Stochastic Block Mirror Descent Methods for Nonsmooth and Stochastic Optimization

In this paper, we present a new stochastic algorithm, namely the stochastic block mirror descent (SBMD) method for solving large-scale nonsmooth and stochastic optimization problems. The basic idea of this algorithm is to incorporate the block-coordinate decomposition and an incremental block averaging scheme into the classic (stochastic) mirror-descent method, in order to significantly reduce the cost per iteration of the latter algorithm. We establish the rate of convergence of the SBMD method along with its associated large-deviation results for solving general nonsmooth and stochastic optimization problems. We also introduce different variants of this method and establish their rate of convergence for solving strongly convex, smooth, and composite optimization problems, as well as certain nonconvex optimization problems. To the best of our knowledge, all these developments related to the SBMD methods are new in the stochastic optimization literature. Moreover, some of our results also seem to be new for block coordinate descent methods for deterministic optimization

Linearly Convergent First-Order Algorithms for Semi-definite Programming

In this paper, we consider two formulations for Linear Matrix Inequalities (LMIs) under Slater type constraint qualification assumption, namely, SDP smooth and non-smooth formulations. We also propose two first-order linearly convergent algorithms for solving these formulations. Moreover, we introduce a bundle-level method which converges linearly uniformly for both smooth and non-smooth problems and does not require any smoothness information. The convergence properties of these algorithms are also discussed. Finally, we consider a special case of LMIs, linear system of inequalities, and show that a linearly convergent algorithm can be obtained under a weaker assumption

Fiber-coupled erbium microlasers on a chip

An erbium-doped, toroid-shaped microlaser fabricated on a silicon chip is described and characterized. Erbium-doped sol-gel films are applied to the surface of a silica toroidal microresonator to create the microcavity lasers. Highly confined whispering gallery modes make possible single-mode and ultralow threshold microlasers

Calculation of wing response to gusts and blast waves with vortex lift effect

A numerical study of the response of aircraft wings to atmospheric gusts and to nuclear explosions when flying at subsonic speeds is presented. The method is based upon unsteady quasi-vortex-lattice method, unsteady suction analogy, and Pade approximate. The calculated results, showing vortex lag effect, yield reasonable agreement with experimental data for incremental lift on wings in gust penetration and due to nuclear blast waves

Calculation of wing response to gusts and blast waves with vortex lift effect

A numerical study of the response of aircraft wings to atmospheric gusts and to nuclear explosions when flying at subsonic speeds is presented. The method is based upon unsteady quasi-vortex lattice method, unsteady suction analogy and Pade approximant. The calculated results, showing vortex lag effect, yield reasonable agreement with experimental data for incremental lift on wings in gust penetration and due to nuclear blast waves

Generalized quantization condition in topological insulator

The topological magnetoelectric effect (TME) is the fundamental quantization effect for topological insulators in units of the fine structure constant $\alpha$. In [Phys. Rev. Lett. 105, 166803(2010)], a topological quantization condition of the TME is given under orthogonal incidence of the optical beam, in which the wave length of the light or the thickness of the TI film must be tuned to some commensurate values. This fine tuning is difficult to realize experimentally. In this article, we give manifestly $SL(2,\mathbb{Z})$ covariant expressions for Kerr and Faraday angles at oblique incidence at a topological insulator thick film. We obtain a generalized quantization condition independent of material details, and propose a more easily realizable optical experiment, in which only the incidence angle is tuned, to directly measure the topological quantization associated with the TME.Comment: 3 figure

Noisy Classical Field Theories with Two Coupled Fields: Dependence of Escape Rates on Relative Field Stiffnesses

Exit times for stochastic Ginzburg-Landau classical field theories with two or more coupled classical fields depend on the interval length on which the fields are defined, the potential in which the fields deterministically evolve, and the relative stiffness of the fields themselves. The latter is of particular importance in that physical applications will generally require different relative stiffnesses, but the effect of varying field stiffnesses has not heretofore been studied. In this paper, we explore the complete phase diagram of escape times as they depend on the various problem parameters. In addition to finding a transition in escape rates as the relative stiffness varies, we also observe a critical slowing down of the string method algorithm as criticality is approached.Comment: 16 pages, 10 figure