**A novel force matrix transformation with optimal load-balance for 3-body
potential based parallel Molecular Dynamics using atom-decomposition in
a heterogeneous cluster environment**

JV Sumanth and D Swanson and H Jiang, HIGH PERFORMANCE COMPUTING - HIPC 2007, PROCEEDINGS, 4873, 552-565 (2007).

Evaluating the Force Matrix constitutes the most computationally intensive part of a Molecular Dynamics (MD) simulation. In three-body MD) simulations, the total energy of the system is determined by the energy of every unique triple in the system and the force matrix is three-dimensional. The execution time of a three-body MD algorithm is thus proportional to the cube of the number of atoms in the system. Fortunately, there exist symmetries in the Force Matrix that can be exploited to improve the running time of the algorithm. While this optimization is straight forward to implement in the case of sequential code, it has proven to be nontrivial for parallel code even in a homogeneous environment. In this paper, we present a force matrix transformation that is capable of exploiting the symmetries in the force matrix in both a homogeneous and a heterogeneous environment while balancing the load among all the participating processors. The proposed transformation distributes the number of interactions to be computed uniformly among all the slices of the force matrix along any of the axes. The transformed matrix can be scheduled using any well known heterogeneous slice-level scheduling technique. We also derive theoretical bounds for efficiency and load balance for prior work in the literature. We then prove some interesting and useful properties of our transformation and evaluate its advantages and disadvantages. A loop reordering optimization for the symmetric transformation is described. The performance of an MPI implementation of the transformation is studied in terms of the Step Time Variation Ratio (STVR) in a homogeneous and heterogeneous environment.

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