T. A. Davis and I. S. Du#. An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM Journal on Matrix Analysis and Applications, 19(1):140--158, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... IBM T.J.W atson Research Center, P.O. Box 218, Yorktown Heights, NY 10598 (anshul watson. ibm.com) 529 matrix for a give peA utation of rows and columns; howeeU the data DAG is a function ofthe sparse factorization algorithm. Multifrontal algorithms [9, 14, 23] for sparse factorization can work with a minimal data DAG(i.e0 a data DAG withthe smalleL possible numb e ofewA9L for a give matrix. Inthe case ofsymmeO0w sparse matricew the minimal task and data DAGs for the factorization pro cew are atre calle the eeLw[OO] tre [22] Howe e for unsymmeDA ] ....

.... 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 ....

[Article contains additional citation context not shown here]

T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J.Matrix Anal.Appl., 18 (1997), pp.140--158.

....the solution of symmetric sparse systems of linear equations, can be obtained from http: www.cs.umn.edu agupta wsmp.html, along with some example programs and technical papers related to the software. For solving general sparse systems, WSMP uses a modified version of the multifrontal algorithm [11, 1] for matrices with an unsymmetric pattern of nonzeros. WSMP supports threshold partial pivoting for general matrices with a user defined threshold. Detailed performance results of WSMP and a comparison of various general sparse solver packages can be found in [9] A discussion of the algorithms ....

Timothy A. Davis and Iain S. Duff. An unsymmetric-pattern multifrontal method for sparse LU factorization. Technical Report TR-93-018, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1993.

....gap. Note that two matrices have been omitted since they fit approximately within the large (8 MB) L2 cache. 4 Related work Sparse triangular solve is a key component in many of the existing serial and parallel direct sparse factorization libraries (e.g. SuperLU [14] MUMPS [2] UMFPACK [13], PSPASES [24] and SPOOLES [5] among others) These libraries have focused primarily on speeding up the factorization step, and employ sophisticated methods for creating dense structure. E#orts to speedup the triangular solve step in these and other work [33, 32, 25, 18, 26, 36, 19, 1, 30] have ....

T. A. Davis and I. S. Du#. An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM Journal on Matrix Analysis and Applications, 19(1):140--158, 1997.

....Berkeley National Laboratory, 1 Cyclotron Rd. Berkeley CA 94720 NERSC, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. Berkeley CA 94720 structures are more costly and complicated to handle than a tree, although they are better at capturing the asymmetry of the matrix. Davis and Du [6] implicitly use this structure to drive their unsymmetricpattern multifrontal approach. We explain, in this article, how to use the simple elimination tree structure of the symmetric matrix M to detect, during the numerical factorization phase, structural asymmetry in the factors. We show that ....

....performance gains (in terms of size of the factors, memory requirement and factorization time) of the new approach with respect to the standard multifrontal code on these matrices. In Section 4, we compare the performance of our approach with the unsymmetricpattern multifrontal approach (UMFPACK, [6, 7]) and with the supernodal partial pivoting code (SuperLU, 9] We add some concluding remarks in Section 5. 2. Description of the multifrontal factorization algorithms. Let A be an unsymmetric matrix and let M denote its symmetrized form. The structure of M is Struct(A) Struct(A ) where ....

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T. A. Davis and I. S. Du. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM Journal on Matrix Analysis and Applications, 18:140-158, 1997.

....for the degree of r. However, the s adjacency sets of the enodes in eadj(r) might overlap, making the bound too loose, and causing a large gap between the real degree and the approximated degree bound of r. This gap can be reduced by computing a quantity diff(e) associated with each enode [1, 2] to remove some of the overlap in the adjacency sets. Let u k be the snode that is eliminated at step k. It is then merged with all of its e neighbors, and the weight of the new giant enode u k in the quotient graph G k becomes the sum of the weights of all the snodes r 2 reach G k Gamma1 (u k ....

T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Anal. Appl., 18 (1996), pp. 140--158.

....through your web browser. Some technical papers related to the software and the example programs in Appendices A C can be obtained from the site http: www.cs.umn.edu agupta wsmp.html. For solving general sparse systems, WSMP uses a modified version of the multifrontal algorithm [11, 1] for matrices with an unsymmetric pattern of nonzeros. WSMP supports threshold partial pivoting for general matrices with a user defined threshold. WSMP automatically exploits SMP parallelism on an RS6000 workstation or SP node with multiple CPUs and this parallelism is transparent to the user. On ....

