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

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

In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through creation of new classes for linear and nonlinear solvers, enhancements to the vector class, and the creation of modern Fortran interfaces that leverage interoperability features of the Fortran 2003 standard. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. However, these changes now allow advanced users to create highly customized solvers that exploit their problem structure, enabling SUNDIALS use on extreme-scale, heterogeneous computational architectures.

Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers.

Finite element methods for Maxwell's equations.

By L I Sedov London: Cleaver-Hume Press Ltd Pp xvi + 363 Price 105s This is really two separate books within the same pair of covers First of all Chapters 1-3, some 145 pages, are devoted to the discussion of similarity and dimensional, methods and their application to a variety of problems in mechanics and fluid mechanics

http://iopscience.iop.org/article/10.1088/0031-9112/11/6/008/pdf

Similarity and Dimensional Methods in Mechanics

We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integration techniques is limited to spatial parallelism. However, current trends in computer architectures are leading toward systems with more, but not faster, processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieving parallelism in time is with multigrid, but extending classical multigrid methods for elliptic operators to this setting is not straightforward. In this paper, we present a nonintrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). We demonstrate optimality of our multigrid-reduction-in-time algorithm (MGRIT) for solving diffusion equations in two and three space dimensions in numerical ex...

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Parallel Time Integration with Multigrid

The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310] and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-ord...

High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics

MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM’s goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library.

MFEM: a modular finite element methods library

This paper investigates the properties of smoothers in the context of algebraic multigrid (AMG) running on parallel computers with potentially millions of processors. The development of multigrid smoothers in this case is challenging, because some of the best relaxation schemes, such as the Gauss-Seidel (GS) algorithm, are inherently sequential. Based on the sharp two-grid multigrid theory from [R. D. Falgout and P. S. Vassilevski, SIAM J. Numer. Anal., 42 (2004), pp. 1669-1693] and [R. D. Falgout, P. S. Vassilevski, and L. T. Zikatanov, Numer. Linear Algebra Appl., 12 (2005), pp. 471-494] we characterize the smoothing properties of a number of practical candidates for parallel smoothers, including several $C$-$F$, polynomial, and hybrid schemes. We show, in particular, that the popular hybrid GS algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications, provided that the problem size per processor is large enough. This is encouraging news for the scalability of AMG on ultraparallel computers. We also introduce the more robust $\ell_1$ smoothers, which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604-623].

/pdf/multigrid-smoothers-for-ultraparallel-computing-2zk7c7gm2s.pdf

Multigrid Smoothers for Ultraparallel Computing

In this paper we review a number of auxiliary space based preconditioners for the second order deflnite and semi-deflnite Maxwell problems discretized with the lowest order Nedelec flnite elements. We discuss the parallel implementation of the most promising of these methods, the ones derived from the recent Hiptmair-Xu (HX) auxiliary space decomposition (Hiptmair and Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509). An extensive set of numerical experiments demonstrate the scalability of our implementation on large-scale H(curl) problems.

Tzanio V. Kolev

Papers

Parallel Time Integration with Multigrid

High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics

MFEM: a modular finite element methods library

Multigrid Smoothers for Ultraparallel Computing

PARALLEL AUXILIARY SPACE AMG FOR H(curl) PROBLEMS