Solving Large Sparse Linear Systems Efficiently on Grid Computers Using an Asynchronous Iterative Method as a Preconditioner

Collignon, T.P.; Gijzen, Martin B. van

doi:10.1007/978-3-642-11795-4_27

Cited by 4 publications

(7 citation statements)

References 22 publications

(20 reference statements)

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“…To reap the benefits and awards of both techniques, the authors propose in [17] to use an asynchronous iterative method as a preconditioner in a flexible iterative method. By combining a slow but coarse-grain asynchronous preconditioning iteration with a fast but fine-grain outer iteration, it is believed that high convergence rates may be achieved on Grid computers.…”

Section: Hybrid Methods: Best Of Both Worldsmentioning

confidence: 99%

“…In the second part of this chapter the various facets of the whole process will be discussed in detail. Most of the concepts given here are taken from [16,17,18].…”

Section: Efficient Numerical Algorithms In Grid Computingmentioning

confidence: 99%

“…Extensive experimental results and specific implementation details pertaining to implementing numerical algorithms on Grid computers may be found in [16,17,18]. …”

Section: Concluding Remarks and Further Readingmentioning

confidence: 99%

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Parallel Scientific Computing on Loosely Coupled Networks of Computers

Collignon

Gijzen

2009

Lecture Notes in Computational Science and Engineering

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Efficiently solving large sparse linear systems on loosely coupled networks of computers is a rich and vibrant field of research. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. Designing efficient numerical algorithms for said systems is a complex process that brings together many different scientific disciplines. This book chapter is divided into two distinct parts. The purpose of the first half (Sect. §2- §4) is to give a bird's view of the issues pertaining to designing efficient numerical algorithms for Grid computing. It kicks off by clearly stating the problem and exposing the various bottlenecks, subsequently followed by the presentation of potential solutions. Thus, the stage is set and Sect. §3 proceeds by detailing classical iterative solution methods, along with the concept of asynchronism, which is a highly favorable quality in the context of Grid computing. The first half is wrapped up by explaining how asynchronism can be introduced into faster but more complicated subspace methods. The general idea is that by using an asynchronous method as a preconditioner, the best of both worlds can be combined. The advantages and disadvantages of this approach are discussed in minute detail. The second half (Sect. §5) contains discussions on the various intricacies related to implementing the proposed algorithm on Grid computers. Section §6 gives some concluding remarks along with suggestions for further reading.

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Section: Hybrid Methods: Best Of Both Worldsmentioning

confidence: 99%

“…In the second part of this chapter the various facets of the whole process will be discussed in detail. Most of the concepts given here are taken from [16,17,18].…”

Section: Efficient Numerical Algorithms In Grid Computingmentioning

confidence: 99%

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Parallel Scientific Computing on Loosely Coupled Networks of Computers

Collignon

Gijzen

2009

Lecture Notes in Computational Science and Engineering

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“…The resulting partially asynchronous inner-outer method is analysed in depth in (Collignon and van Gijzen, 2008;2009) …”

Section: Related Workmentioning

confidence: 99%

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

Collignon¹,

Gijzen²

2010

International Journal of Applied Mathematics and Computer Science

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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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“…This application involves the solution of large sparse symmetric and positive definite systems, which leaves the flexible Conjugate Gradient method (Axelsson, 1994; Notay, 2000) as the method of choice for the outer iteration. Nevertheless, the proposed approach can be used for non-symmetric systems as well by using a flexible method such as Generalized Conjugate Residual (GCR) (Eisenstat et al, 1983; van der Vorst and Vuik, 1994) as the outer iteration (Collignon and van Gijzen, 2010).…”

Section: Introductionmentioning

confidence: 99%

Fast iterative solution of large sparse linear systems on geographically separated clusters

Collignon

Gijzen

2011

The International Journal of High Performance Computing Applica

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Parallel asynchronous iterative algorithms exhibit features that are extremely well-suited for Grid computing, such as lack of synchronisation points. Unfortunately, they also suffer from slow convergence rates. In this paper we propose using asynchronous methods as a coarse-grain preconditioner in a flexible iterative method, where the preconditioner is allowed to change in each iteration step. A full implementation of the algorithm is presented using Grid middleware that allows for both synchronous and asynchronous communication. Advantages and disadvantages of the approach are discussed. Numerical experiments on heterogeneous computing hardware demonstrate the effectiveness of the proposed algorithm on Grid computers, with application to large 2D and 3D bubbly flow problems.

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Solving Large Sparse Linear Systems Efficiently on Grid Computers Using an Asynchronous Iterative Method as a Preconditioner

Cited by 4 publications

References 22 publications

Parallel Scientific Computing on Loosely Coupled Networks of Computers

Parallel Scientific Computing on Loosely Coupled Networks of Computers

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

Fast iterative solution of large sparse linear systems on geographically separated clusters

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