S. C. Eisenstat and J. W. H. Liu, Exploiting structural symmetry in unsymmetric sparse symbolic factorization, SIAM J.Matrix Anal.Appl., 13 (1992), pp.202--211.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... symbolic factorization.Thet etorizati structure are two DAGs thatare transitive reitivwA] ofthe graphs ofthe factormatrice L and U ,re5 e5D e , and can be use todeAU e a task DAG forsparse LU factorization.Some reeor hee have argue that computing anewA0 transitive rensitiv can be tooeU eUD3 e [9, 15] and have prop using subminimal DAGs withmore ere thannenwUD35 . HoweeD trave093D unne[DU]9O DAGeA95 during numeUO3w factorization can be asource of ove0A95w More veA in aparalle implee tation,etio DAGeUD] can be pote tial source ofunneD3w[D] synchronization or communication. In this pap we ....

....resit Struct(L i,# ) 2. Compute 2, i 1 . 3. Transitiven resit . 4. Compute ) # j:j#i#E U O 2, i 1 . Fig 3.2. Amodified symbolic factorization algorithm. hasprompte repte hep toseA alte09w[L el such as computing fast butincomple[ transitive resitivw [9, 15].The use of suchalteLw[L0 e toG withmore eore thanG U O ,re0 e05 e , canincre]O the cost ofsteD 1 and 3, asweD as that of numeL3Aw factorization. 3.1. A modification to Gilbert and Liu s algorithm. We nowde0O9D e a reDDA] e simple modification tothe algorithm shown inFigure 3.1. We start ....

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S. C. Eisenstat and J. W. H. Liu, Exploiting structural symmetry in unsymmetric sparse symbolic factorization, SIAM J.Matrix Anal.Appl., 13 (1992), pp.202--211.

.... Library code ma41 [2, 3] and in the distributed memory code MUMPS developed in the context of the PARASOL project (EU ESPRIT IV LTR project 20160) 5, 4] Another way to represent the symbolic LU factorization of a structurally unsymmetric matrix is to use directed acyclic graphs (see for example [17, 18]) These This work was supported by the Director, Oce of Science, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy under contract number DE AC03 76SF00098. amestoy enseeiht.fr, ENSEEIHT IRIT, 2 rue Camichel 31071 Toulouse (France) and NERSC, ....

S. C. Eisenstat and J. W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. SIAM Journal on Matrix Analysis and Applications, 13:202-211, 1992.

....phases of the scheduling algorithm use the dependencies between tasks. In the case of an unsymmetric matrix A, several tools can be used to represent the dependencies: the elimination tree of A A [14] the elimination DAGs of L and U [8] the symmetric pruned graphs of L and U [4, 5]. The elimination tree of A A is a structure o#ering simple manipulation and access. This tree can be used to represent the dependencies between supernodes, but it overestimates these dependencies. The elimination DAGs of L and U are the transitive reductions of the graphs of L and U , and ....

....U , and are a compact way of representing all dependencies between computations, and only those dependencies. But a major disadvantage is their significant amount of construction time. A good tradeo# is to use the symmetric pruned graphs of L and U . These graphs can be built very e#ciently [4, 5]. Even if they introduce redundant dependencies compared to the elimination DAGs, we experimentally observed that they contain few redundant edges. Thus, they can be used e#ectively. However, due to their simplicity, we still use the elimination DAGs to develop the theoretical results. Knowing ....

S. C. Eisenstat and J. W. H. Liu. Exploiting Structural Symmetry in Unsymmetric Sparse Symbolic Factorization. SIAM J. Matrix Anal. Appl., 13(1):202--211, 1992.

....These elimination structures are two DAGs that are transitive reductions of the graphs of the factor matrices L and U , respectively, and can be used to derive a task DAG for sparse LU factorization. Some researchers have argued that computing an exact transitive reduction can be too expensive [8, 14] and have proposed using subminimal DAGs with more edges than necessary. Traversing unnecessary DAG edges during numerical factorization can be a source of overhead. Moreover, in a parallel implementation, extra DAG edges can be potential sources of unnecessary synchronization or communication. ....

....and the i th step could potentially traverse all edges in G L O i and G U O i . Steps 2 and 4 of Gilbert and Liu s algorithm are much costlier than steps 1 and 3. The cost of these steps has prompted researchers to seek alternatives, such as computing fast but incomplete transitive reduction [8, 14]. Such use of alternatives to G L O and G U O with more edges than G L O and G U O , respectively, can increase the cost of Steps 1 and 3, as well as that of numerical factorization. 3.1. A modification to Gilbert and Liu s algorithm. We now describe a relatively straightforward modification ....

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Stanley C. Eisenstat and Joseph W.-H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. SIAM Journal on Matrix Analysis and Applications, 13(1):202--211, 1992.

.... code #### [2, 3] and in the distributed memory code ##### developed in the context of the PARASOL project (EU ESPRIT IV LTR project 20160) 4, 5] Another way to represent the symbolic LU factorization of a structurally unsymmetric matrix is to use directed acyclic graphs (see for example [14, 15]) These structures more costly and complicated to handle than a tree, capture better the asymmetry of the matrix. Davis and Du [6] implicitly use this structure to drive their unsymmetric pattern multifrontal approach. We explain, in this article, how to use the simple elimination tree ....

