Consensus on homogeneous manifolds

The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative state according to an interconnection graph. The paper first formalizes the consensus problem for synchronization (i.e. maximizing the consensus) and balancing (i.e. minimizing the consensus); it thereby introduces the induced arithmetic mean, an easily computable mean position on CCH manifolds. Then it proposes and analyzes various consensus algorithms on manifolds: natural gradient algorithms which reach local consensus equilibria; an adaptation using auxiliary variables for almost-global synchronization or balancing; and a stochastic gossip setting for global synchronization. It closes by investigating the dependence of synchronization properties on the attraction function between interacting agents on the circle. The theory is also illustrated on SO(n) and on the Grassmann manifolds. ©2009 IEEE.

Consensus optimization on manifolds

The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO (n) and the Grassmann manifold Grass (p, n) are treated as original examples. A link is also drawn with the many existing results on the circle. © 2009 Society for Industrial and Applied Mathematics.

Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in ℝn. In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.

Optimization Algorithms on Matrix Manifolds

This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms.

Fano resonance in Raman scattering of graphene

Fano resonances and their strong doping dependence are observed in Raman scattering of single-layer graphene (SLG). As the Fermi level is varied by a back-gate bias, the Raman G band of SLG exhibits an asymmetric line shape near the charge neutrality point as a manifestation of a Fano resonance, whereas the line shape is symmetric when the graphene sample is electron or hole doped. However, the G band of bilayer graphene (BLG) does not exhibit any Fano resonance regardless of doping. The observed Fano resonance can be interpreted as interferences between the phonon and excitonic many-body spectra in SLG. The absence of a Fano resonance in the Raman G band of BLG can be explained in the same framework since excitonic interactions are not expected in BLG. © 2013 Elsevier Ltd. All rights reserved.

Optimization on manifolds: Methods and applications

This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however,we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton's method in some detail as well as referencing the more important of the other approaches.There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail. © 2010 Springer -Verlag Berlin Heidelberg.

Manopt, a matlab toolbox for optimization on manifolds

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design effcient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. By dealing internally with most of the differential geometry, the package aims particularly at lowering the entrance barrier. © 2014 Nicolas Boumal.

ac gate-induced Fano resonance and peak splitting of conductance in a quantum dot

We have investigated the conductance of a quantum dot system suffering an anti-symmetric ac gate voltage which induces the transition between dot levels in the linear regime at zero temperature in the rotating wave approximation. Interesting Fano resonances appear on one side of the displaced resonant tunnelling peaks for the nonresonant case or the peak splitting for the resonant case. The line shape of conductance (vs Fermi energy) near each level of the quantum dot can be decomposed into two profiles: a Breit-Wigner peak and a Fano profile, or a Breit-Wigner peak and a dip in both cases.

Photon-assisted Fano resonance and corresponding shot noise in a quantum dot

We have studied the Fano resonance in photon-assisted transport through a quantum dot. Both the coherent current and the spectral density of shot noise have been calculated. It is predicted that the shape of the Fano profile will also appear in satellite peaks. It is found that the variations of Fano profiles with the strengths of nonresonant transmissions are not synchronous in absorption and emission sidebands. The effect of interference on photon-assisted pumped current has also been investigated. We further predict the current and spectral density of shot noise as a periodic function of the phase, which exhibits an intrinsic property of resonant and nonresonant channels in the structures.

Mesoscopic Fano effect modulated by Rashba spin-orbit coupling and external magnetic field

By employing non-equilibrium Green's function method, the mesoscopic Fano effect modulated by Rashba spin-orbit (SO) coupling and external magnetic field has been elucidated for electron transport through a hybrid system composed of a quantum dot (QD) and an Aharonov-Bohm (AB) ring. The results show that the orientation of the Fano line shape is modulated by the Rashba spin-orbit interaction k(R)L variation, which reveals that the Fano parameter q will be extended to a complex number, although the system maintains time-reversal symmetry (TRS) under the Rashba SO interaction. Furthermore, it is shown that the modulation of the external magnetic field, which is applied not only inside the frame, but also on the QD, leads to the Fano resonance split due to Zeeman effect, which indicates that the hybrid is an ideal candidate for the spin readout device. (C) 2007 Elsevier B.V All rights reserved.

Tunable Fano resonance in photonic crystal slabs

A three dimensional analysis of a special class of anisotropic materials is presented. We introduce an extension of the Scattering Matrix Method (SMM) to investigate the behavior of anisotropic Photonic Crystal Slabs (PhCS) subject to external radiation. We show how the Fano effect can play a fundamental role in the realization of tunable optical devices. Moreover, we show how to utilize electron injection, electric field and temperature as parameters to control the Fano resonance shift in both isotropic and anisotropic materials as Si and Potassium Titanium Oxide Phosphate (KTP). We will see that because Fano modes are sensitive and controllable, a broad range of applications can be considered. (c) 2006 Optical Society of America