J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334352, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and data DAGs for the factorization pro cew are atre calle the eeLw[OO] tre [22] Howe e for unsymmeDA ] sparse matriceA the task and data DAGsare ge535 DAGs.More veD the eeU]A5 ofthe minimal data DAG forunsymmew[O sparse factorization can be a supepw9 ofthe eeL05L of a task DAG. Gilbel and Liu [16]de]wUD e e]wUD9U]w[ structure forunsymmew[O sparse LU factors and give an algorithm forsparse unsymmesym 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 ....

....by asingle rowand column g = #( q : r] in .HeA total numb of sup epw dew Furthew[LU5 if g = #( q : r] h = #( s : t] and r s,the g h; that is,the column and rowindice maintainthe reewD e orde of sup de in L and U . 3. Computing a task DAG and the structures of L and U . and Liu [16]preA] t anunsymmeA0A symbolic factorization algorithm tocompute the structure ofthe factors L and U andtheL transitive resitivwU L . Figure 3.1 summarize Gilbel and Liu s algorithm.The algorithmcompute the structure of L, U , and L rowby rowandcompute the structure by columns. The ....

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J. R. Gilbert and J. W. H. Liu, Elimination structures for unsymmetric sparse LU factors, SIAM J.Matrix Anal.Appl., 14 (1993), pp.334--352.

....supernodal algorithm used in SuperLU are two representative algorithms to perform sparse Gaussian elimination. In this section, we brie y describe the main characteristics of the algorithms and highlight the major di erences between them. Both algorithms can be described by a computational graph [7] whose nodes represent computations and whose edges represent transfer of data. In the case of the multifrontal method, MUMPS, this graph is a tree. Some steps of Gaussian elimination are performed on a dense frontal matrix at each node of the tree and the Schur complement (or contribution block) ....

.... and 2D for the root) asynchronous, dynamic distributed scheduling (in part because of the fact that numerical pivoting causes a delay in pivot selection with consequent modi cation of data structures during the numerical factorization) SuperLU supernodal fan out based on elimination DAGs [7] static pivoting with possible half precision perturbations on the diagonal static 2D irregular block cyclic mapping using supernode structure loosely synchronous scheduling with pipelining. Throughout this paper, we will use a set of test problems to illustrate the performance of our ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334352, 1993.

.... 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, ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334-352, 1993.

....to each other in the lled pattern. Unlike the Cholesky factor whose minimum graph representation is a tree (called the elimination tree, or etree for short) 48] the minimum graph representations of the L and U factors are directed acyclic graphs (called elimination DAGs, or edags for short) [31, 32]. Despite these diculties, researchers have been addressing these issues successfully for sequential and shared memory machines; available codes include MA41 [6, 5] PARDISO [57] SPOOLES [9] SuperLU [19] SuperLU MT [20] UMFPACK MA38 [15] and WSMP [34] In our earlier codes SuperLU (serial) ....

John R. Gilbert and Joseph W.H. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM J. Matrix Anal. Appl., 14(2):334-352, April 1993.

....from its candidate processors set. The two 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 ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM J. Matrix Anal. Appl., 14(2):334--352, April 1993.

....algorithm used in SuperLU are two representative algorithms to perform sparse Gaussian elimination. In this section, 1 we brie y describe the main characteristics of the algorithms and highlight the major di erences between them. Both algorithms can be described by a computational graph [8] whose nodes represent computations and whose edges represent transfer of data. In the case of the multifrontal method, MUMPS, this graph is a tree. Some steps of Gaussian elimination are performed on a dense frontal matrix at each node of the tree and the Schur complement (or contribution block) ....

.... and 2D for the root) asynchronous, dynamic distributed scheduling (in part because of the fact that numerical pivoting causes a delay in pivot selection with consequent modi cation of data structures during the numerical factorization) SuperLU supernodal fan out based on elimination DAGs [8] static pivoting with possible half precision perturbations on the diagonal static 2D irregular block cyclic mapping using supernode structure loosely synchronous scheduling with pipelining. Throughout this paper, we will use a set of test problems to illustrate the performance of our ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334-352, 1993.