S. C. Eisenstat and J. W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. #### ####### ## ###### ######## ### ############, 13:202-211, 1992.

.... Library code ma41 [2, 3] and in the distributed memory code MUMPS developed in the context of the PARASOL project (EU ESPRIT IV LTR project 20160) 4, 5] Another way to represent the symbolic LU factorization of a structurally unsymmetric matrix is to use directed acyclic graphs (see for example [14, 15]) These structures more costly and complicated to handle than a tree, capture better the asymmetry of the matrix. Davis and Du [6] implicitly use this structure to drive their unsymmetric pattern multifrontal approach. We explain, in this article, how to use the simple elimination tree ....

S. C. Eisenstat and J. W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. SIAM Journal on Matrix Analysis and Applications, 13:202-211, 1992.

....European Center for Research and Advanced Training in Scientific Computation (CERFACS) Toulouse, France. 1 2 T. A. DAVIS AND I. S. DUFF an assembly tree and the more general structure of an assembly dag (directed acyclic graph) 5] similar to that of Gilbert and Liu [22] and Eisenstat and Liu [17, 18] is required. In the current work we do not explicitly use this structure. We have developed a new unsymmetric pattern multifrontal approach [4, 5] As in the symmetric multifrontal case, advantage is taken of repetitive structure in the matrix by factorizing more than one pivot in each frontal ....

S. C. Eisenstat and J. W. H. Liu, Exploiting structural symmetry in unsymmetric sparse symbolic factorization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 202--211.

....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 similar to the assembly dag presented in this paper. Moreover, we ....

....into E j , but the contribution that E i makes to row j must be assembled into E j . i k j row j row k Figure 1: A possible subgraph that characterizes Equation 7 If the assembly tree is unsuitable, what kind of graph can guide the unsymmetric pattern multifrontal method The elimination dag [18, 23] is one possibility. Define G(L) as the directed graph associated with L. That is, hi; ji is an edge of G(L) if and only if l ji is nonzero. Similarly, hi; ji is an edge of G(U T ) if and only if u ij is nonzero. Several reductions to these graphs are described in [18, 23] such as transitive ....

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S. C. Eisenstat and J. W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. SIAM J. Matrix Anal. Appl., 13(1):202--211, 1992.

.... and GU reduce to a pair of directed acyclic graphs called elimination dags[10] Unfortunately the elimination dags can be fairly expensive to compute, and so somewhat denser but cheaper graphs, intermediate between the triangular factors and their transitive reductions, have been investigated in [8]. For this application, an alternative route is to use graphs whose transitive closures contain the structures of W T and Z but may be a little denser still for example, the elimination tree of the symmetrized A. With these cases in mind, the unsymmetric generalization of the previous theorem ....

....if G ffi L and G ffi U are chosen to be the elimination dags or other intermediate structures between the elimination dags and the triangular factors. For example, the inner product rZ j at step i is just L ij D jj , and the above characterization is the same as that shown for the rows of L in [10, 8]. Table 2 compares the regular form of AINV with the symbolic factorization enhanced version, with a drop tolerance of 0.1 for each test matrix as before. The timing counts are from a C implementation running on an Apple Macintosh workstation with a 233MHz PowerPC 750 processor. For the matrices ....

S. Eisenstat and J. Liu, Exploiting structural symmetry in unsymmetric sparse symbolic factorization, SIAM J. Matrix Anal. Appl., 13 (1992), no. 1, pp. 202--211.

....we have a coarse grained algorithm that does not need complicated communication. The direct solver we used (for the direct part of our method) was our own implementation of GP Mod. This is the sparse LU method of Gilbert and Peierls [9] extended with the symmetric reductions of Eisenstat and Liu [8], 7] see also [4] Any sparse LU method for the Schur complement is allowed as a direct solver, including, for example, multifrontal methods. We choosed GP Mod because it is relatively easy to implement. Moreover, in [4] it is reported that, for circuit simulation problem memplus , GP Mod is ....

S.C. Eisenstat, J.W.H. Liu, Exploiting structural symmetry in unsymmetric sparse symbolic factorization. SIAM J. Matrix. Anal. Appl., 13(1) (1992), pp. 202-211.

....A is n by n with m nonzeros. They also showed that G (A) can be computed in time asymptotically the same as that to compute G (A) so a faster algorithm to compute G (A) would give a faster algorithm to compute transitive closures than the best currently known. Eisenstat, Gilbert, and Liu [11, 22] give algorithms to compute G (A) that are more efficient in practice than transitive closure, by using various transitively reduced graphs. 7 Remark 4.2. A nonsingular square matrix may have an LU factorization even though it has zeros on the diagonal. In this case, Theorem 4.1(i) still ....

Stanley C. Eisenstat and Joseph W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. Technical Report CS--90--12, York University, 1990. To appear in SIMAX.

