M.T. Heath, E. Ng and B.W. Peyton, Parallel Algorithms for Sparse Linear Systems, SIAM Review 33.3: 420-460, Sep. 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to obtain a di erent solution vector x in each case. Thus, in problems which involve solution of multiple b vectors, the time taken by repeated execution of substitution phase dominates the overall solution time. Although ecient parallel algorithms exist for the numerical factorization phase [2, 1, 23, 6, 4, 10, 15], not much progress has been made in the case of substitution phase [6, 11, 14] due to the limited amount of parallelism inherent in this phase. In Part I of this paper, we developed a bidirectional algorithm that is suitable for the solution of sparse symmetric linear systems with multiple ....

M.T.Heath, E.Ng and B.W.Peyton, Parallel algorithms for sparse linear systems, SIAM Review, Vol. 33, 1991, pp. 420-460.

....the time taken by the substitution phase, will contribute signi cantly to enhanced performance of the entire process. Although traditional approaches to parallel solution of sparse symmetric system of linear equations have yielded ecient parallel algorithms for the numerical factorization phase [1, 2, 7, 11, 15, 21], not much progress has been made in the case of substitution phase due to the limited amount of parallelism inherent in this phase. Moreover, the forward and backward substitution components of the substitution phase require di erent parallel algorithms due to the manner in which data is ....

M.T.Heath, E.Ng and B.W.Peyton, Parallel algorithms for sparse linear systems, SIAM Review, Vol. 33, 1991, pp. 420-460.

....such as Jacobi, Gauss Seidel, SOR, conjugate gradient and Chebyshev. Semi iterative methods to approximately solve general sparse linear systems. A succinct introduction to these iter ative methods is given in [6] also see [28 32] for further details, see [33] for sequential implementations, see [34,35] for parallel implementations, and see [36 38] for detailed analysis of the Conjugate Gradient method. These methods have the advantage that the most costly work per iteration consists of an inner product of the input matrix with a vector, but have the disadvantage of requiring potentially more ....

M.T. Heath, E. Ng and B.W. Peyton, Parallel algorithms for sparse linear systems, In Parallel Algorithms for Matrix Computations, SIAM, (1990).

....which is inherent in the matrix computations of the original, flat algorithm; and topological parallelism, which arises from the partial ordering of node updates in the computational tree. Similar computational structure can be found in, for example, direct sparse Cholesky factorization [35, 42, 63] and query evaluation in a belief network [54] In those applications, parallel processes are statically assigned to nodes in the tree in some top down fashion, based on the workload distribution over the computational hierarchy. We first demonstrate a parallel implementation, with a similar ....

....computation of all the children of a node before we may compute that node itself; however, nodes with no ancestor descendant relationship may be processed in any order. Similar computational patterns can be found in other applications as well. For example, in direct sparse Cholesky factorization [35, 42, 63], an elimination tree represents a partial ordering of columns (and corresponding rows) of the sparse matrix. Each node in the tree represents a column of the matrix. Before a node can be eliminated, all of its descendants must have been eliminated; nodes with no ancestor descendant relationship ....

M. T. Heath, E. Ng, and B. W. Peyton, "Parallel Algorithms for Sparse Linear Systems", SIAM Review, vol. 33, no. 3, pp. 420-460, 1991.

....graph G is partitioned into p parts. Thus, the problem of performing a p way partition is reduced to that of performing a sequence of 2 way partitions or bisections. Even though this scheme does not necessarily lead to optimal partition [27, 15] it is used extensively due to its simplicity [8, 10]. The basic structure of the multilevel bisection algorithm is very simple. The graph G = V, E) is first coarsened down to a few thousand vertices (coarsening phase) a bisection of this much smaller graph is computed (initial partitioning phase) and then this partition is projected back ....

M.T. Heath, E. G.-Y. Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:42046.

....factorization used extensively in practice, their use for solving large sparse systems has been mostly confined to big vector supercomputers due to its high time and memory requirements. As a result, parallelization of sparse Cholesky factorization has been the subject of intensive research [26, 55, 12, 15, 14, 18, 54, 40, 41, 3, 49, 50, 57, 9, 28, 26, 27, 51, 2, 1, 44, 58, 16, 55, 43, 33, 5, 42, 4, 59]. We have developed highly scalable formulations of sparse Cholesky factorization that substantially improve the state of the art in parallel direct solution of sparse linear systems both in terms of scalability and overall performance. It is well known that dense matrix factorization can be ....

