On deflation and singular symmetric positive semi-definite matrices

Tang, Jinyu; Vuik, C.

doi:10.1016/j.cam.2006.08.015

Cited by 19 publications

(17 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper the test problem from [23,21] is considered. It is a two-phase bubbly flow problem with two separate fluids Γ 0 and Γ 1 , representing water (high-density phase) and vapour (low-density phase) respectively.…”

Section: Numerical Experiments 41 Motivationmentioning

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

View full text Add to dashboard Cite

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.

show abstract

Section: Numerical Experiments 41 Motivationmentioning

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

View full text Add to dashboard Cite

show abstract

“…In this paper we will consider the 3D test problem taken from (van der Pijl et al, 2005;Tang and Vuik, 2007a). It is a two-phase bubbly flow problem where we have two separate fluids Γ 0 and Γ 1 , representing water (high-density phase) and water vapour (low-density phase), respectively.…”

Section: Heterogeneous Sparse Linear Solvers In Gridsolvementioning

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

View full text Add to dashboard Cite

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.

show abstract

“…In many applications, DPCG has been shown to be an efficient method, such as in porous-media [50] and ground-water flows [27]. Recently, DPCG has also been applied successfully to bubbly flows [40][41][42].…”

Section: Introductionmentioning

confidence: 99%

“…While several previous papers have detailed the application of deflated PCG methods to bubbly flows [40][41][42], the use of advanced multigrid methods for these flow simulations has, to our knowledge, not been previously considered in the literature. As well, the systems of equations that arise in three-dimensional multiphase flows offer a good opportunity for comparison of these two families of solvers.…”

Section: Introductionmentioning

confidence: 99%