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

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

This text is a modern treatment of the theory of solids. The core of the book deals with the physics of electron and phonon states in crystals and how they determine the structure and properties of the solid. The discussion uses density functional theory as a starting point and covers electronic and optical phenomena, magnetism and superconductivity. There is also an extensive treatment of defects in solids, including point defects, dislocations, surfaces and interfaces. A number of modern topics where the theory of solids applies are also explored, including quasicrystals, amorphous solids, polymers, metal and semiconductor clusters, carbon nanotubes and biological macromolecules. Numerous examples are presented in detail and each chapter is accompanied by problems and suggested further readings. An extensive set of appendices provides all the necessary background for deriving all the results discussed in the main body of the text.

Atomic and Electronic Structure of Solids

Amp: A modular approach to machine learning in atomistic simulations☆

Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the field of electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

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The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

https://arxiv.org/pdf/physics/0607081.pdf

New approach for the electronic energies of the hydrogen molecular ion

Herein, we present a sequel to earlier work on a generalization of the Lambert W function. In particular, we examine series expansions of the generalized version providing computational means for evaluating this function in various regimes and further confirming the notion that this generalization is a natural nextension of the standard Lambert W function.

Asymptotic series of generalized Lambert W function

Institut für Physikalische Chemie, RWTH Aachen University, 52056 Aachen, Germany 2 Centre for Experimental and Constructive Mathematics (CECM), Simon-Fraser University, Burnaby, BC V5A 1S6 Canada 3 Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, Germany 4 College of Physics and Optoelectronics, Taiyuan University of Technology, Shanxi 030024, China tcscott@gmail.com

Numerics of the generalized Lambert W function

edited by Johannes Grotendorst, Godehard Sutmann, Gerhard Gompper, Dominik Marx The focus of this Winter School was on hierarchical methods for dynamical problems having primarily in mind systems described in terms of many atoms or molecules. One end of relevant time scales certainly is nonadiabatic quantum dynamics methods, which operate on the subfemtosecond time scale but influence dynamical events that are orders of magnitude slower. Examples for such phenomena might be photoinduced switching of individual molecules, which results into large-amplitude relaxation in liquids or photodriven phase transitions of liquid crystals. On the other end of the relevant time scales methods are important to investigate and understand the non-equilibrium dynamics of complex fluids, with typical time scales in the range from microseconds to seconds. Examples are the flow of polymer solutions, or the flow of blood through microvessels.

Hierarchical Methods for Dynamics in Complex Molecular Systems

edited by Godehard Sutmann, Johannes Grotendorst, Gerhard Gompper, Dominik Marx The majority of chemical reactions including many important industrial processes and virtually all biological activities take place within a liquid environment. Solvents, of which water is certainly the most important, are able to “solvate” molecules, thereby transferring these as “solutes” into the liquid state. Transport processes and solute-solute interactions in the solvent are then supporting structure formation, selforganization or chemical reactions. Solvents are not only able to provide a liquid phase for simple chemical reagents and the much more complex proteins; they have the additional ability to wet extended surfaces such as lipid membranes or metal electrodes, thereby creating interfaces. An in-depth understanding of solvation at a fundamental level of chemistry, physics and engineering is essential to enable major advances in key technologies for environmentally friendly technologies, e.g. to reduce pollution, to increase energy efficiency or to prevent corrosion to name but a few challenges to our modern day society. In biophysics and life sciences, water is the most important and dominant solvent, providing the basic environment for the complexity of life. Therefore, an understanding of solvation is crucial to unravel biological function in a comprehensive way.

Johannes Grotendorst

Papers

New approach for the electronic energies of the hydrogen molecular ion

Asymptotic series of generalized Lambert W function

Numerics of the generalized Lambert W function

Hierarchical Methods for Dynamics in Complex Molecular Systems

Computational Trends in Solvation and Transport in Liquids - Lecture Notes