442 research outputs found
Scalable Resilience Against Node Failures for Communication-Hiding Preconditioned Conjugate Gradient and Conjugate Residual Methods
The observed and expected continued growth in the number of nodes in
large-scale parallel computers gives rise to two major challenges: global
communication operations are becoming major bottlenecks due to their limited
scalability, and the likelihood of node failures is increasing. We study an
approach for addressing these challenges in the context of solving large sparse
linear systems. In particular, we focus on the pipelined preconditioned
conjugate gradient (PPCG) method, which has been shown to successfully deal
with the first of these challenges. In this paper, we address the second
challenge. We present extensions to the PPCG solver and two of its variants
which make them resilient against the failure of a compute node while fully
preserving their communication-hiding properties and thus their scalability.
The basic idea is to efficiently communicate a few redundant copies of local
vector elements to neighboring nodes with very little overhead. In case a node
fails, these redundant copies are gathered at a replacement node, which can
then accurately reconstruct the lost parts of the solver's state. After that,
the parallel solver can continue as in the failure-free scenario. Experimental
evaluations of our approach illustrate on average very low runtime overheads
compared to the standard non-resilient algorithms. This shows that scalable
algorithmic resilience can be achieved at low extra cost.Comment: 12 pages, 2 figures, 2 table
Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method
Pipelined Krylov subspace methods (also referred to as communication-hiding
methods) have been proposed in the literature as a scalable alternative to
classic Krylov subspace algorithms for iteratively computing the solution to a
large linear system in parallel. For symmetric and positive definite system
matrices the pipelined Conjugate Gradient method outperforms its classic
Conjugate Gradient counterpart on large scale distributed memory hardware by
overlapping global communication with essential computations like the
matrix-vector product, thus hiding global communication. A well-known drawback
of the pipelining technique is the (possibly significant) loss of numerical
stability. In this work a numerically stable variant of the pipelined Conjugate
Gradient algorithm is presented that avoids the propagation of local rounding
errors in the finite precision recurrence relations that construct the Krylov
subspace basis. The multi-term recurrence relation for the basis vector is
replaced by two-term recurrences, improving stability without increasing the
overall computational cost of the algorithm. The proposed modification ensures
that the pipelined Conjugate Gradient method is able to attain a highly
accurate solution independently of the pipeline length. Numerical experiments
demonstrate a combination of excellent parallel performance and improved
maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm.
This work thus resolves one of the major practical restrictions for the
useability of pipelined Krylov subspace methods.Comment: 15 pages, 5 figures, 1 table, 2 algorithm
Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is
demonstrated to efficiently solve eigenvalue problems for graph Laplacians that
appear in spectral clustering. For static graph partitioning, 10-20 iterations
of LOBPCG without preconditioning result in ~10x error reduction, enough to
achieve 100% correctness for all Challenge datasets with known truth
partitions, e.g., for graphs with 5K/.1M (50K/1M) Vertices/Edges in 2 (7)
seconds, compared to over 5,000 (30,000) seconds needed by the baseline Python
code. Our Python code 100% correctly determines 98 (160) clusters from the
Challenge static graphs with 0.5M (2M) vertices in 270 (1,700) seconds using
10GB (50GB) of memory. Our single-precision MATLAB code calculates the same
clusters at half time and memory. For streaming graph partitioning, LOBPCG is
initiated with approximate eigenvectors of the graph Laplacian already computed
for the previous graph, in many cases reducing 2-3 times the number of required
LOBPCG iterations, compared to the static case. Our spectral clustering is
generic, i.e. assuming nothing specific of the block model or streaming, used
to generate the graphs for the Challenge, in contrast to the base code.
Nevertheless, in 10-stage streaming comparison with the base code for the 5K
graph, the quality of our clusters is similar or better starting at stage 4 (7)
for emerging edging (snowballing) streaming, while the computations are over
100-1000 faster.Comment: 6 pages. To appear in Proceedings of the 2017 IEEE High Performance
Extreme Computing Conference. Student Innovation Award Streaming Graph
Challenge: Stochastic Block Partition, see
http://graphchallenge.mit.edu/champion
KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners
[EN] Contemporary applications in computational science and engineering often require the solution of linear systems which may be of different sizes, shapes, and structures. The goal of this paper is to explain how two libraries, PETSc and HPDDM, have been interfaced in order to offer end-users robust overlapping Schwarz preconditioners and advanced Krylov methods featuring recycling and the ability to deal with multiple right-hand sides. The flexibility of the implementation is showcased and explained with minimalist, easy-to-run, and reproducible examples, to ease the integration of these algorithms into more advanced frameworks. The examples provided cover applications from eigenanalysis, elasticity, combustion, and electromagnetism.Jose E. Roman was supported by the Spanish Agencia Estatal de Investigacion (AEI) under project SLEPc-DA (PID2019-107379RB-I00)Jolivet, P.; Roman, JE.; Zampini, S. (2021). KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners. Computers & Mathematics with Applications. 84:277-295. https://doi.org/10.1016/j.camwa.2021.01.0032772958
Scalable iterative methods for sampling from massive Gaussian random vectors
Sampling from Gaussian Markov random fields (GMRFs), that is multivariate
Gaussian ran- dom vectors that are parameterised by the inverse of their
covariance matrix, is a fundamental problem in computational statistics. In
this paper, we show how we can exploit arbitrarily accu- rate approximations to
a GMRF to speed up Krylov subspace sampling methods. We also show that these
methods can be used when computing the normalising constant of a large
multivariate Gaussian distribution, which is needed for both any
likelihood-based inference method. The method we derive is also applicable to
other structured Gaussian random vectors and, in particu- lar, we show that
when the precision matrix is a perturbation of a (block) circulant matrix, it
is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure
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