....SuperLU time spent in AMD is very similar in both cases, this gives a good estimation of the cost difference of the analysis phase of the two solvers. During the SuperLU analysis phase, all the unsymmetric structures involved during the factorization are computed and the directed acyclic graph [17] of the unsymmetric matrix must be built and mapped onto the processors. Path searches in the directed acyclic graph are used to reduce communications. With MUMPS, the main data structure handled during analysis is the assembly tree which is produced directly as a by product of the ordering ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334--352, 1993.

....graphs. Moreover, the edge set of the minimal data DAG for unsymmetric sparse factorization Mathematical Sciences Department, IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598 (anshul watson.ibm.com) 1 2 ANSHUL GUPTA can be a superset of the edge set of a task DAG. In [15], Gilbert and Liu describe elimination structures for unsymmetric sparse LU factors and give an algorithm for sparse unsymmetric symbolic factorization. These elimination structures are two DAGs that are transitive reductions of the graphs of the factor matrices L and U , respectively, and can be ....

....i . 3. Compute Struct(U i; j:j i2E L 0O i Struct(U j; Struct(A i; Gamma f1; 2; i Gamma 1g. 4. Transitively reduce Struct(U ;i ) using G U 0O i Gamma1 and extend it to G U 0O i . end for Fig. 3.1. Gilbert and Liu s unsymmetric symbolic factorization algorithm [15]. by i;j. The graph GM O is a transitive reduction of the graph GM if and only if GM has a directed path i ; j and there is no other graph with fewer edges than GM O that satisfies this condition. Since we are primarily dealing with the nonzero structure of matrices rather than the actual ....

[Article contains additional citation context not shown here]

John R. Gilbert and Joseph W.-H. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14(2):334--352, 1993.

....kind of independence comes from not only the sparsity but also the 2D process to matrix mapping. An even more interesting study would be to formalize these 2D task dependencies into a task graph, and perform some optimal scheduling on it. Parallelism from the directed acyclic elimination graphs [15] often referred to as elimination dags or edags. Consider another matrix with 6 by 6 blocks mapped onto a 2 by 3 process mesh 2 6 6 6 6 6 6 6 6 4 0 1 0 2 4 3 2 0 2 4 3 5 0 1 2 3 5 5 3 7 7 7 7 7 7 7 7 5 : Columns 1 and 3 are independent in the elimination dags. The ....

....0.8 SuperLU AMD 2.4 0.7 MC64 AMD 2.6 0.2 0.7 Table 6: In uence of permuting large entries onto the diagonal (using MC64) on the time (in seconds) for the analysis phase of MUMPS and SuperLU. 16 asymmetric structures needed for the factorization are computed and the directed acyclic graph [15] of the unsymmetric matrix must be built and mapped onto the processors. With MUMPS, the main data structure handled during analysis is the assembly tree which is produced directly as a by product of the ordering phase. No further data structures are introduced during this phase. Dynamic ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334-352, 1993.

....storage space for them. This process can be performed solely based on the matrix structure information without taking matrix values into consideration. Substantial performance improvements are obtained in this phase if graph theoretic concepts such as elimination trees and elimination dags [12] are eciently utilized. Then, supernodes are found which are sets of columns that have a similar sparsity structure. They are used in the next stage: the numerical factorization. Supernodes enable the use of BLAS routines to improve the performance of the last phase. In case of SuperLU, the ....

John R. Gilbert, and Joseph W. H. Liu, Elimination structures for unsymmetric sparse LU factors, SIAM J. Matrix Anal. Appl. ##(2), pp. 334-352, April, 1993.

.... 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 ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse lu factors. #### ####### ## ###### ######## ### ############, 14:334-352, 1993.

