Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

Solving existence problems via -contractions

In this paper, we study the following Hamiltonian elliptic system with gradient term and inverse square potential \begin{document}$ \left\{ \begin{array}{ll}-\Delta u +\vec{b}(x)\cdot \nabla u +V(x)u-\frac{\mu}{|x|^{2}}v=H_{v}(x,u,v)\\-\Delta v -\vec{b}(x)\cdot \nabla v +V(x)v-\frac{\mu}{|x|^{2}}u=H_{u}(x,u,v)\\\end{array} \right.$\end{document} for $x\in\mathbb{R}^{N}$, where $N\geq3$, $\mu\in\mathbb{R}$, and $V(x)$, $\vec{b}(x)$ and $H(x, u, v)$ are $1$-periodic in $x$. Under suitable conditions, we prove that the system possesses a ground state solution via variational methods for sufficiently small $\mu\geq0$. Moreover, we provide the comparison of the energy of ground state solutions for the case $\mu>0$ and $\mu=0$. Finally, we also give the convergence property of ground state solutions as $\mu\to0^+$.

Ground state solutions for Hamiltonian elliptic system with inverse square potential

We find solutions $${E : \Omega \to \mathbb{R}^3}$$
 of the problem

 $$\left\{\begin{aligned}&\nabla \times(\nabla \times E) + \lambda E = \partial_E F(x, E) &&\quad \text{in }\quad \Omega \\& \nu \times E = 0 &&\quad \text{on }\quad \partial \Omega \end{aligned} \right.$$
 on a simply connected, smooth, bounded domain $${\Omega \subset \mathbb{R}^3}$$
 with connected boundary and exterior normal $${\nu : \partial \Omega \to \mathbb{R}^3}$$
 . Here $${\nabla \times}$$
 denotes the curl operator in $${\mathbb{R}^3}$$
 , the nonlinearity $${F : \Omega \times \mathbb{R}^3 \to \mathbb{R}}$$
 is superquadratic and subcritical in E. The model nonlinearity is of the form $${F(x, E)=\Gamma(x) |E|^p}$$
 for $${\Gamma \in L^\infty(\Omega)}$$
 positive, some 2 < p < 6. It need not be radial nor even in the E-variable. The problem comes from the time-harmonic Maxwell equations, the boundary conditions are those for Ω surrounded by a perfect conductor.

Ground and Bound State Solutions of Semilinear Time-Harmonic Maxwell Equations in a Bounded Domain

We investigate the existence of solutions ${E:\mathbb{R}^3 \to \mathbb{R}^3}$ of the time-harmonic semilinear Maxwell equation 

$$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$

where ${V:\mathbb{R}^3 \to \mathbb{R}}$, ${V(x) \leqq 0}$ almost everywhere on ${\mathbb{R}^3}$, ${\nabla \times}$ denotes the curl operator in ${\mathbb{R}^3}$ and ${F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}$ is a nonlinear function in E. In particular we find a ground state solution provided that suitable growth conditions on F are imposed and the ${L^{3/2}}$ -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and ${V(x) = -\mu\omega^2 \varepsilon(x)}$ where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field ${\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}$ and ${\varepsilon}$ is the linear part of the permittivity in an inhomogeneous medium.

/pdf/ground-states-of-time-harmonic-semilinear-maxwell-equations-4jnhsrjh8k.pdf

Ground States of Time-Harmonic Semilinear Maxwell Equations in ${\mathbb{R}^3}$ with Vanishing Permittivity

Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials

We are concerned with a system of coupled Schrodinger equations $$-\Delta u_i + V_i(x)u_i = \partial_{u_i}F(x,u)\hbox{ on }\mathbb{R}^N,\,i=1,2,...,K,$$ where $F$ and $V_i$ are periodic in $x$ and $0\notin \sigma(-\Delta+V_i)$ for $i=1,2,...,K$, where $\sigma(-\Delta+V_i)$ stands for the spectrum of the Schrodinger operator $-\Delta+V_i$. We impose general assumptions on the nonlinearity $F$ with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. Our approach is based on a new linking-type result involving the Nehari-Pankov manifold.

Ground states of a system of nonlinear Schr\"odinger equations with periodic potentials

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} 
-\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\ \int_{\mathbb{R}^N} |u|^2 \, dx = \rho > 0, \end{cases} where $\rho$ is prescribed and $(\lambda, u) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints is demonstrated, which allows to provide general growth assumptions imposed on $g$. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at $0$.

Jarosław Mederski

Papers

Ground and Bound State Solutions of Semilinear Time-Harmonic Maxwell Equations in a Bounded Domain

Ground States of Time-Harmonic Semilinear Maxwell Equations in \({\mathbb{R}^3}\) with Vanishing Permittivity

Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials

Ground states of a system of nonlinear Schr\"odinger equations with periodic potentials

Normalized ground states of the nonlinear Schr\"{o}dinger equation with at least mass critical growth