T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Rep. TR-91-023, CISE Dept., Univ. of Florida, Gainesville, FL, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....has been some work on solving less regular problems. Research has recently been published in [32] that describes load balancing techniques to support the work in [31] Also, research has been ongoing to examine techniques that can efficiently factor irregular matrices using multifrontal techniques [5, 6, 7]. Techniques for sparse Choleski factorization have even been developed for single instructionmultiple data (SIMD) computers like the Thinking Machines CM 1 and the MasPar MPP [29] This discussion is by no means an exhaustive literature survey, although it does represent a significant portion of ....

T. A. David and I. S. Duff. Unsymmetric-Pattern Multifrontal Methods for Parallel Sparse LU Factorization. Technical Report TR-91-23, University of Florida, Computer and Information Sciences Department, September 1991.

....matrix, L, and an upper triangular matrix, U. Triangular solves (forward and backward substitutions) can then be used to solve the system. One approach to LU factorization that holds significant potential for parallel implementation is the unsymmetric pattern multifrontal method of Davis and Duff [34, 35]. Like most sparse matrix factorization algorithms, the unsymmetric pattern multifrontal method has two principal operations, analyze and factorize. The analyze operation selects matrix entries to act as pivots for the numerical factorization with objectives of reducing computations and ....

.... Specifically, the parallel algorithms developed will perform the numerical factorization of sequences of identically structured sparse matrices using a directed acyclic graph (DAG) structure produced by a sequential implementation of Davis and Duff s Unsymmetric Multifrontal Package (UMFPACK) [34, 35, 32]. This DAG structure, known as the assembly DAG, defines the necessary computations in terms of the partial factorizations of small, dense submatrices, which are known as frontal matrices and represented by nodes in the assembly DAG. The edges of the assembly DAG define the necessary data ....

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T. A. Davis and I. S. Duff. An unsymmetric-pattern multifrontal method for parallel sparse LU factorization. Technical Report TR-93-018, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1993. 211

....matrix, L, and an upper triangular matrix, U. Triangular solves (forward and backward substitutions) can then be used to solve the system. One approach to LU factorization that holds significant potential for parallel implementation is the unsymmetric pattern multifrontal method of Davis and Duff [34, 35]. Like most sparse matrix factorization algorithms, the unsymmetric pattern multifrontal method has two principal operations, analyze and factorize. The analyze operation selects matrix entries to act as pivots for the numerical factorization with objectives of reducing computations and ....

.... Specifically, the parallel algorithms developed will perform the numerical factorization of sequences of identically structured sparse matrices using a directed acyclic graph (DAG) structure produced by a sequential implementation of Davis and Duff s Unsymmetric Multifrontal Package (UMFPACK) [34, 35, 32]. This DAG structure, known as the assembly DAG, defines the necessary computations in terms of the partial factorizations of small, dense submatrices, which are known as frontal matrices and represented by nodes in the assembly DAG. The edges of the assembly DAG define the necessary data ....

[Article contains additional citation context not shown here]

T. A. Davis and I. S. Duff. Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization. Technical Report TR-91-023, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1991.

.... been reported in the power systems community journals to solve the special very sparse irregular power systems network matrices, there has been significant research into efficient general sparse linear solvers for general matrices, always larger and less sparse than power systems network matrices [8, 9, 10, 19, 20, 21, 25, 29, 46, 47, 55, 56, 57, 58, 64]. In the research presented in this thesis, we have developed specialized, efficient parallel sparse linear solvers for linear systems derived from power systems networks. The performance of our parallel linear solvers is significantly better than the performance of linear solvers reported in the ....

....has been some work on solving less regular problems. Research has recently been published in [47] that describes load balancing techniques to support the work in [46] Also, research has been ongoing to examine techniques that can efficiently factor irregular matrices using multifrontal techniques [8, 9, 10]. Techniques for sparse Choleski factorization have even been developed for single instructionmultiple data (SIMD) computers like the Thinking Machines CM 1 and the MasPar MPP [44] These techniques rely on regularity in the data to avoid processor load imbalance. Developing efficient parallel ....

T. A. David and I. S. Duff. Unsymmetric-Pattern Multifrontal Methods for Parallel Sparse LU Factorization. Technical Report TR-91-23, University of Florida, Computer and Information Sciences Department, September 1991.

....method, in which portions of the matrix are gathered into fronts for factoring, and these fronts make contributions to other fronts. The tasks in the DTG represent the fronts, and the links represent the contributions passed between fronts. In some analysisfactor multifrontal algorithms [4, 5], the tasks and their dependencies are determined during execution. In section 2, we present the basic concurrent DTG algorithm, and some extensions. In section 3, we explore algorithms for scheduling eligible tasks, and in section 4 we examine some performance issues. Finally, in section 5 we ....

T.A. Davis and I.S. Duff. Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization. Technical report, University of Florida, Dept. of CIS TR-91-23, 1991. Available at anonymous ftp site cis.ufl.edu:cis/tech-reports.

....the vector processing supercomputers to achieve Gigaflop processing rates. Little work has been performed on vectorizing data structures, so that symbolic processing is typically executed using only the scalar processor. Many algorithms, such those for as asymmetric sparse matrix factorization [3, 4, 6] require a significant amount of symbolic processing and use many data structures. Much of this code typically is executed using the scalar processor, and as a result can be relatively very time consuming. available via anonymous ftp at cis.ufl.edu:cis tech reports In this paper, we present a ....

....the matrix rows can shrink, as their entries are zeroed, or can grow due to fill in (non zeros created by the factorization) In Davis D2 algorithm [3] rows are stored in linked blocks, and hence could benefit from a vectorized buffer allocator. Other algorithms, known as multi frontal methods [4], build dense submatrices to factor. Allocation of the space for these submatrices can benefit from a vectorized heap memory allocator. Asymmetric sparse matrix factorization algorithms which calculate the pivoting sequence while they factor the matrix typically make extensive use of linked list ....

[Article contains additional citation context not shown here]

T.A. Davis and I.S. Duff. Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization. Technical report, University of Florida, Dept. of CIS TR-91-23, 1991. Available at anonymous ftp site cis.ufl.edu:cis/tech-reports.

....of these algorithms. One example is the work of Zlatev [41] who reduced the search space for the Markowitz heuristic by limiting the pivot search to rows of low degree. Recent work has been done to reduce the cost of calculating degrees in the minimum degree algorithm by using approximate degrees [5, 6, 25]. In other cases, attempts have been made to improve the quality of heuristics. Tinney and Walker [39] proposed a minimum de ciency version of the minimum degree algorithm. This actually minimizes real ll in at each stage. However, despite slight quality improvements, the time costs are ....

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Rep. TR-91-023, CISE Dept., Univ. of Florida, Gainesville, FL, 1991.

....19] and Liu [22] More recently, several researchers have relaxed this heuristic by computing upper bounds on the degrees, rather than the exact degrees, and selecting a node of minimum upper bound on the degree. This work includes that of Gilbert, Moler, and Schreiber [21] and Davis and Duff [2, 3]. Davis and Duff use degree bounds in the unsymmetric pattern multifrontal method (UMFPACK) an unsymmetric Markowitz style algorithm. In this paper, we describe an approximate minimum degree ordering algorithm based on the symmetric analogue of the degree bounds used in UMFPACK. Section 2 ....

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Report TR-91-023, CIS Dept., Univ. of Florida, Gainesville, FL, 1991.

....Appleton Laboratory, Chilton, Didcot, 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 ....

....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 matrix. Thus the algorithm can use higher level dense matrix kernels in its innermost loops (Level 3 BLAS [6] We refer to the unsymmetric pattern ....

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Report TR-91-023, CIS Dept., Univ. of Florida, Gainesville, FL, 1991.

....square and nonsingular. When the additional assumption of sparsity can be made, algorithmic modifications are possible that can significantly enhance execution times. Dr. Tim Davis, working with Dr. Iain Duff, has developed a very promising algorithm for solving such systems of linear equations [4]. Their method focuses on large, sparse, nonsingular systems that have an unsymmetric nonzero pattern. A most significant aspect of the method is the potential it has for parallel implementation. Current implementations of the method operate on sequential and vectorized processors. Parallel ....

