Scheduling Many-Body Short Range MD Simulations on a Cluster of Workstations and Custom VLSI

Hardware

Sumanth J.V, David R. Swanson and Hong Jiang

University of Nebraska-Lincoln

We thank ONR, RCF and SDI(NSF 0091900) for funding this research andDr. Kenji Yasuoka and Dr. Takahiro Koishi for the many useful talks we had.

Introduction

MD is very computationally intensive. Luckily, MD is parallelizable hence can be

efficiently implemented on a cluster. Custom VLSI solutions like the MD-GRAPE 2

are another approach, but is more limited to kinds of potential functions that can be evaluated.

We combine the above two techniques to combine the advantages of both.

Computational Aspects of MD

Perform time integration of following equation

Forces are computed as

Potential Function

We restrict ourselves to 2-body and 3-body potentials.

The MD-GRAPE 2 is designed to compute only 2-body potentials.

The cluster can however be programmed to evaluate any kind of potential.

We use a combination of the cluster and the MD-GRAPE 2 board to evaluate a 3-body potential.

Lennard-Jones(LJ) Potential Is a very simple empirical 2-body potential

Reactive Bond Order (REBO) Potential

Simple MD Algorithm

Can improve execution time by using cut-off radius, neighbor lists, link cell or combination of these. Cut-off radius introduces discontinuities. Can be overcome by smoothing the potential function. Velocity-Verlet Integration.

Boundary Conditions

Minimum Image ConventionPeriodic Boundary Conditions

Neighbor Lists

Link Cell Method

Parallel MD – Atom Decomposition

Involves dividing up the N atoms into sets of N/P atoms and assigning each set to one of the P processors.

At every time-step, two global communication operations are required (one for updating positions and the other for updating forces).

Runs in time proportional to the square of the number of atoms N.

Very good efficiency but long running times. Is a suitable technique if the system is dense.

Parallel MD – Spatial Decomposition

Involves dividing up the simulation box into domains and assigning domains to processors.

Communication is local. Efficiency is worse, but has lower running time. Works better when the system is not very dense. Load balancing can be performed by dynamically

varying the volume of the domains. When the system is not very dense, the running

time is nearly linear.

Efficiency of Atom and Spatial Decomposition

Atom Decomposition Spatial Decomposition

MD-GRAPE 2 for MD simulations

Parallel pipelined special purpose hardware for computing non-bonded forces.

Bonded forces and time integrations are performed on the host machine.

Can compute forces either in the all-pairs method or link-cell method.

If there are more than half a million atoms in the system, they must be split into batches of at most half a million before being sent to the MD-GRAPE 2.

Peak Performance of 64Gflops.

MD-GRAPE 2 Calculations

The forces and potentials are computed using the following two equations.

The function G(x) is evaluated using a segmented fourth order polynomial interpolation.

MD-GRAPE 2 Architecture

Link Cell Method on MD-GRAPE 2

Number of Processors vs. ExecutionTime for MD-GRAPE 2 link cell methodand domain decomposition method.

Relative Error in Computing TotalEnergy with MD-GRAPE2

Scheduling MD on a cluster and MD-GRAPE 2 simultaneously

The REBO is a three-body potential. It comprises of three two-body components VR, VA

and VvdW and a three-body component Bij. The MD-GRAPE 2 is not capable of computing

three-body potentials due to its architecture. The custom function evaluation table does not allow

for conditional statements to be placed in the function, but this feature is required to evaluate VR

and VA. This allows us to compute the VvdW on the MD-

GRAPE 2 and all the other components on the cluster.

Scheduling MD on a cluster and MD-GRAPE 2 simultaneously

contd. The motivation for doing so is that the Vvdw

has a cut-off that is roughly twice that of the other components.

This can however be efficiently computed on the MD-GRAPE 2 while the other components are being evaluated on the cluster simultaneously.

To aid in communication between the cluster and the machine hosting the MD-GRAPE 2, we implement a server.

Scheduling MD on a cluster and MD-GRAPE 2 simultaneously

contd. The server accepts a position vector and outputs a partial forces vector and a partial potential vector.

They are called partial since they only contain contributions due to the Vvdw component.

This has to be added to the other contributions that are computed on the cluster.

Scheduling MD on a cluster and MD-GRAPE 2 simultaneously

contd. At every time step, before the parallel code begins its computations, it sends a copy of the position vector to the MD-GRAPE 2.

Now the cluster and the MD-GRAPE 2 compute partial forces/potentials simultaneously.

When the MD-GRAPE 2 completes its computations, it returns the partial forces/potentials to the host and the host sums them to give the actual forces and total potential energy.

The cut-off for the computations on the cluster is now 2.5 instead of 5.5 which is required if all the components of the REBO potential were computed on it.

Scheduling MD on a cluster and MD-GRAPE 2 simultaneously

contd. The execution time of an MD simulation using the atom-decomposition method can be well approximated by a second degree polynomial tc(N).

The execution time on the MD-GRAPE 2 can also be approximated by a second degree polynomial tg(N).

The total time it takes to run such a simulation on the cluster and the MD-GRAPE 2 simultaneously is given by

Scheduling MD on a cluster and MD-GRAPE 2 simultaneously

contd. The optimum number of processors to use can be determined by

Experimentally, we have determined an optimal p to be 35 for our setup. With this setup, we found the speedup to gradually approach 1.4 and nearly remain constant

after that. We used the atom-decomposition method since the system being simulated is very dense.

Plot of speedup when using a cluster and MD-GRAPE 2 simultaneously vs. using a

cluster alone

Conclusion

At the time of writing, cost per processor including network was 1500 USD.

Cost of MD-GRAPE 2 was 15000 USD. For long range potentials it is more cost effective to

use MD-GRAPE 2 since it takes 61 cluster CPUs to equal its performance.

For short range potentials, it is more effective to use a cluster since it takes only 12 cluster CPUs to match performance.

However, using a combination of a cluster and MD-GRAPE 2 to solve more complex potentials can yield a significant gain.

Future Work

Incorporate multiple MD-GRAPE 2 boards into the current setup.

Schedule MD simulations on larger scale systems with Globus.

Custom FPGA solutions to solve more than just pair potentials.

Using GPUs to perform MD. Hand optimizing energy calculations to use

SSE/SSE2/SSE3 instructions for optimal performance.