....of independence comes from not only the sparsity but also the 2D process to matrix mapping. An even more interesting study would be to formalize these 2D task dependencies into a task graph, and perform some optimal scheduling on it. ffl Parallelism from the directed acyclic elimination graphs (Gilbert and Liu 1993) often referred to as elimination dags or edags. Consider another matrix with 6 by 6 blocks mapped onto a 2 by 3 process mesh 2 6 6 6 6 6 6 6 6 4 0 1 0 2 4 3 2 0 2 4 3 5 0 1 2 3 5 5 3 7 7 7 7 7 7 7 7 5 : Columns 1 and 3 are independent in the elimination dags. The ....

....from an ordering algorithm. A tighter coupling with an ordering, as is the case with MUMPS and AMD, should reduce the analysis time for SuperLU. However, during the analysis phase of SuperLU, all the asymmetric structures needed for the factorization are computed and the directed acyclic graph (Gilbert and Liu 1993) of the unsymmetric matrix must be built and mapped onto the processors. With MUMPS, the main data structure handled during analysis is the assembly tree which is produced directly as a by product of the ordering phase. No further data structures are introduced during this phase. Dynamic ....

Gilbert, J. R. and Liu, J. W. H. (1993), `Elimination structures for unsymmetric sparse LU factors', SIAM J. Matrix Analysis and Applications 14, 334--354.

.... 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 ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse lu factors. SIAM Journal on Matrix Analysis and Applications, 14:334-352, 1993.

....Oxon. 0X11 0QX England, and 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 ....

J. R. Gilbert and J. W. H. Liu, Elimination structures for unsymmetric sparse LU factors, SIAM Journal on Matrix Analysis and Applications, 14 (1993), pp. 334--354.

....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 similar to the assembly dag presented in ....

....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 ....

[Article contains additional citation context not shown here]

J. R. Gilbert and J. W. H. Liu. Elimination structures for unsymmetric sparse LU factors. Technical Report CS-90-11, Dept. of Computer Sci., York Univ., North York, Ontario, Feb. 1990.

....SuperLU time spent in AMD is very similar in both cases, this gives a good estimation of the cost di erence of the analysis phase of the two solvers. During the SuperLU analysis phase, all the unsymmetric structures involved during the factorization are computed and the directed acyclic graph [17] of the unsymmetric matrix must be built and mapped onto the processors. Path searches in the directed acyclic graph are used to reduce communications. With MUMPS, the main data structure handled during analysis is the assembly tree which is produced directly as a by product of the ordering ....

J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334-352, 1993.

....the transitive closure. If all such redundant edges are deleted, the result is called the transitive reduction. If A was structurally symmetric, this turns out to be the elimination tree mentioned above[15] otherwise GL 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 ....

....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 ....

J. Gilbert and J. Liu, Elimination structures for unsymmetric sparse LU factors, SIAM J. Matrix Anal. Appl., 14 (1993), no. 2, pp. 334--352.

....matrix of order n, the elimination tree is a tree on n nodes such that node j is the father of node i if entry (i; j) j i is the first entry below the diagonal in column i of the lower triangular factor. An analogous graph for an unsymmetric patterned sparse matrix is the directed acyclic graph [24, 54]. Sparse Cholesky factorization by columns can be represented by an elimination tree. This can either be a left looking (or fan in) algorithm, where updates are performed on each column in turn by all the previous columns that contribute to it, then the pivot is chosen in that column and the ....

J. R. Gilbert and J. W. H. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM J. Matrix Analysis and Applications, 14:334--354, 1993.

....the need for packing and unpacking in a send receive operation or sending many more smaller messages. In each send receive pair, two messages are exchanged, one for index[ and another for nzval[ To further reduce the amount of communication, we employ the notion of elimination dags (EDAGs) [7]. That is, we send the K th column of L rowwise to the process owning the J th column of L only if there exists a path between (super)nodes K and J in the elimination dags. This is done similarly for the columnwise communication of rows of U . Therefore, each block in L may be sent to fewer than P ....

