J. R. Gilbert, "An efficient parallel sparse partial pivoting algorithm", Report No. 88/45052-1. Chr. Michelsen Institute, Department of Science and Technology, Centre for Computer Science, Fantoftvegen 38, N-5036 Fantoft, Bergen, Norway, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....edges incident on only one node in the subgraph become incident on the resulting supernode. The process repeats until the desired assembly tree is obtained. Elimination and assembly trees are typically defined only for symmetric patterned LU factors, with the exception of partial pivoting methods [21, 22]. A more general graph is needed for the unsymmetric pattern multifrontal method. The classical multifrontal method [2, 9, 10, 15, 16] is based on the assembly tree. It has a similar formulation as the general frontal matrix formulation described in the previous section, except that the analysis ....

J. R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report 88/45052-1, Center for Computer Science, Chr. Michelsen Institute, Bergen, Norway, 1988.

....memory implementations of both dense and sparse matrix factorizations. Distributed Memory Factorization: In distributed memory environments, algorithms are heavily dependent upon the storage allocation scheme for the matrix. The two most common schemes are row oriented [22] and column oriented [64, 40, 27, 76]. Blocking schemes of rows or columns are frequently used and Dongarra and Ostrouchov [40] discuss such methods together with the need for adaptive blocking mechanisms. Choi, Dongarra, Pozo, and Walker explore the various data allocation schemes and associated algorithms and develop a generalized ....

.... Cholesky factorization) are the fan in methods [9, 117, 88] These methods accumulate contributions to the updating of the active submatrix and send fewer but larger messages [9] An increasingly popular approach to sparse matrix factorization are the symmetricpattern, multifrontal methods [105, 76, 51, 121, 122, 77]. Large grain parallelism is available in these methods via independent subtrees in the assembly trees. However, as subtrees combine and parallelism at that level decreases, a switch is made to exploit parallelism at a finer grain within the factorization of a particular frontal matrix. This is ....

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J. R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report CMI No. 88/45052-1, Christian Michelsen Institute, 1988.

....vectorization within each frontal matrix. However, this method is based on an assumption of a symmetric nonzero pattern, and so has a poor performance on matrices whose patterns are very unsymmetric. None of the previous parallel methods for unsymmetric patterned matrices use dense matrix kernels [2, 10, 11], with the exception of a multifrontal QR factorization algorithm [15] which will be compared later on with the algorithms we develop) available via anonymous ftp to cis.ufl.edu as cis tech reports tr92 tr92 014.ps.Z This paper presents a new unsymmetric pattern multifrontal method that ....

J. R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report 88/45052-1, Center for Computer Science, Chr. Michelsen Institute, Bergen, Norway, 1988.

....[3, 9, 14] will be referred to as the classical multifrontal method. It will be described in Section 2 to contrast it with the unsymmetric pattern multifrontal method proposed here. Other parallel algorithms for unsymmetric sparse matrices include the D2 algorithm [5] partial pivoting methods [20, 21, 22, 24], the PSolve algorithm [4] and others [2, 27] The D2 algorithm is based on a nondeterministic parallel pivot search that constructs a set of independent pivots (m, say) followed by a parallel rank m update of the active submatrix. Near the end of the factorization, the active submatrix is ....

....(s)g father (s) minft j s t; l ts 6= 0g: If a node a appears in the unique path from node d to the root, then node a is an ancestor of its descendant node d. The elimination tree is typically defined only for symmetric patterned LU factors, with the exception of partial pivoting methods [21, 22]. The classical multifrontal method [13] is one method based on the elimination tree. It has a similar formulation as the general frontal matrix formulation described in the previous section, except that the analysis is performed on the pattern of A A T . Frontal matrices are square. The method ....

J. R. Gilbert, An efficient parallel sparse partial pivoting algorithm, Report 88/45052-1, Center for Computer Science, Chr. Michelsen Institute, Bergen, Norway, 1988.

