There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

https://www.sandia.gov/~sjplimp/papers/jcompphys95.pdf

Fast parallel algorithms for short-range molecular dynamics

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.

An N⋅log(N) method for evaluating electrostatic energies and forces of large periodic systems is presented. The method is based on interpolation of the reciprocal space Ewald sums and evaluation of the resulting convolutions using fast Fourier transforms. Timings and accuracies are presented for three large crystalline ionic systems.

Particle mesh Ewald: An N⋅log(N) method for Ewald sums in large systems

The previously developed particle mesh Ewald method is reformulated in terms of efficient B‐spline interpolation of the structure factors This reformulation allows a natural extension of the method to potentials of the form 1/rp with p≥1 Furthermore, efficient calculation of the virial tensor follows Use of B‐splines in place of Lagrange interpolation leads to analytic gradients as well as a significant improvement in the accuracy We demonstrate that arbitrary accuracy can be achieved, independent of system size N, at a cost that scales as N log(N) For biomolecular systems with many thousands of atoms this method permits the use of Ewald summation at a computational cost comparable to that of a simple truncation method of 10 A or less

A smooth particle mesh Ewald method

A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy

A fast multipole method for the three-dimensional Stokes equations

Spectral Approximation of the Free-Space Heat Kernel

A steering column for absorbing impact energy comprising a steering shaft mounted for rotation in a vehicle by an upper bearing. The steering shaft can be turned by a steering wheel operatively connected to the upper end of the shaft by first energy absorbing device which deforms in a controlled manner when impacted. The steering shaft also has an integral helical portion between the upper and lower ends which collapses under a predetermined load if the vehicle is impacted so that the steering wheel remains in position and so that impact energy is absorbed.

/pdf/the-numerical-solution-of-the-n-body-problem-2f0hap3lml.pdf

The numerical solution of the N-body problem

We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the Schrodinger equation in one dimension. The exact nonreflecting boundary condition contains a nonlocal term which is a convolution integral in time, with a kernel proportional to 1 √t. The key observation is that this integral can be split into two parts: a local part and a history part, each of which allows for separate treatment. The local part is computed by a quadrature suited for square-root singularities. For the history part, we approximate the convolution kernel uniformly by a sum of exponentials. The integral can then be evaluated recursively. As a result, the computation of the nonreflecting boundary conditions is both accurate and efficient.

Leslie Greengard

Papers

A Direct Adaptive Poisson Solver of Arbitrary Order Accuracy

A fast multipole method for the three-dimensional Stokes equations

Spectral Approximation of the Free-Space Heat Kernel

The numerical solution of the N-body problem

Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension