127,938 research outputs found
Increasing the Efficiency of Sparse Matrix-Matrix Multiplication with a 2.5D Algorithm and One-Sided MPI
Matrix-matrix multiplication is a basic operation in linear algebra and an
essential building block for a wide range of algorithms in various scientific
fields. Theory and implementation for the dense, square matrix case are
well-developed. If matrices are sparse, with application-specific sparsity
patterns, the optimal implementation remains an open question. Here, we explore
the performance of communication reducing 2.5D algorithms and one-sided MPI
communication in the context of linear scaling electronic structure theory. In
particular, we extend the DBCSR sparse matrix library, which is the basic
building block for linear scaling electronic structure theory and low scaling
correlated methods in CP2K. The library is specifically designed to efficiently
perform block-sparse matrix-matrix multiplication of matrices with a relatively
large occupation. Here, we compare the performance of the original
implementation based on Cannon's algorithm and MPI point-to-point
communication, with an implementation based on MPI one-sided communications
(RMA), in both a 2D and a 2.5D approach. The 2.5D approach trades memory and
auxiliary operations for reduced communication, which can lead to a speedup if
communication is dominant. The 2.5D algorithm is somewhat easier to implement
with one-sided communications. A detailed description of the implementation is
provided, also for non ideal processor topologies, since this is important for
actual applications. Given the importance of the precise sparsity pattern, and
even the actual matrix data, which decides the effective fill-in upon
multiplication, the tests are performed within the CP2K package with
application benchmarks. Results show a substantial boost in performance for the
RMA based 2.5D algorithm, up to 1.80x, which is observed to increase with the
number of involved processes in the parallelization.Comment: In Proceedings of PASC '17, Lugano, Switzerland, June 26-28, 2017, 10
pages, 4 figure
Algebras with ternary law of composition and their realization by cubic matrices
We study partially and totally associative ternary algebras of first and
second kind. Assuming the vector space underlying a ternary algebra to be a
topological space and a triple product to be continuous mapping we consider the
trivial vector bundle over a ternary algebra and show that a triple product
induces a structure of binary algebra in each fiber of this vector bundle. We
find the sufficient and necessary condition for a ternary multiplication to
induce a structure of associative binary algebra in each fiber of this vector
bundle. Given two modules over the algebras with involutions we construct a
ternary algebra which is used as a building block for a Lie algebra. We
construct ternary algebras of cubic matrices and find four different totally
associative ternary multiplications of second kind of cubic matrices. It is
proved that these are the only totally associative ternary multiplications of
second kind in the case of cubic matrices. We describe a ternary analog of Lie
algebra of cubic matrices of second order which is based on a notion of
j-commutator and find all commutation relations of generators of this algebra.Comment: 17 pages, 1 figure, to appear in "Journal of Generalized Lie Theory
and Applications
Communication-Avoiding Optimization Methods for Distributed Massive-Scale Sparse Inverse Covariance Estimation
Across a variety of scientific disciplines, sparse inverse covariance
estimation is a popular tool for capturing the underlying dependency
relationships in multivariate data. Unfortunately, most estimators are not
scalable enough to handle the sizes of modern high-dimensional data sets (often
on the order of terabytes), and assume Gaussian samples. To address these
deficiencies, we introduce HP-CONCORD, a highly scalable optimization method
for estimating a sparse inverse covariance matrix based on a regularized
pseudolikelihood framework, without assuming Gaussianity. Our parallel proximal
gradient method uses a novel communication-avoiding linear algebra algorithm
and runs across a multi-node cluster with up to 1k nodes (24k cores), achieving
parallel scalability on problems with up to ~819 billion parameters (1.28
million dimensions); even on a single node, HP-CONCORD demonstrates
scalability, outperforming a state-of-the-art method. We also use HP-CONCORD to
estimate the underlying dependency structure of the brain from fMRI data, and
use the result to identify functional regions automatically. The results show
good agreement with a clustering from the neuroscience literature.Comment: Main paper: 15 pages, appendix: 24 page
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