clusters, described as finite-mass point-like singularities of the density. This description is similar in spirit to the description of shocks in ordinary ideal gas dynamics.Comment: 4 pages, 3 figures, final versio

Interaction of a vortex ring with the free surface of ideal fluid

The interaction of a small vortex ring with the free surface of a perfect fluid is considered. In the frame of the point ring approximation the asymptotic expression for the Fourier-components of radiated surface waves is obtained in the case when the vortex ring comes from infinity and has both horizontal and vertical components of the velocity. The non-conservative corrections to the equations of motion of the ring, due to Cherenkov radiation, are derived.Comment: LaTeX, 15 pages, 1 eps figur

Dimension reduction for systems with slow relaxation

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize `optimal' model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof