1,047 research outputs found
Efficient Iterative Solution of Large Linear Systems on Heterogeneous Computing Systems
No full textThis dissertation deals mainly with the design, implementation, and analysis of efficient iterative solution methods for large sparse linear systems on distributed and heterogeneous computing systems as found in Grid computing. First, a case study is performed on iteratively solving large symmetric linear systems on both a multi–cluster and a local heterogeneous cluster using standard block Jacobi precondi- tioning within the software constraints of standard Grid middleware and within the algorith- mic constraints of preconditioned Conjugate Gradient–type methods. This shows that global synchronisation is a key bottleneck operation and in order to alleviate this bottleneck, three main strategies are proposed: exploiting the hierarchical structure of multi–clusters, using asyn- chronous iterative methods as preconditioners, and minimising the number of inner products in Krylov subspace methods. Asynchronous iterative methods have never really been successfully applied to the solution of extremely large sparse linear systems. The main reason is that the slow convergence rates limit the applicability of these methods. Nevertheless, the lack of global synchronisation points in these methods is a highly favourable property in heterogeneous computing systems. Krylov subspace methods offer significantly improved convergence rates, but the global synchronisation points induced by the inner product operations in each iteration step limits the applicability. By using an asynchronous iterative method as a preconditioner in a flexible Krylov subspace method, the best of both worlds is combined. It is shown that this hybrid combination of a slow but asynchronous inner iteration and a fast but synchronous outer iteration results in high convergence rates on heterogeneous networks of computers. Since the preconditionering iteration is performed on heterogeneous computing hardware, it varies in each iteration step. Therefore, a flexible iterative method which can handle a varying preconditioner has to be employed. This partially asynchronous algorithm is implemented on two different types of Grid hardware applied to two different applications using two different types of Grid middleware. The IDR(s) method and its variants are new and powerful algorithms for iteratively solving large nonsymmetric linear systems. Four techniques are used to construct an efficient IDR(s) variant for parallel computing and in particular for Grid computing. Firstly, an efficient and robust IDR(s) variant is presented that has a single global synchronisation point per matrix– vector multiplication step. Secondly, the so–called IDR test matrix in IDR(s) can be chosen freely and this matrix is constructed such that the work, communication, and storage involving this matrix are minimised in the context of multi–clusters. Thirdly, a methodology is presented for a priori estimation of the optimal value of s in IDR(s). Finally, the proposed IDR(s) variant is combined with an asynchronous preconditioning iteration. By using an asynchronous preconditioner in IDR(s), the IDR(s) method is treated as a flexible method, where the preconditioner changes in each iteration step. In order to begin analysing mathematically the effect of a varying preconditioning operator on the convergence properties of IDR(s), the IDR(s) method is interpreted as a special type of deflation method. This leads to a better understanding of the core structure of IDR(s) methods. In particular, it provides an intuitive explanation for the excellent convergence properties of IDR(s). Two applications from computational fluid dynamics are considered: large bubbly flow prob- lems and large (more general) convection–diffussion problems, both in 2D and 3D. However, the techniques presented can be applied to a wide range of scientific applications. Large numerical experiments are performed on two heterogeneous computing platforms: (i) local networks of non–dedicated computers and (ii) a dedicated cluster of clusters linked by a high–speed network. The numerical experiments not only demonstrate the effectiveness of the techniques, but they also illustrate the theoretical results.Numerical AnalysisElectrical Engineering, Mathematics and Computer Scienc
Variations sur Cadet Roussel [music] /
No full textJ.W.C. 3835 (Publisher number). For voice and piano.; Title from list t.p.; "Harmonisées par Arnold Bax, Frank Bridge, Eugène Goossens, John Ireland."; Individual sections within the work by each of the composers are marked with their initials.; Pl. no.: J.W.C. 3835.; Also available online http://nla.gov.au/nla.mus-vn6005741
I'm owre young to marry yet [music] /
