A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices

Abstract

We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. We follow the idea of Approximate Computing, allowing imprecision in the final result in order to be able to utilize the sparsity of the input matrix and to allow massively parallel execution. For an n × n matrix, the proposed algorithm allows to distribute the calculations over n nodes with only little communication overhead. The approximate result matrix exhibits the same sparsity pattern as the input matrix, allowing for efficient reuse of allocated data structures.We evaluate the algorithm with respect to the error that it introduces into calculated results, as well as its performance and scalability. We demonstrate that the error is relatively limited for well-conditioned matrices and that results are still valuable for errorresilient applications like preconditioning even for ill-conditioned matrices. We discuss the execution time and scaling of the algorithm on a theoretical level and present a distributed implementation of the algorithm using MPI and OpenMP. We demonstrate the scalability of this implementation by running it on a high-performance compute cluster comprised of 1024 CPU cores, showing a speedup of 665× compared to single-threaded execution.

Accurate Sampling with Noisy Forces from Approximate Computing

“…We conclude by noting that the present method has been recently implemented in the universal force engine i-PI [Kapil et al 2019], which can be generally applied to all sorts of forces affected by stochastic noise such as those computed by GPUs or other hardware accelerators [Abraham et al 2015;Anderson et al 2008;Brown et al 2012;Colberg and Höfling 2011;Eastman and Pande 2010;Le Grand et al 2013;Stone et al 2010], and potentially even quantum computing devices [Benhelm et al 2008;Chow et al 2014;Knill 2005;Steane 1999]. The possibility to apply similar ideas to N-body simulations [Efstathiou et al 1985;Hernquist et al 1993] and to combine it with further algorithmic approximations [Lass et al 2018b] is to be underlined and will be presented elsewhere.…”

Towards Electronic Structure-Based Ab-Initio Molecular Dynamics Simulations with Hundreds of Millions of Atoms

“…However, due to the distributed nature of the required matrix multiplications, large-scale applications are usually limited by a communication bottleneck. To avoid this bottleneck, we have extended the recently developed submatrix method [21], [22], which transforms calculations on large distributed sparse matrices into computations on small local dense matrices and combined it with the second-generation Car-Parrinello method of Kühne et al [9], [10] to bypass the previously mentioned self-consistent solution. This transformation opens the door to employ hardware-accelerated low-precision linear algebra without compromising the accuracy of the eventual results.…”

“…In contrast, we view the purification as a matrix function and approximate it with our submatrix method so that no global matrix multiplications are required that would otherwise lead to a communicationbound algorithm. The submatrix method [21], [22], recently developed by some of the authors, approximates a matrix function of a large sparse matrix by evaluating it on a series of much smaller and denser matrices. The underlying idea of the submatrix method is described by Fig.…”

Breaking the exascale barrier for the electronic structure problem in ab-initio molecular dynamics

“…The submatrix method (Lass, Mohr, et al 2018; Lass, Schade, et al 2020) instead views the density matrix as a matrix function to be evaluated. Therein, the evaluation of a matrix function f ( A ) is performed in three steps.…”