....[110, 111] In addition to demonstrating the efficiency of our parallel algorithm on these machines, we also study the (theoretical) upper bound on performance of this algorithm. Several methods have been proposed to perform sparse Cholesky factorization [49, 73, 90] and sparse LU factorization [6, 57, 65] on shared memory machines. A common practice is to organize the program as a self scheduling loop, interacting with a global pool of tasks that are ready to be executed. Each processor repeatedly takes a task from the pool, executes it, and puts new ready task(s) in the pool. This pool of tasks ....

....this model as well. Our parallel algorithm resembles the supernode panel sparse Cholesky factorization studied in [73] in the way we define the basic tasks and the computational primitives. The way we handle the coordination of the dependent tasks is reminiscent of the approach used by Gilbert [65] in his column wise sparse LU factorization. However, our algorithm represents a non trivial extension to the earlier work in that we have incorporated several new mechanisms, such as unsymmetric supernodes and symmetric structure reduction. We begin this chapter by looking at the architectural ....

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John R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report CMI No. 88/45052-1, Computer Science Department, University of Bergen, Norway, 8 1988.

....by the numeric values in the second phase. Also, in many applications a sequence of problems with the same nonzero structure must be solved, and the structural phase can be done just once. The structural phase may also be used to schedule the numerical phase efficiently on a parallel machine [20, 29]. Structure prediction can be used to save time as well as space in sparse Gaussian elimination. The asymptotically fastest algorithms to compute the Cholesky factorization of a symmetric positive definite matrix are those of the Yale Sparse Matrix Package [12] and Sparspak [15] which predict ....

....integers. Permuting the rows and columns of A only relabels the vertices of the bipartite graph: if P and Q are row and column permutation matrices, then H(PAQ T ) is isomorphic to H(A) Several structure prediction problems use matchings and alternating paths in the bipartite graph of a matrix [4, 6, 7, 20, 23, 21, 32]. This paper does not consider such problems in detail, but we include enough definitions here to state some of these results in later sections. Let A be an m by n matrix with m n. We say that A has the Hall property if, for every k with 0 k n, every set of k columns of A contains nonzeros in ....

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John R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report 88/45052-1, Christian Michelsen Institute, 1988.

....by a method analogous to block back substitution; see the references for more details. The decomposition is named for A. L. Dulmage and N. S. Mendelsohn, whose work on matchings and decompositions of bipartite graphs in the late 1950s has found application in several sparse matrix settings [5, 7, 16, 23]. Pothen and Fan [24] survey this theory in sparse matrix language. In graphtheoretic terms, dmperm computes a maximum matching on the bipartite graph of A, permutes the matched edges onto the diagonal of the matrix, and then finds the strongly connected components of the directed graph of the ....

John R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report 88/45052-1, Christian Michelsen Institute, 1988.

....permutations P r and P c , the directed graph G(P r AP c ) is strongly connected. We conclude this subsection by proving a theorem (Theorem 2.9) about strong Hall matrices that is useful in several structure prediction results. The theorem first appeared in a technical report by Gilbert [15]; other proofs have been given by Hare, Johnson, Olesky, and van den Driessche [21] and Brualdi and Shader [4] First we need two technical lemmas. Lemma 2.7. Let H be a strong Hall graph and let hr 0 ; ci be an edge of H. Then there is a column complete matching that includes hr 0 ; ci, and ....

....and Ng s static data structure [13] is the tightest possible for Gaussian elimination with partial pivoting. This is a one at a time result; as we will see, no all at once result is possible. The case r c of Theorem 4. 5 (that is, the proof for U ) first appeared in a technical report by Gilbert [15]; the case r c (for L) has not appeared before. Gilbert actually related U to G (A) rather than G Theta (A) but the U parts of those graphs are the same for strong Hall A by Corollary 3.9. Theorem 4.5. Let H be the structure of a square strong Hall matrix with nonzero diagonal. Let ....

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John R. Gilbert. An efficient parallel sparse partial pivoting algorithm. Technical Report 88/45052-1, Christian Michelsen Institute, 1988.

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J. R. Gilbert, "An efficient parallel sparse partial pivoting algorithm", Report No. 88/45052-1. Chr. Michelsen Institute, Department of Science and Technology, Centre for Computer Science, Fantoftvegen 38, N-5036 Fantoft, Bergen, Norway, 1988.

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