....other known classes of algorithms for this problem. 3 the column based schemes represented in box A has been improved using smarter ways of mapping the matrix columns onto processors, such as, the subtree to subcube mapping [14] box B) A number of column based parallel factorization algorithms [40, 41, 3, 49, 50, 57, 12, 9, 28, 26, 55, 43, 5] have a lower bound of O(Np) on the total communication volume [15] Since the overall computation is only O(N 1:5 ) 13] the ratio of communication to computation of column based schemes is quite high. As a result, these column cased schemes scale very poorly as the number of processors is ....

M. T. Heath, Esmond G.-Y. Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420--460,

....feel though that the experience we have gained in this task will be useful in extending the comparisons in the future. In this section, we summarize the major characteristics of the parallel sparse direct codes of which we are aware. A clear description of the terms used in the tables is given by [19]. Code Technique Scope Availability Ref CAPSS Multifrontal SPD www.netlib.org scalapack [20] MUMPS Multifrontal SYM UNS www.enseeiht.fr apo MUMPS [3] PaStiX Fan in SPD see caption ( 21] PSPASES Multifrontal SPD www.cs.umn.edu mjoshi pspases [18] SPOOLES Fan in SYM UNS ....

M. T. Heath, E. Ng, and B. W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420--460, 1991.

....the entire processing graph breaks down. When this occurs, the partial matching technique allows the processing as defined by the graph to continue by seeking out a similar service. 5 Caching Mapping a task graph to a system graph is an area of high interest in parallel and distributed computing [4, 15, 17, 20, 23, 31]. The goal of such approaches is to reduce processing times while also more e#ectively utilizing resources across the network. Calculating such graphs is an NP Complete problem, and heuristic approaches still result in super linear solutions. The I XML approach allows for dependencies between ....

M.T. Heath, G.Y. Ng, and B.W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420--460, 1991.

....that the experience we have gained in this task will be useful in extending the comparisons in the future. In the following tables, we summarize the major characteristics of the parallel sparse direct codes of which we are aware. A clear description of the terms used in the tables is given by [18]. Code Technique Scope Availability Ref CAPSS Multifrontal SPD www.netlib.org scalapack [19] MUMPS Multifrontal SYM UNS www.enseeiht.fr apo MUMPS [3] PaStiX Fan in SPD see caption x [20] PSPASES Multifrontal SPD www.cs.umn.edu mjoshi pspases [17] SPOOLES Fan in SYM UNS ....

M. T. Heath, E. Ng, and B. W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420-460, 1991.

....that it is theoretically possible to achieve tighter bounds on the amount of work performed by PND algorithms. We also show how our algorithms generalize to solve all pairs minimal cost path problems, within the same complexity. Supported by National Science Foundation Grant Number NSF IRI 91 00681 and Army Research Office contract DAAH 04 96 1 0448. A preliminary version of this paper appeared as An Optimal Space and Efficient Parallel Nested Dissection Algorithm, 4th Annual ACM Symposium on Parallel Algorithms and Architectures, San Diego, CA, July 1992. 1 Introduction The ....

....tighter bounds on the amount of work performed by PND algorithms. We also show how our algorithms generalize to solve all pairs minimal cost path problems, within the same complexity. Supported by National Science Foundation Grant Number NSF IRI 91 00681 and Army Research Office contract DAAH 04 96 1 0448. A preliminary version of this paper appeared as An Optimal Space and Efficient Parallel Nested Dissection Algorithm, 4th Annual ACM Symposium on Parallel Algorithms and Architectures, San Diego, CA, July 1992. 1 Introduction The problem of solving very large sparse linear systems ....

[Article contains additional citation context not shown here]

M.T. Heath, E. Ng and B.W. Peyton. Parallel Algorithms for Sparse Linear Systems. SIAM Review 33:420--460, 1991.

....Gradient and Chebyshev Semiiterative methods to approximately solve general sparse linear systems. A succinct introduction to these iterative methods is given in [GV 89] also see [V 62, Y 71, A 84, HY 81, H 94] for further details, see [PFTV 88] for sequential implementations, see [GO 93, HNP 90] for parallel implementations, and see [HS 52, R 71, GO 89] for detailed analysis of the Conjugate Gradient method. These methods have the advantage that the most costly work per iteration consists of an inner product of the input matrix with a vector, but have the disadvantage of requiring ....

M.T. Heath, E. Ng, and B.W. Peyton, Parallel algorithms for sparse linear systems, in Parallel Algorithms for Matrix Computations SIAM, (1990).