No full textJ.W.C. 3840 (Publisher number). For voice and piano.; Title from list t.p.; Lyrics from Robert Burns' poem of the same name.; Pl. no.: J.W.C. 3840.; Also available online http://nla.gov.au/nla.mus-vn6005755
Fast solution of nonsymmetric linear systems on Grid computers using parallel variants of IDR(s)
No full textIDR(s) is a family of fast algorithms for iteratively solving large nonsymmetric linear systems [14]. With cluster computing and in particular with Grid computing, the inner product is a bottleneck operation. In this paper, three techniques are combined in order to alleviate this bottleneck. Firstly, the efficient and stable IDR(s) algorithm from [16] is reformulated in such a way that it has a single global synchronisation point per iteration step. Secondly, the so–called test matrix is chosen so that the work, communication, and storage involving this matrix is minimised in multi–cluster environments. Finally, a methodology is presented for a–priori estimation of the optimal value of s using only problem and machine–based parameters. Numerical experiments applied to a 3D convection–diffusion problem are performed on the DAS–3 Grid computer, demonstrating the effectiveness of these three techniques.Electrical Engineering, Mathematics and Computer Scienc
Solving large sparse linear systems efficiently on grid computers using an asynchronous iterative method as a preconditioner
No full textElectrical Engineering, Mathematics and Computer Scienc
Fast iterative solution of large sparse linear systems on geographically separated clusters
No full textElectrical Engineering, Mathematics and Computer Scienc
Implementing the conjugate gradient method on a grid computer
No full textElectrical Engineering, Mathematics and Computer Scienc
Exploiting the flexibility of IDR(s) for grid computing
No full textThe IDR(s) method that is proposed in [26] is an efficient limited memory method for solving large nonsymmetric systems of linear equations. In [11] an IDR(s) variant is described that has a single synchronisation point per iteration step, which makes this variant well-suited for parallel and grid computing. In this paper, we combine this IDR(s) variant with an a-synchronous preconditioning iteration to further improve the performance of IDR(s) on a grid computer. A-synchronous preconditioners do not require expensive synchronisation and adapt to volatile computational resources, and are therefore well-suited for such a computational environment. However, an a-synchronous preconditioning operation is also non-constant by nature: the preconditioner changes in every iteration. The success of the combination of IDR(s) with an a-synchronous preconditioner therefore depends on the flexibility of IDR(s). We will explain why IDR(s) can be used as a flexible method, and we will successfully use the combination of IDR(s) with an a-synchronous preconditioner for solving large convection-diffusion problems. The numerical experiments are performed on the DAS-3 grid computer, which is composed of five geographically separated parallel clusters.Electrical Engineering, Mathematics and Computer Scienc
Interpreting IDR(s) as a deflation method
No full textIn this paper the IDR(s) method is interpreted in the context of deflation methods. It is shown that IDR(s) can be seen as a Richardson iteration preconditioned by a variable deflation–type preconditioner. The main result of this paper is the IDR projection theorem, which relates the spectrum of the deflated system in each IDR(s) cycle to all previous cycles. The theorem shows that this so–called active spectrum becomes increasingly more clustered. This clustering property may serve as an intuitive explanation for the excellent convergence properties of IDR(s). These remarkable spectral properties exist whilst using a deflation subspace matrix of fixed rank. Variants of explicitly deflated IDR(s) are compared to IDR(s) in which the IDR deflation subspace matrix is augmented with “traditional” deflation vectors. The theoretical results are illustrated by numerical experiments.Electrical Engineering, Mathematics and Computer Scienc
A funeral handkerchief [electronic resource] : in two parts : I. Part, containing arguments to comfort us at death of friends, II. Part, containing several uses which we ought to make of such losses : to which is added, Three sermons preached at Coventry /
No full textThe 2nd part has special t.p., continuous paging, and imprint: London : Printed for the author, 1691.Errata: prelim. p. [15].Imperfect: t.p. to the "Three sermons preached at Coventry" lacking.Includes bibliographical references.Reproduction of original in Huntington Library.WingElectronic reproduction
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