Timothy A. Davis and Iain S. Duff. An unsymmetric-pattern multifrontal method for sparse LU factorization. Technical Report TR-93-018, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1993.

....matrix. An edge from a vertex i to a vertex j in the data DAG denotes that at least some of the output data of task i is required as input by task j. While the task DAG is unique to a given sparse matrix, the data DAG is a function of the sparse factorization algorithm. Multifrontal algorithms [13, 23, 8] for sparse factorization can work with a minimal data DAG (i.e. a data DAG with the smallest possible number of edges) for a given matrix. In the case of symmetric sparse matrices, the minimal task and data DAG for the factorization process is a tree called the elimination tree [22] However, ....

....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. ....

[Article contains additional citation context not shown here]

Timothy A. Davis and Iain S. Duff. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM Journal on Matrix Analysis and Applications, 18(1):140--158, January 1997.

....and let numerical consideration dictate the row ordering. Since columns are reordered before the row permutation is known, we need to order the columns such that fill is minimized no matter how rows are exchanged. Some nonsymmetric factorization codes that employ pivoting, such as umfpack ma38 [4, 5], determine the column permutation during the numerical factorization; such codes do not preorder columns so the technique in this paper does not apply to them. A result by George and Ng [8] suggests one e#ective way to preorder the columns to reduce fill. They have shown that the fill of the LU ....

T. A. Davis and I. S. Du#. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM Journal on Matrix Analysis and Applications, 19:140--158, 1997.

....it is the block diagonal matrix. We conclude this section remarking that the considered technique may be applied to parallelization of any method which is dicult or impossible to parallelize. For instance, in IPARS, the frequency correction is applied to parallelization of a sparse factorization [6] for the pressure block, the AMG preconditioner [5] 19] and the MLILU method [2] 5 Numerical experiments In this section, we consider the parallel performance of solvers applied to the hydrology model. We have made 15 day simulation (15 time steps) on a mesh with 1000000 grid cells (40 200 ....

Davis T. and Du I., An Unsymmetric-Pattern Multifrontal Method for Sparse LU Factorization, SIAM J. Matrix Analysis and Applications, V.19, No.1, 140-158, 1997

....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 structure of the symmetric matrix M to detect, during the numerical factorization phase, structural asymmetry in the factors. We show that, ....

....structure. Therefore, it produces LU factors such that the matrix F = L U is symmetric in structure. This approach is currently used in the context of two publically available packages (#### [2,3] and ##### [5, 4] and has the advantage, with respect to other unsymmetric factorization algorithm [6, 7, 17], of having the LU factorization based on the processing of an assembly tree, while the other approaches explicitly or implicitly use a more complex to handle graph structure. Wehave demonstrated that, based on the same assembly tree, one can derive a new multifrontal algorithm that will ....

T. A. Davis and I. S. Du. An unsymmetric-pattern multifrontal method for sparse LU factorization. #### ####### ## ###### ######## ### ############, 18:140-158, 1997.

....and let numerical consideration dictate the row ordering. Since columns are reordered before the row permutation is known, we need to order the columns such that fill is minimized no matter how rows are exchanged. Some nonsymmetric factorization codes that employ pivoting, such as UMFPACK MA38 [2, 3], determine the column permutation during the numerical factorization; such codes do not preorder columns so the technique in this paper does not apply to them. A result by George and Ng [6] suggests one e#ective way to preorder the columns to reduce fill. They have shown that the fill of the LU ....

T. A. Davis and I. S. Du#. An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM Journal on Matrix Analysis and Applications, 19:140--158, 1997.

....and let numerical consideration dictate the row ordering. Since columns are reordered before the row permutation is known, we need to order the columns such that fill is minimized no matter how rows are exchanged. Some nonsymmetric factorization codes that employ pivoting, such as UMFPACK MA38 [3, 4], determine the column permutation during the numerical factorization; such codes do not preorder columns so the technique in this paper does not apply to them. A result by George and Ng [7] suggests one e#ective way to preorder the columns to reduce fill. They have shown that the fill of the LU ....

T. A. Davis and I. S. Du#. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM Journal on Matrix Analysis and Applications, 19:140--158, 1997.

....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 structure of the symmetric matrix M to detect, during the numerical factorization phase, structural asymmetry in the factors. We show that, ....

....structure. Therefore, it produces LU factors such that the matrix F = L U is symmetric in structure. This approach is currently used in the context of two publically available packages (ma41 [2, 3] and MUMPS [5, 4] and has the advantage, with respect to other unsymmetric factorization algorithms [6, 7, 17], of having the LU factorization based on the processing of an assembly tree, while the other approaches explicitly or implicitly use a more complex to handle graph structure. We have demonstrated that, based on the same assembly tree, one can derive a new multifrontal algorithm that will ....

T. A. Davis and I. S. Du. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM Journal on Matrix Analysis and Applications, 18:140-158, 1997.

....are generally smaller and denser than in the frontal method. The classical multifrontal approach (Duff and Reid, 1984) has met with only limited success when the pattern of nonzeros is highly unsymmetric. However, recently a new unsymmetric pattern multifrontal algorithm has been described by Davis and Duff (1993,1996) and implemented in the code 3 UMFPACK (Version 1.1) Our experiments (Zitney et al. 1996) have shown that the multifrontal algorithm used by UMFPACK V1.1 does not perform as well as the frontal algorithm (FAMP) on many equilibrium stage separation problems, primarily because it is not able to ....

....a lower triangular matrix and U is an upper triangular matrix. Thus, Ax = LU)x = L(Ux) b, and the system can be solved by a simple forward substitution to solve Ly = b for y, followed by a back substitution to find the solution vector x from Ux = y. In an unsymmetric pattern multifrontal method (Davis and Duff, 1993,1996), a frontal matrix, consisting of pivot row(s) and column(s) their entries from the original matrix A, and contributions to them from previous frontal matrices, is assembled at each stage of the factorization process. The frontal matrix E k for steps k through k g k Gamma 1 of the LU ....

[Article contains additional citation context not shown here]

Davis, T. A.; Duff, I. S. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl. 1996, in press, (available as Technical Report TR-94-038; see http://www.cise.ufl.edu/research/tech-reports).

....are generally smaller and denser than in the frontal method. The classical multifrontal approach (Duff and Reid, 1984) has met with only limited success when the pattern of nonzeros is highly unsymmetric. However, recently a new unsymmetric pattern multifrontal algorithm has been described by Davis and Duff (1993,1996) and implemented in the code 3 UMFPACK (Version 1.1) Our experiments (Zitney et al. 1996) have shown that the multifrontal algorithm used by UMFPACK V1.1 does not perform as well as the frontal algorithm (FAMP) on many equilibrium stage separation problems, primarily because it is not ....

....a lower triangular matrix and U is an upper triangular matrix. Thus, Ax = LU)x = L(Ux) b, and the system can be solved by a simple forward substitution to solve Ly = b for y, followed by a back substitution to find the solution vector x from Ux = y. In an unsymmetric pattern multifrontal method (Davis and Duff, 1993,1996) a frontal matrix, consisting of pivot row(s) and column(s) their entries from the original matrix A, and contributions to them from previous frontal matrices, is assembled at each stage of the factorization process. The frontal matrix E k for steps k through k g k Gamma 1 of the LU ....

[Article contains additional citation context not shown here]

Davis, T. A.; Duff, I. S. An unsymmetric-pattern multifrontal method for sparse LU factorization. Tech. Rep. TR-93-018, CIS Department, University of Florida, Gainesville, FL, 1993 (see http://www.cise.ufl.edu/research/tech-reports).

....the context of fill reduction. In the fill reduction context, there has been considerable effort (e.g. Duff and Reid, 1983; Eisenstat et al. 1981; George and Liu, 1980a,b; Liu, 1985) spent on reducing the work required to keep track of the degrees, and recently work (e.g. Gilbert et al. 1992; Davis and Duff, 1997; Davis et al. 1996) has 17 concentrated on using approximations of (generally upper bounds on) the degrees, in order to further reduce computational requirements. Our implementations do not employ these useful approximations and should not be considered as the best that could be achieved with ....

Davis, T. A. and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl., 18, 140--158 (1997).

