l1 solution of linear inequalities

The numerical solution of a possible inconsistent system oflinear inequalities in the l₁ sense is considered. The non-differentiablel₁ norm minimization problem is approximated by a piecewisequadratic Huber smooth function. A continuation algorithm isdesigned to find an l₁ solution of the inequality system. Inthe case where the linear inequality system is consistent, asolution is obtained by solving any smoothed problem. Otherwise,the algorithm is shown to terminate in a finite number of iterations.We also consider an alternative smoothing scheme which sharessimilar properties with the first one, but results in an improvedcomputational performance of the continuation algorithm on inconsistentsystems. Numerical experiments are conducted to test the efficiencyof the algorithm.     

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted \(L^2\) space, and where dependencies on the mean-free path \(\tau \) and the Debye length \(\delta \) are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions \(\tau \rightarrow \infty \) to the strongly collisional regime \(\tau \rightarrow 0\). As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the \(\tau \)-dependent constraint on \(\delta \) ensuring exponential decay with explicit \(\tau \)-dependent rates towards the stationary solution. In the strongly collisional limit \(\tau \rightarrow 0\), we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a \(L^2\) space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.     

Pulse compression in a longitudinally inhomogeneous fiber

A single-mode fiber with a linear or sinusoidal variation in the group-velocity dispersion is fabricated. The nonreciprocal effects and pulse compression due to the longitudinal oscillations of the fiber dispersion are experimentally demonstrated. The periodic modulation of the dispersion can be used to control the pulse dynamics.