....be handled by explicitly holding zero entries. It is more complicated to design an efficient multifrontal scheme for matrices that are asymmetric in structure. The main difference is that the elimination cannot be represented by an (undirected) tree but a directed acyclic graph (DAG) is required (Eisenstat and Liu 1992). The frontal matrices are, of course, no longer square and, as in the case discussed in the previous section, they are not necessarily absorbed at the parent node and can persist in the DAG. Finally the complication of a posteriori numerical pivoting is even more of a problem with this scheme so ....

Eisenstat, S. C. and Liu, J. W. H. (1992), `Exploiting structural symmetry in unsymmetric sparse symbolic factorization', SIAM J. Matrix Analysis and Applications 13, 202--211.

....and symmetric in pattern, but rather rectangular and unsymmetric in pattern. Secondly, as a consequence of the rectangular, unsymmetric structure of the frontal matrices, the factorization can no longer be represented by an assembly tree but, rather, by an assembly dag (directed acyclic graph) [24,28]. The contribution matrix of a frontal matrix may now have to be assembled into more than one subsequent frontal matrices, a situation quite different from that in the classical method where a contribution matrix can always be fully assembled into its parent node in the assemble tree. We discuss ....

S. C. Eisenstat and J. W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. Technical Report CS-90-12, Dept. of Computer Science, York Univ., North York, Ontario, Nov. 1990.

.... and son in the elimination tree (in the symmetric pattern case) The disruptions are more severe in the unsymmetric pattern multifrontal method if the pivot ordering is not known to be numerically acceptable (unsymmetric symbolic factorization when the pivot ordering is known is described in [8, 9, 12]) If a single phase is used, the numerical values are available to the pivot search heuristic. This limits the disruption, but requires a dynamic parallelism where the inter dependence between parallel tasks is unknown until the pivot search is complete. 2 Formulation of the unsymmetric pattern ....

....2 E L and (s; t) 62 E U . Node s is a Uson of its Ufather node t if (s; t) 62 E L and (s; t) 2 E U . Finally, node s is an LUson of its LUfather node t if (s; t) 2 E L and (s; t) 2 E U . Similar directed acyclic graphs been used for symbolic factorization of unsymmetric sparse matrices [8, 9, 12]. If the pattern of the LU factors is symmetric, the contribution block of s is always completely assembled into its single LUfather node f in the elimination tree. In general, the elimination tree cannot be used in the unsymmetric pattern multifrontal method because of the incomplete assembly ....

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S. C. Eisenstant and J. W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. Technical Report CS-90-12, Dept. of Computer Sci., York Univ., North York, Ontario, Nov. 1990.

....with dense matrix kernels) Of all these algorithms for unsymmetric sparse matrices, the classical multifrontal method takes most advantage of dense matrix kernels, but is unsuitable when the pattern of the matrix is very unsymmetric. Most recently, Gilbert and Liu [23] and Eisenstat and Liu [16] 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 similar to the reduced data flow and control flow graphs presented ....

.... = f(s; t) j t 2 U 2s g E complete = E L complete [ E U complete E L complete control j E L complete data j E L complete E U complete control j E U complete data j E U complete : Similar directed acyclic graphs have been used for symbolic LU factorization of unsymmetric sparse matrices [16, 23]. Most parallel sparse matrix factorization algorithms are based on the elimination tree, T , T = V; E tree ) E tree = f(s; t) j t = father (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 ....

[Article contains additional citation context not shown here]

S. C. Eisenstat and J. W. H. Liu, Exploiting structural symmetry in unsymmetric sparse symbolic factorization, Report CS-90-12, Dept. of Computer Science, York University, North York, Ontario, Canada, Nov. 1990.

....of codes for unsymmetric systems has been the observation of Gilbert and Peierls (1988) that partial pivoting can be performed in time proportional to the number of arithmetic operations, so avoiding any potentially costly sorting operations. A nice refinement of their technique was provided by Eisenstat and Liu (1992), who suggested ways of pruning a search tree to reduce work in the symbolic phase. Variants of this technique are used in nearly all sparse partial pivoting codes, for example Duff and Reid (1993) and Demmel et al. 1995) Frontal, multifrontal, and supernodal approaches for the solution of ....

Eisenstat, S. C. and Liu, J. W. H. (1992), `Exploiting structural symmetry in unsymmetric sparse symbolic factorization', SIAM J. Matrix Analysis and Applications 13, 202--211.

....of LU factors in a way that parallels, and generalizes, the elimination tree model of the Cholesky factor. We presented a new algorithm for sparse symbolic LU factorization, using edags, and showed experimentally that it compares favorably with existing symbolic schemes. Eisenstat and Liu [5] have recently extended this work by defining a pair of dags they call the symmetric reductions of a matrix. These dags are intermediate in size between the edags and the triangular factors, but are easier to find than edags; Eisenstat and Liu s experiments show that a symbolic factorization ....

Stanley C. Eisenstat and Joseph W. H. Liu. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. Technical Report CS--90--12, York University, 1990.

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