....amalgamation clusters columns and rows simultaneously using structural containment information implied by an elimination forest. Our previous design S [10, 11] does not consider the bounding of nonzeros in the U part. We compare 4 We did not compare with another well optimized package UMFPACK [2] because SuperLU has been shown competitive to UMFPACK [4] 15 Matrix P=8 P=16 P=32 P=64 P=128 S S S S S S S S S S goodwin 215.2 403.5 344.6 603.4 496.3 736.0 599.2 797.3 715.2 826.8 e40r0100 205.1 443.2 342.9 727.8 515.8 992.8 748.0 1204.8 930.8 1272.8 raefsky4 391.2 ....

T. Davis and I. S. Duff, An Unsymmetric-pattern Multifrontal Method for Sparse LU factorization, SIAM Matrix Analysis & Applications, (1997).

....symbolic analysis, and numerical factorization are performed at the same time. The tree is replaced by a directed acyclic graph (DAG) A contribution block may be assembled into more than one subsequent frontal matrix. The rst pivot within a frontal matrix (called the seed pivot by Davis and Du [22]) de nes its size. This new frontal matrix is held in a larger working array, to allow room for the assembly of subsequent pivot rows and columns. The next pivots can either be taken from the fully summed part, or the non fully summed part. If a potential pivot lies in the non fully summed part of ....

T. A. Davis, I. S. Du: An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Applic. 18 (1997), pp. 140{ 158. 95

....disappointing in this respect. In Version 2 of Umfpack a combined unifrontal multifrontal approach (see Section 2.4.2) is used to combine the advantages of both methods. Version 1 was based on a multifrontal method. A detailed description of the algorithms used in Umfpack is given in Davis, Du [21]. Umfpack can be obtained from the Netlib; the directory linalg contains the le umfpack.shar. Umfpack is also part of the Harwell Subroutine Library [1] Umfpack Version 2 implements the same algorithm as MA38 from the Harwell library. Although the routines have di erent names, the user ....

T. A. Davis, I. S. Du: An Unsymmetric-Pattern Multifrontal Method for Sparse LU Factorization. Technical Report TR-94-038, Computer and Information Science Dept., University of Florida, 1994.

....to other minimum external degree algorithms in terms of llin and the number of oating point operations needed to compute the Cholesky factorization. A full account of this method is given by Amestoy et al. 3] MC47 is the symmetric analogue of the ordering used in the code of Davis and Du [20]. 1.3.2 METIS Metis is a software package for partitioning large irregular graphs, partitioning large meshes, and computing ll in reducing orderings of sparse matrices. The algorithms in Metis are based on multilevel graph partitioning techniques described in Karypis et al. 62] Metis is ....

T. A. Davis, I. S. Du: An unsymmetric-pattern multifrontal method for sparse LU factorization. Technical Report RAL 93-036, Rutherford Appleton Laboratory, 1993.

....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 faster than SuperLU and also faster than the multi frontal code UMFPACK [2]. The same parameters are used as in section 8.1. But for the non circuit simulation matrices 12 Liu s minimum degree ordering [19] is applied to C instead of Matlab s symmmd. The results on the SGI Power Challenge are showed in Table 4. Table 5 shows the parallel results for the SGI Origin 200. ....

T.A. Davis, I.S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl., 18(1) (1997), pp. 140-158.

.... algorithm for ordering symmetric matrices prior to a Cholesky factorization is based on a bound on the external row degree that is tighter than the Matlab bound [1] It was rst used in the nonsymmetricpattern multifrontal method (UMFPACK) to compute the ordering during numerical factorization [8, 9]. In the context of a column ordering algorithm to bound the ll in for the sparse partial pivoting method, the bound on kR r k is kR r k kR s n fcgk X i2Ccnfsg (kR i n R s k) 3) where R s is the most recent pivot row that modi ed C c in the symbolic update, and thus s 2 C c . To compute ....

T. A. Davis and I. S. Du. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Applic., 18(1):140{ 158, 1997.

No context found.

T. A. Davis and I. S. Du#. An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM Journal on Matrix Analysis and Applications, 19(1):140--158, 1997.

No context found.

T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Anal. Appl., 18 (1996), pp. 140--158.

No context found.

T. Davis and I. S. Duff. An Unsymmetric-Pattern Multifrontal Method for Sparse LU Factorization. SIAM Matrix Analysis & Applications, January 1997.

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