A scalable iterative dense linear system solver for multiple right-hand sides in data analytics

Vassilis Kalantzis, A. Cristiano I. Malossi, Costas Bekas, Alessandro Curioni, Efstratios Gallopoulos, Yousef Saad

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


We describe Parallel-Projection Block Conjugate Gradient (PP-BCG), a distributed iterative solver for the solution of dense and symmetric positive definite linear systems with multiple right-hand sides. In particular, we focus on linear systems appearing in the context of stochastic estimation of the diagonal of the matrix inverse in Uncertainty Quantification. PP-BCG is based on the block Conjugate Gradient algorithm combined with Galerkin projections to accelerate the convergence rate of the solution process of the linear systems. Numerical experiments on massively parallel architectures illustrate the performance of the proposed scheme in terms of efficiency and convergence rate, as well as its effectiveness relative to the (block) Conjugate Gradient and the Cholesky-based ScaLAPACK solver. In particular, on a 4 rack BG/Q with up to 65,536 processor cores using dense matrices of order as high as 524,288 and 800 right-hand sides, PP-BCG can be 2x-3x faster than the aforementioned techniques.

Original languageEnglish (US)
Pages (from-to)136-153
Number of pages18
JournalParallel Computing
StatePublished - May 2018

Bibliographical note

Publisher Copyright:
© 2018 Elsevier B.V.


  • (Block) Conjugate Gradient
  • Deflation
  • Galerkin projections
  • Massively parallel architectures
  • Multiple right-hand sides


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