Another interesting paper came out last week on the preprint archive, “A matrix generalization of Euler identity.” The Euler identity reads,
and is traditionally derived from the Taylor expansion of sine and
cosine. In this paper, the author generalizes the Euler identity to
matrices. He does this by introducing a complex matrix, known as the
imaginary unit matrix
Just like 
, 
.
He then proves the Euler identity for these matrices as:
for all real x.
A friend brought this paper(pdf) to my attention “A common, avoidable source of error in molecular dynamics integrators” recently published in the Journal of Chemical Physics by D. E. Shaw Research. I had previously written about their award winning paper at SC06.
In constrained molecular dynamics simulations using some of the most popular molecular dynamics codes, calculation of the velocities of constrained particles is based solely on the differences in particle positions during two successive time steps. This creates a numerical instability that the authors’ show to be significant in a typical single-precision floating-point simulation. They describe a simple modification that eliminates this source of instability and demonstrate that this change substantially reduces the energy drift of a sample single-precision NVE simulation.
The paper is really short (2 pages) and describes recipes for
modifying constraint algorithms SHAKE, RATTLE and SETTLE. Below
is a graph of the energy drift in double-precision, single-precision
and modified versions of GROMACS and Desmond (that’s D.E. Shaw’s MD
engine.) They simulated a 30A cube of 901 water molecules with
periodic boundary conditions for one nanosecond with a one femtosecond
timestep. See how bad single-precision GROMACS is for NVE
simulations?
One of the papers nominated for the best paper award at Supercomputing 06 is titled “Scalable Algorithms for Molecular Dynamics Simulations on Commodity Clusters” from the DE Shaw Research group. If your standard of a modern molecular dynamics code is Computer Simulations of Liquids, then this paper is definitely worth a look. I’ve summarized the innovations below:
Published earlier this year in the Journal of Chemical Physics, this is a parallelizing algorithm which is a hybrid between spatial decompositions and force decompositions. The major innovation of this method is the reduced communication bandwidth, which can be a problem when you scale simulations to hundreds of computers.
One of the most important requirements for an integrator solving Newton’s equations of motion is symplecticity. One way to check to check for this is by running your integrator in negative time. Doing so should return the system to the initial state. Though not an absolute requirement, this is a nice thing to have for minimizing the energy drift, which ultimately leads to a more realistic simulation. These guys have an integrator that does this.
To reduce high frequency motions (like vibrations of Carbon-Hydrogen bonds for example), simulations typically make use of constraints to hold bond lengths or atom velocities fixed. Typical codes RATTLE and SHAKE need to be done in double precision for accuracy and to maintain energy conservation. This paper talks about the use of an (unpublished) algorithm that is stable in single precision.
Instead of representing coordinates in an absolute global frame, these guys normalize coordinates. This gives them more precision (due to operations done on floating points of the same magnitudes.) It also makes handling periodic boundary conditions simple. Another advantage they mention is that doing pressure control is time reversible.
Personally, I think this is the cornerstone of the paper. The paper
mentions that instead of interactions being a function of distance 
between the particles, they compute it as a function of
. The advantage is that the same set of polynomial
coefficients can be used for computing both forces and energies.
I’m just mentioning this for completeness. As far as I know, every high performance engine makes use of these instructions. In fact, GROMACS has hand tuned kernels written using these instructions. In theory, this makes your code four times faster.
In conclusion, I was pretty impressed with the quality of work and the innovations these guys have brought to the community. Download the paper here.