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

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order O(log(T?)(log(1??)+loglog(T?)))$O(\log (\frac {T}{\delta }) (\log (\frac {1}{\epsilon })+\log \log (\frac {T}{\delta })))$ for any x??$x\in \mathbb R$ and ?≤t≤T, where ?? is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only O(log2(T?))$O(\log ^{2}(\frac {T}{\delta }))$ terms for fixed accuracy ??. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For NS points in the spatial discretization and NT time steps, the cost is O(NSNTlog2T?)$O(N_{S} N_{T} \log ^{2} \frac {T}{\delta })$ in terms of both memory and CPU time for fixed accuracy ??. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.

Efficient sum-of-exponentials approximations for the heat kernel and their applications

The issue of accuracy and convergence for numerical calculations in situations in which the exact solution of the Navier–Stokes equations (under the imposed initial and boundary conditions) is unstable is discussed. If one relies on roundoff error or truncation error (or the randomness of a stochastic scheme) in order to perturb the computed solution away from the unstable one, then the computed result cannot be accurate. Instead, one must explicitly provide perturbations in order to compute accurately flows of physical interest. Computational results for flow past an impulsively started cylinder are presented.

On the accurate calculation of vortex shedding

A fluctuating boundary integral method for Brownian suspensions

http://www.cmap.polytechnique.fr/~jingrebeccali/papers/wavejcp2.pdf

High order marching schemes for the wave equation in complex geometry

The present disclosure methods and systems that provide heat, via at least Photon-Electron resonance, also known as excitation, of at least a particle utilized, at least in part, to initiate and/or drive at least one catalytic chemical reaction. In some implementations, the particles are structures or metallic structures, such as nanostructures. The one or more metallic structures are heat at least as a result of interaction of incident electromagnetic radiation, having particular frequencies and/or frequency ranges, with delocalized surface electrons of the one or more particles. This provides a control of catalytic chemical reactions, via spatial and temporal control of generated heat, on the scale of nanometers as well as a method by which catalytic chemical reaction temperatures are provided.

Leslie Greengard

Papers

Efficient sum-of-exponentials approximations for the heat kernel and their applications

On the accurate calculation of vortex shedding

A fluctuating boundary integral method for Brownian suspensions

High order marching schemes for the wave equation in complex geometry

Electromagnetic control of chemical catalysis