J. R. Gilbert and J. W. Liu, Elimination structures for unsymmetric sparse LU factors, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 334--352.

....the need for packing and unpacking in a send receive operation or sending many more smaller messages. In each send receive pair, two messages are exchanged, one for index[ and another for nzval[ To further reduce the amount of communication, we employ the notion of elimination dags (EDAGs) [18]. That is, we send the K th column of L rowwise to the process owning the J th column of L only if there exists a path between (super)nodes K and J in the elimination dags. This is done similarly for the columnwise communication of rows of U . Therefore, each block in L may be sent to fewer than P ....

John R. Gilbert and Joseph W.H. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM J. Matrix Anal. Appl., 14(2):334--352, April 1993.

....denoted: E T . As E T really consists of two types of edges (L edges and U edges) it can be decomposed into two disjoint subsets: E L T , which contains only the L edges, and E U T , which contains only the U edges. This is similar to a formulation of elimination DAGs done by Gilbert and Liu [75], which is used to predict the structure of the LU factors. The edge set used to define the assembly DAG (which is provided as input) is called the assembly edge set and denoted by EA . In general, EA is a transitive 158 reduction of E T and it is frequently useful to think of it that way. ....

J. Gilbert and J. Liu. Elimination structures for unsymmetric sparse LU factors. Technical Report CS-90-11, Department of Computer Science, York University, Ontario, Canada, 1991.

....the set of vertices of G(A) from which there are paths to x. The structure of a vector v is defined as Struct(v) fijv i 6= 0g. In the following we will state results for the factor L Gamma1 . Similar results hold, of course, also for the factor U Gamma1 . These results were given in [17] [18]. The usual no cancellation assumption is made throughout. We denote by L Gamma1 ( i) the ith column of L Gamma1 . Proposition 3.1. Struct(L Gamma1 ( i) cl G(L) i) Let G ffi (L) denote the transitive reduction of the directed acyclic graph (dag) G(L) This is a graph with a ....

....(L) denote the transitive reduction of the directed acyclic graph (dag) G(L) This is a graph with a minimal number of edges which satisfies the following condition: G ffi (L) has a directed path from i to j if and only if G(L) has a directed path from i to j. Then the following result holds [18]. Proposition 3.2. Struct(L Gamma1 ( i) cl G ffi (L) i) We mention two simple consequences of this relation. Proposition 3.3. cl G ffi (L) i) cl G ffi (L) j) Struct(L Gamma1 ( i) Struct(L Gamma1 ( j) Proposition 3.4. Let K = cl G ffi (L) i) cl G ....

J. R. Gilbert and J. W. H. Liu, Elimination structures for unsymmetric sparse LU factors, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 334--352.

....matrix of order n, the elimination tree is a tree on n nodes such that node j is the father of node i if entry (i; j) j i is the first entry below the diagonal in column i of the triangular factors. An analogous graph for an unsymmetric patterned sparse matrix is the directed acyclic graph [55, 102]. The main property that we exploit in this tree is that computations corresponding to nodes that are not ancestors or descendants of each other are independent (see, for example, 77, 140] The tree can thus be used to schedule parallel tasks. For shared memory machines, this can be accomplished ....

J. R. Gilbert and J. W. H. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM J. Matrix Analysis and Applications, 14:334--354, 1993.

....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 ....

John R. Gilbert and Joseph W. H. Liu. Elimination structures for unsymmetric sparse LU factors. Technical Report CSL 90--11, Xerox Palo Alto Research Center, 1991. To appear in SIMAX.

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J. R. Gilbert and J. W. Liu. Elimination structures for unsymmetric sparse LU factors. SIAM Journal on Matrix Analysis and Applications, 14:334352, 1993.

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