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A conjugate gradient method with quasi-Newton approximation

100%
Koko J.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 2
153-165
EN
The conjugate gradient method of Liu and Storey is an efficient minimization algorithm which uses second derivatives information, without saving matrices, by finite difference approximation. It is shown that the finite difference scheme can be removed by using a quasi-Newton approximation for computing a search direction, without loss of convergence. A conjugate gradient method based on BFGS approximation is proposed and compared with existing methods of the same class.
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Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids

75%
Collignon T., Van Gijzen M.
International Journal of Applied Mathematics and Computer Science
|
2010
|
tom 20
|
nr 1
109-121
EN
Efficient 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.
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