## Abstract

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 language | English (US) |
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Pages (from-to) | 136-153 |

Number of pages | 18 |

Journal | Parallel Computing |

Volume | 74 |

DOIs | |

State | Published - May 2018 |

### Bibliographical note

Funding Information:Vassilis Kalantzis was partially supported by a Gerondelis Foundation Fellowship. This work was completed while E. Gallopoulos was on sabbatical leave at the Dept. CSE, University of Minnesota. We thank the University of Minnesota Supercomputing Institute for providing the computational resources needed in the initial stage of this work. We also thank the anonymous referees for their comments which greatly improved the clarity and presentation of this paper.

Publisher Copyright:

© 2018 Elsevier B.V.

## Keywords

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