Abstract

We present a multi-scale modeling and computational scheme for optical- mechanical responses of nano-structures. The multi-physical nature of the problem is a result of the interaction between the electromagnetic (EM) field, the molecular motion, and the electronic excitation. To balance accuracy and complexity, we adopt the semi-classical approach that the EM field is described classically by the Maxwell equations, and the charged particles follow the Schrödinger equations quantum mechanically. To overcome the numerical challenge of solving the high dimensional multi-component many- body Schrödinger equations, we further simplify the model with the Ehrenfest molecular dynamics to determine the motion of the nuclei, and use the Time- Dependent Current Density Functional Theory (TD-CDFT) to calculate the excitation of the electrons. This leads to a system of coupled equations that computes the electromagnetic field, the nuclear positions, and the electronic current and charge densities simultaneously. In the regime of linear responses, the resonant frequencies initiating the out-of-equilibrium optical-mechanical responses can be formulated as an eigenvalue problem. A self-consistent multi-scale method is designed to deal with the well separated space scales. The isomerization of Azobenzene is presented as a numerical example.

In this talk, I will begin with introducing the method of
pseudo-holomorphic curves (which are defined by Cauchy-Riemnn type elliptic
systems) in the study of symplectic and contact topology. Then I will focus
on discussing the one studied by Yong-Geun Oh and myself recently, including
its potential, drawbacks and possible improvements towards the goal of a
better understanding in contact topology. (The similar elliptic system named
the generalized pseudo-holomorphic curves in symplectizations was introduced
by Hofer and studied by Abbas-Cieliebak-Hofer, Abbas in the proposal of
proving the Weinstein conjecture for dimension three.)

Non-cooperative games are closely associated with multi-agent optimization wherein a large number of selfish players compete non-cooperatively to optimize their individual objectives under various constraints. Unlike centralized algorithms that require a certain system mechanism to coordinate the players' actions, distributed algorithms have the advantage that the players, either individually or in subgroups, can each make their best responses without full information of their rivals' actions. These distributed algorithms by nature are particularly suited for solving huge size games where the large number of players in the game makes the coordination of the players almost impossible. The distributed algorithms are distinguished by several features: parallel versus sequential implementations, scheduled versus randomized player selections, synchronized versus asynchronous transfer of information, and individual versus multiple player updates. There are two general approaches to establish the convergence of distributed algorithms: contraction versus potential based, each requiring different properties of the players' objective functions. We present convergence results based on these two approaches and discuss randomized extensions of the algorithms that require less coordination and hence are more suitable for big data problems.