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

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

In the field of machine learning, semi-supervised learning (SSL) occupies the middle ground, between supervised learning (in which all training examples are labeled) and unsupervised learning (in which no label data are given). Interest in SSL has increased in recent years, particularly because of application domains in which unlabeled data are plentiful, such as images, text, and bioinformatics. This first comprehensive overview of SSL presents state-of-the-art algorithms, a taxonomy of the field, selected applications, benchmark experiments, and perspectives on ongoing and future research. Semi-Supervised Learning first presents the key assumptions and ideas underlying the field: smoothness, cluster or low-density separation, manifold structure, and transduction. The core of the book is the presentation of SSL methods, organized according to algorithmic strategies. After an examination of generative models, the book describes algorithms that implement the low-density separation assumption, graph-based methods, and algorithms that perform two-step learning. The book then discusses SSL applications and offers guidelines for SSL practitioners by analyzing the results of extensive benchmark experiments. Finally, the book looks at interesting directions for SSL research. The book closes with a discussion of the relationship between semi-supervised learning and transduction. Adaptive Computation and Machine Learning series

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Semi-Supervised Learning

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Reconstructing Volume Tracking

Introduction Fernando Grinstein, Len Margolin and William Rider Part I. Motivation: 1. Historical introduction Jay Boris 2. ILES for turbulent flows: a rationale Fernando Grinstein, Len Margolin and William Rider Part II. Capturing Physics with Numerics: 3. Subgrid scale modeling: issues and approaches Pierre Sagaut 4. Numerics for ILES 4a. Limiting algorithms Dimitris Drikakis, Marco Hahn, Fernando Grinstein, Carl DeVore, Christer Fureby, Mattias Liefvendahl and David Youngs 4b. Piecewise parabolic method Paul Woodward 4c. Lagrangean remap method David Youngs 4d. MPDATA Piotr Smolarkiewicz and Len Margolin 4e. Vorticity confinement John Steinhoff, Nicholas Lynn and Lesong Wang 5. Numerical regularization Len Margolin and William Rider 6. Approximate deconvolution Nikolaus Adams and J. A. Domaradzki Part III. Verification and Validation: 7. Homogeneous turbulence David Porter and Paul Woodward 8. Vortex dynamics and transition in free shear flows Fernando Grinstein 9. Symmetry bifurcation and instabilities Dimitris Drikakis 10. Incompressible wall bounded flows Christer Fureby, Mattias Liefvendahl, Urban Svennberg, Leif Persson and Tobias Persson 11. Compressible turbulent shear flows Christer Fureby and Doyle Knight 12. Studies based on vorticity confinement John Steinhoff, Nicholas Lynn, Wenren Yonghu, Meng Fan, Lesong Wang and Bill Dietz 13. Rayleigh-Taylor and Richtmyer-Meshkov mixing David Youngs Part IV. Frontier Flows: 14. Studies of geophysics Piotr Smolarkiewicz and Len Margolin 15. Studies of astrophysics David Porter and Paul Woodward 16. Complex engineering turbulent flows Niklas Alin, Magnus Berglund, Christer Fureby, Eric Lillberg and Urban Svennberg 17. Large scale urban simulations Gopal Patnaik, Fernando Grinstein, Jay Boris, Ted Young and Oskar Parmhed 18. Outlook and open research issues Fernando Grinstein, Len Margolin and William Rider.

Implicit large eddy simulation : computing turbulent fluid dynamics

A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows

Solution algorithms are presented for tracking interfaces with piecewise linear (PLIC) volume-of-fluid (VOF) methods on fixed (Eulerian) two-dimensional (2-D) structured and three-dimensional (3-D) structured and unstructured grids. We review the theory of volume tracking methods, derive appropriate volume evolution equations, identify and present solutions to the basic geometric functions needed for interface reconstruction and volume fluxing, and provide detailed algorithm templates for modern 2-D and 3-D PLIC VOF interface tracking methods. We discuss some key outstanding issues for PLIC VOF methods, namely the method used for time integration of fluid volumes (operator splitting, unsplit, Runge-Kutta, etc.) and the estimation of interface normals. We also present our latest developments in the continuum surface force (CSF) model for surface tension, namely extension to 3-D and variable surface tension effects. We identify and focus on key outstanding CSF model issues that become especially critical on fine meshes with high density ratio interfacial flows, namely the surface delta function approximation, the estimation of interfacial curvature, and the continuum surface force scaling and/or smoothing model. Numerical results in two and three dimensions are used to illustrate the properties of these methods.

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Volume tracking of interfaces having surface tension in two and three dimensions

We present a rationale for the success of nonoscillatory finite volume (NFV) difference schemes in modeling turbulent flows without need of subgrid scale models. Our exposition focuses on certain truncation terms that appear in the modified equation of one particular NFV scheme, MPDATA. We demonstrate that these truncation terms have physical justification, representing the modifications to the governing equations that arise when one considers the motion of finite volumes of fluid over finite intervals of time.

William J. Rider

Papers

Reconstructing Volume Tracking

Implicit large eddy simulation : computing turbulent fluid dynamics

A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows

Volume tracking of interfaces having surface tension in two and three dimensions

A rationale for implicit turbulence modelling