On the solution of symmetric linear systems by the preconditioned conjugate gradient method

1994 ◽  
Vol 52 (3-4) ◽  
pp. 225-241
Author(s):  
Anna Psimarni
Keyword(s):  
Linear Systems ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Preconditioned Conjugate Gradient Method ◽  
Preconditioned Conjugate Gradient

scholarly journals Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids

2010 ◽  
Vol 20 (1) ◽  
pp. 109-121 ◽  
Author(s):  
Tijmen Collignon ◽  
Martin Van Gijzen
Keyword(s):  
Linear Systems ◽  
Iterative Algorithm ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Heterogeneous Computing ◽  
Iterative Solution ◽  
Preconditioned Conjugate Gradient Method ◽  
Preconditioned Conjugate Gradient ◽  
Communication Overhead

Two implementations of the preconditioned conjugate gradient method on heterogeneous computing gridsEfficient iterative solution of large linear systems on grid computers is a complex problem. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. This paper describes a case study regarding iterative solution of large sparse linear systems on grid computers within the software constraints of the grid middleware GridSolve and within the algorithmic constraints of preconditioned Conjugate Gradient (CG) type methods. We identify the various bottlenecks induced by the middleware and the iterative algorithm. We consider the standard CG algorithm of Hestenes and Stiefel, and as an alternative the Chronopoulos/Gear variant, a formulation that is potentially better suited for grid computing since it requires only one synchronisation point per iteration, instead of two for standard CG. In addition, we improve the computation-to-communication ratio by maximising the work in the preconditioner. In addition to these algorithmic improvements, we also try to minimise the communication overhead within the communication model currently used by the GridSolve middleware. We present numerical experiments on 3D bubbly flow problems using heterogeneous computing hardware that show lower computing times and better speed-up for the Chronopoulos/Gear variant of conjugate gradients. Finally, we suggest extensions to both the iterative algorithm and the middleware for improving granularity.

scholarly journals A backward/forward recovery approach for the preconditioned conjugate gradient method

2016 ◽  
Vol 17 ◽  
pp. 522-534 ◽  
Author(s):  
Massimiliano Fasi ◽  
Julien Langou ◽  
Yves Robert ◽  
Bora Uçar
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Preconditioned Conjugate Gradient Method ◽  
Preconditioned Conjugate Gradient ◽  
Recovery Approach
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