....that the experience we have gained in this task will be useful in extending the comparisons in the future. In the following tables, we summarize the major characteristics of the parallel sparse direct codes of which we are aware. A clear description of the terms used in the tables is given by Heath, Ng and Peyton (1991). Code Technique Scope Availability Reference CAPSS Multifrontal SPD www.netlib.org scalapack (Heath and Raghavan 1997) MUMPS Multifrontal SYM UNS www.enseeiht.fr apo MUMPS (Amestoy et al. 1999) PaStiX Fan in SPD see caption x (Henon et al. 1999) PSPASES Multifrontal SPD ....

Heath, M. T., Ng, E. G. Y. and Peyton, B. W. (1991), `Parallel algorithms for sparse linear systems', SIAM Review 33, 420--460.

....(MA37) 2, 9, 10, 16] will be referred to as the classical multifrontal method. The method takes more advantage of dense matrix kernels than D2, but is unsuitable when the pattern of the matrix is very unsymmetric. Many methods for symmetric matrices use dense kernels; a survey may be found in [25]. Most recently, Gilbert and Liu [23] and Eisenstat and Liu [18] have presented symbolic factorization algorithms for unsymmetric matrices, assuming that the pivot ordering is known a priori. The algorithms are based on the elimination directed acyclic graph (dag) and its reductions, which are ....

....just before step 4. This assembly is performed by the edge reductions described in Section 4.2. 3 Elimination and assembly trees The elimination tree, T , 26] and its variants (such as the assembly tree [15] are used either explicitly or implicitly in most parallel sparse matrix algorithms [25]: T = T V ; T E ) T V = 1 : n T E = fhi; ji j j = parent (i)g 10 parent (i) minfj j i j; l ji 6= 0g: Starting with the elimination tree, T , an assembly tree is constructed by amalgamating a connected (node induced) subgraph into a single supernode. The minimum label of the nodes in ....

M. T. Heath, E. Ng, and B. W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33(3):420--460, 1991.

....case, LU factorization is necessary [4] While sparsity in the coefficient matrix allows a significant reduction of required computations, it also provides opportunities for the exploitation of parallelism. However, much of the investigation of parallelism has focused on Cholesky factorization [11]. The multifrontal method originally proposed by Duff and Reid [6, 7] has application to both Cholesky and LU factorization [12] and significant parallel potential [3] With the multifrontal approach, the sparse matrix is decomposed into a set of partially overlapping dense submatrices and a ....

Michael T. Heath, Esmond Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. In R. J. Plemmons, editor, Parallel Algorithms for Matrix Computations, pages 83--124. SIAM, Philadelphia, PA, 1990.

....feel though that the experience we have gained in this task will be useful in extending the comparisons in the future. In this section, we summarize the major characteristics of the parallel sparse direct codes of which we are aware. A clear description of the terms used in the tables is given by [19]. Code Technique Scope Availability Ref CAPSS Multifrontal SPD www.netlib.org scalapack [20] MUMPS Multifrontal SYM UNS www.enseeiht.fr apo MUMPS [3] PaStiX Fan in SPD see caption ( 21] PSPASES Multifrontal SPD www.cs.umn.edu mjoshi pspases [18] SPOOLES Fan in SYM UNS ....

M. T. Heath, E. Ng, and B. W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420-460, 1991.

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M.T. Heath, E. Ng and B.W. Peyton, Parallel Algorithms for Sparse Linear Systems, SIAM Review 33.3: 420-460, Sep. 1991.

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M. T. Heath, E. G.-Y. Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420--460, 1991.

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M. T. Heath, E. G.-Y. Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420-- 460, 1991.

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M. T. Heath, Esmond G.-Y. Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420--460, 1991.

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Michael T. Heath, Esmond Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM REVIEW, 33(3):420--460, September 1991.

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M.T.Heath, E.Ng and B.W.Peyton, Parallel algorithms for sparse linear systems, SIAM Review, Vol. 33, 1991, pp. 420-460.

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Michael T. Heath, Esmond Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33(3):420-460, September 1991.

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M. T. Heath, E. Ng, and B. W. Peyton, "Parallel algorithms for sparse linear systems," SIAM Review, Vol. 33, pp. 420-460, 1991.

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M. T. Heath, Esmond G.-Y. Ng, and Barry W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33:420--460,

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IEEE. #75# M.T. Heath, E. Ng., and B.W. Peyton. Parallel algorithms for sparse linear systems. SIAM Review, 33#3#:420#460, September 1991.

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