Abstract
The simulation of the physical movement of multi-body systems at an atomistic level, with forces calculated from a quantum mechanical description of the electrons, motivates a graph partitioning problem studied in this article. Several advanced algorithms relying on evaluations of matrix polynomials have been published in the literature for such simulations. We aim to use a special type of graph partitioning to efficiently parallelize these computations. For this, we create a graph representing the zero–nonzero structure of a thresholded density matrix, and partition that graph into several components. Each separate submatrix (corresponding to each subgraph) is then substituted into the matrix polynomial, and the result for the full matrix polynomial is reassembled at the end from the individual polynomials. This paper starts by introducing a rigorous definition as well as a mathematical justification of this partitioning problem. We assess the performance of several methods to compute graph partitions with respect to both the quality of the partitioning and their runtime.
Highlights
The physical movements of multi-body systems on an atomistic level is at the core of molecular dynamics (MD) simulations
This section uses a selection of physical test systems to evaluate all algorithms of Section 3, which were chosen to represent a variety of realistic scenarios where graph partitioning can be applied to MD simulations
This paper speeds up the computation of the density matrix in MD simulations through parallelization, informed by graph partitioning applied to the structure graph underlying a molecule
Summary
The physical movements of multi-body systems on an atomistic level is at the core of molecular dynamics (MD) simulations. In [11], the authors consider a similar set-up in which the graph encoding the molecular structure is partitioned (using various techniques) for faster heuristic computation; their article crucially differs from ours in that there are no generally applicable results pertaining to lossless partitioning of the matrix polynomial using overlapping blocks. This article is an extension of the work of [12], and includes proofs of all theoretical results of Section 2 in Appendix A, pseudo-code of our simulated annealing approach, a visualization of the relationship between the graph structure of a molecule and its partitioned graph representation, and more detailed performance data used for all experiments in Appendix B This article is an extension of the work of [12], and includes proofs of all theoretical results of Section 2 in Appendix A, pseudo-code of our simulated annealing approach in Section 3.2, a visualization of the relationship between the graph structure of a molecule and its partitioned graph representation in Section 4.3, and more detailed performance data used for all experiments in Appendix B
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