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

One of the most powerful approaches to imaging at the nanometer or subnanometer length scale is coherent diffraction imaging using X-ray sources. For amorphous (non-crystalline) samples, the raw data can be interpreted as the modulus of the continuous Fourier transform of the unknown object. Making use of prior information about the sample (such as its support), a natural goal is to recover the phase through computational means, after which the unknown object can be visualized at high resolution. While many algorithms have been proposed for this phase retrieval problem, careful analysis of its well-posedness has received relatively little attention. In this paper, we show that the problem is, in general, not well-posed and describe some of the underlying issues that are responsible for the ill-posedness. We then show how this analysis can be used to develop experimental protocols that lead to better conditioned inverse problems.

Geometry of the phase retrieval problem

We have developed easy to use fast multipole method (FMM) libraries for the Laplace, low-frequency Helmholtz, and Stokes equations in two and three dimensions. The codes are based on a new method for applying translation operators and provide reasonable performance on either single core processors, or small multi-core systems using OpenMP.

Computational Software: Simple FMM Libraries for Electrostatics, Slow Viscous Flow, and Frequency-Domain Wave Propagation

A method is presented for calculating potential flows in infinite channels. Given a collection of N sources in the channel and a zero normal flow boundary condition, the method requires an amount of work proportional to N to evaluate the induced velocity field at each source position. It is accurate to within machine precision and for its performance does not depend on the distribution of the sources. Like the Fast Multipole Method developed by Greengard and Rokhlin [J. Comput. Phys., 73 (1987), pp. 325–348], it is based on a recursive subdivision of space, knowledge of the governing Green's function, and the use of asymptotic representations of the potential field. Previous schemes have been based either on conformal mapping, which experiences numerical difficulties with the domain boundary, or direct evaluation of Green's function. Both require $O(N^2 )$ work.

/pdf/potential-flow-in-channels-57r15j6t2g.pdf

Potential Flow in Channels

We describe a suite of validation metrics that assess the credibility of a given automatic spike sorting algorithm applied to a given electrophysiological recording, when ground-truth is unavailable. By rerunning the spike sorter two or more times, the metrics measure stability under various perturbations consistent with variations in the data itself, making no assumptions about the noise model, nor about the internal workings of the sorting algorithm. Such stability is a prerequisite for reproducibility of results. We illustrate the metrics on standard sorting algorithms for both in vivo and ex vivo recordings. We believe that such metrics could reduce the significant human labor currently spent on validation, and should form an essential part of large-scale automated spike sorting and systematic benchmarking of algorithms.

Validation of neural spike sorting algorithms without ground-truth information

We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without addi- tional rank-completing constraints. Such problems arise in a variety of applications, such as the computation of the eigenvectors of a matrix corresponding to a known eigenvalue. The method is based on elementary linear algebra combined with the ob- servation that if the matrix is rank-k deficient, then a random rank-k perturbation yields a nonsingular matrix with probability 1.

/pdf/randomized-methods-for-rank-deficient-linear-systems-2zh5iwpelh.pdf

Leslie Greengard

Papers

Geometry of the phase retrieval problem

Computational Software: Simple FMM Libraries for Electrostatics, Slow Viscous Flow, and Frequency-Domain Wave Propagation

Potential Flow in Channels

Validation of neural spike sorting algorithms without ground-truth information

Randomized methods for rank-deficient linear systems