Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration

Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.Comment: 5 Figure

Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators

We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems

Stochastic Quasi-Fej\'er Block-Coordinate Fixed Point Iterations with Random Sweeping

This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis relies on a notion of stochastic quasi-Fej\'er monotonicity, which is thoroughly investigated. The iterative methods under consideration feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. Algorithms using quasinonexpansive operators or compositions of averaged nonexpansive operators are constructed, and weak and strong convergence results are established for the sequences they generate. As a by-product, novel block-coordinate operator splitting methods are obtained for solving structured monotone inclusion and convex minimization problems. In particular, the proposed framework leads to random block-coordinate versions of the Douglas-Rachford and forward-backward algorithms and of some of their variants. In the standard case of $m=1$ block, our results remain new as they incorporate stochastic perturbations

Free-energy model for fluid helium at high density

We present a semi-analytical free-energy model aimed at characterizing the thermodynamic properties of dense fluid helium, from the low-density atomic phase to the high-density fully ionized regime. The model is based on a free-energy minimization method and includes various different contributions representative of the correlations between atomic and ionic species and electrons. This model allows the computation of the thermodynamic properties of dense helium over an extended range of density and temperature and leads to the computation of the phase diagram of dense fluid helium, with its various temperature and pressure ionization contours. One of the predictions of the model is that pressure ionization occurs abruptly at \rho \simgr 10 g cm$^{-3}$, {\it i.e.} P\simgr 20 Mbar, from atomic helium He to fully ionized helium He$^{2+}$, or at least to a strongly ionized state, without He$^{+}$ stage, except at high enough temperature for temperature ionization to become dominant. These predictions and this phase diagram provide a guide for future dynamical experiments or numerical first-principle calculations aimed at studying the properties of helium at very high density, in particular its metallization. Indeed, the characterization of the helium phase diagram bears important consequences for the thermodynamic, magnetic and transport properties of cool and dense astrophysical objects, among which the solar and the numerous recently discovered extrasolar giant planets.Comment: Accepted for publication in Phys. Rev.

Observability and Synchronization of Neuron Models

Observability is the property that enables to distinguish two different locations in $n$-dimensional state space from a reduced number of measured variables, usually just one. In high-dimensional systems it is therefore important to make sure that the variable recorded to perform the analysis conveys good observability of the system dynamics. In the case of networks composed of neuron models, the observability of the network depends nontrivially on the observability of the node dynamics and on the topology of the network. The aim of this paper is twofold. First, a study of observability is conducted using four well-known neuron models by computing three different observability coefficients. This not only clarifies observability properties of the models but also shows the limitations of applicability of each type of coefficients in the context of such models. Second, a multivariate singular spectrum analysis (M-SSA) is performed to detect phase synchronization in networks composed by neuron models. This tool, to the best of the authors' knowledge has not been used in the context of networks of neuron models. It is shown that it is possible to detect phase synchronization i)~without having to measure all the state variables, but only one from each node, and ii)~without having to estimate the phase

Terahertz Magnetoplasmon Energy Concentration and Splitting in Graphene PN Junctions

Terahertz plasmons and magnetoplasmons propagating along electrically and chemically doped graphene p-n junctions are investigated. It is shown that such junctions support non-reciprocal magnetoplasmonic modes which get concentrated at the middle of the junction in one direction and split away from the middle of the junction in the other direction under the application of an external static magnetic field. This phenomenon follows from the combined effects of circular birefringence and carrier density non-uniformity. It can be exploited for the realization of plasmonic isolators.Comment: 6 Pages, 10 figure

Effect of connecting wires on the decoherence due to electron-electron interaction in a metallic ring

We consider the weak localization in a ring connected to reservoirs through leads of finite length and submitted to a magnetic field. The effect of decoherence due to electron-electron interaction on the harmonics of AAS oscillations is studied, and more specifically the effect of the leads. Two results are obtained for short and long leads regimes. The scale at which the crossover occurs is discussed. The long leads regime is shown to be more realistic experimentally.Comment: LaTeX, 4 pages, 4 eps figure