A Variational Perspective on Accelerated Methods in Optimization

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods correspond to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.Comment: 38 pages. Subsumes an earlier working draft arXiv:1509.0361

The effect of equine hyperimmune sera on TNF alpha activity in a L929 cell bioassay

In this pilot study we examined the effect of sera from a proprietary hyperimmune plasma product (Equiplas) on the activity of TNF: in an in vitro L929 cell bioassay. In brief, we report observations from 2 accessions of sera. Accession 1 describes the antiTNF: activity of 3 hyperimmune sera and an untreated serum sample that were provided blind to the study. Accession 2 reports a comparison of antiTNFalpha activity found in 3 paired hyperimmune sera collected following a multiple endotoxin vaccination regimen

Critical velocity ionisation in substellar atmospheres

The observation of radio, X-ray and Hα emission from substellar objects indicates the presence of plasma regions and associated high-energy processes in their surrounding envelopes. This paper numerically simulates and characterises Critical Velocity Ionisation, a potential ionisation process, that can efficiently generate plasma as a result of neutral gas flows interacting with seed magnetized plasmas. By coupling a Gas-MHD interactions code (to simulate the ionisation mechanism) with a substellar global circulation model (to provide the required gas flows) we quantify the spatial extent of the resulting plasma regions, their degree of ionisation and their lifetime for a typical substellar atmosphere. It is found that the typical average ionisation fraction reached at equilibrium (where the ionisation and recombination rates are equal and opposite) ranges from 10-5 to 10-8, at pressures between 10-1 and 10-3 bar, with a trend of increasing ionisation fraction with decreasing atmospheric pressure. The ionisation fractions reached as a result of Critical Velocity Ionisation are sufficient to allow magnetic fields to couple to gas flows in the atmosphere

Josephson effect in ballistic graphene

We solve the Dirac-Bogoliubov-De-Gennes equation in an impurity-free superconductor-normal-superconductor (SNS) junction, to determine the maximal supercurrent that can flow through an undoped strip of graphene with heavily doped superconducting electrodes. The result is determined by the superconducting gap and by the aspect ratio of the junction (length L, small relative to the width W and to the superconducting coherence length). Moving away from the Dirac point of zero doping, we recover the usual ballistic result in which the Fermi wave length takes over from L. The product of critical current and normal-state resistance retains its universal value (up to a numerical prefactor) on approaching the Dirac point.Comment: 4 pages, 2 figure