Abstract
Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work where we examine some recent developments in a package of AMG preconditioners to improve efficiency, scalability, and robustness on extreme-scale problems. The main novelty is the design and implementation of a parallel coarsening algorithm based on aggregation of unknowns employing weighted graph matching techniques; this is a completely automated procedure, requiring no information from the user, and applicable to general symmetric positive definite (s.p.d.) matrices. The new coarsening algorithm improves in terms of numerical scalability at low operator complexity over decoupled aggregation algorithms available in previous releases of the package. The preconditioners package is built on the parallel software framework \texttt{PSBLAS}, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputers in Europe, for linear systems with sizes up to $O(10^{10})$ unknowns.
Highlights
Solving algebraic linear systems of the form: (1.1)Ax = b, where A ∈ Rn×n is a very large and sparse matrix and x, b ∈ Rn are vectors, is the most time consuming computational kernel in many areas of computational science and engineering, including more recent fields such as data and network analysis
In [15, 19, 4] we proposed a package of Algebraic MultiGrid (AMG) preconditioners built on top of the PSBLAS framework; the first version of the package implemented a multilevel version of some domain decomposition preconditioners of additive-Schwarz type and was based on a parallel decoupled version of the smoothed aggregation method described in [39, 37] to generate the multilevel hierarchy of coarser matrices
AMG preconditioners introduced in this paper we focus on both algorithmic and implementation scalability
Summary
Coarsening by aggregation uses disjoint aggregates of fine unknowns to form the coarse unknowns and, in general, the prolongation matrices are piecewise-constant interpolation matrices (unsmoothed aggregation [33, 22, 12]) or a smoothed variant of them (smoothed aggregation [39]). In both cases, the way to select aggregates of fine level variables normally exploits heuristics measures of affinity ( known as strength of connection) among the variables; these measures have been constructed for systems arising from scalar elliptic PDEs, and often loose their robustness for more general systems. This paper deals with aggregation-based approaches; in particular, we present a parallel aggregation scheme exploiting maximum weight matching in the weighted adjacency graph of the sparse matrices at each level
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