....the scheduling of tasks. The results of the analysis just outlined illustrate a strong potential for parallelism with practical and realizable processor set sizes. 3 Chapter 2 Background This section briefly overviews some of the basic concepts of the unsymmetric pattern multifrontal method [4]. 2.1 LU Factorization LU factorization is a numerical algorithm based on Gaussian elimination for solving square, nonsingular systems of linear equations of the form: Ax = b, where A is an n by n real matrix and x; b are n dimensional real column vectors [7] Specifically, the coefficient ....

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Davis, T. A. and I. S. Duff, Unsymmetric-Pattern Multifrontal Methods for Parallel Sparse LU Factorization, Technical Report TR-91-23, Computer and Information Sciences Department, Univ. of Florida, Gainesville, FL 1991.

.... : 35 8 Performance results 36 9 Final remarks 38 10 Acknowledgments 46 3 List of Figures 1 A possible subgraph that characterizes Equation 7 : 12 2 Assembly graph D [2] A [2] G [2] F [2] for matrix A in Equation 6 : 15 3 Assembly graph D [4] = A [4] G [4] F [4] for matrix A in Equation 6 : 17 4 Assembly graph D [5] A [5] G [5] F [5] for matrix A in Equation 6 : 18 5 Fill in due to amalgamation between LU child and LU parent : 22 6 Fill in due to amalgamation between ....

.... results 36 9 Final remarks 38 10 Acknowledgments 46 3 List of Figures 1 A possible subgraph that characterizes Equation 7 : 12 2 Assembly graph D [2] A [2] G [2] F [2] for matrix A in Equation 6 : 15 3 Assembly graph D [4] A [4] ; G [4] F [4] for matrix A in Equation 6 : 17 4 Assembly graph D [5] A [5] G [5] F [5] for matrix A in Equation 6 : 18 5 Fill in due to amalgamation between LU child and LU parent : 22 6 Fill in due to amalgamation between L child and ....

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T. A. Davis and I. S. Duff. Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization. Technical Report TR-91-023, CIS Dept., Univ. of Florida (anonymous ftp to cis.ufl.edu:cis/tech-reports/tr91/tr91-023.ps.Z), Gainesville, FL, 1991.

.... davis cis.ufl.edu Technical Report TR 94 027, Computer and Information Sciences Department, University of Florida April 19, 1994 This project is supported the National Science Foundation (ASC 9111263, DMS 9223088) 1 2 Abstract The unsymmetric pattern multifrontal method of Davis and Duff [2] generalizes earlier multifrontal approaches to LU factorization by removing the assumption of a symmetric pattern of nonzeros in the sparse matrix. As a result, the underlying computational structure becomes a directed acyclic graph (DAG) instead of a tree. This research explores the potential ....

....method applied to symmetric pattern matrices using analytical models based on unbounded parallelism. In this work, we explore the both the available and achievable parallelism of a new multifrontal method for the LU factorization of unsymmetric pattern matrices developed by Davis and Duff [2]. First some of the key concepts of the unsymmetric pattern multifrontal method are outlined. Then the available parallelism is determined using unbounded parallelism models similar to those of Duff and Johnsson [5] These models are refined into bounded parallelism models to determine how much ....

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T. A. Davis and I. S. Duff. An unsymmetric-pattern multifrontal method for parallel sparse LU factorization. Technical Report TR-93-018, Computer and Info. Sci. Dept., University of Florida, Gainesville, FL, 1993.

....Parallelism in the Unsymmetric Pattern Multifrontal LU Factorization Method for Sparse Matrices Tech. Report TR 94 006; Proc. Fifth SIAM Conf. on Applied Linear Algebra, June 1994 Steven M. Hadfield and Timothy A. Davis y Abstract The unsymmetric pattern multifrontal method of Davis and Duff [1] generalizes earlier multifrontal approaches to LU factorization by removing the assumption of a symmetric pattern of nonzeros in the sparse matrix. As a result, the underlying computational structure becomes a directed acyclic graph (DAG) instead of a tree. This research explores the potential ....

....to investigate the achievable parallelism. Finally, a factorization only version of the method is implemented on the nCUBE 2 and its achieved parallelism and scalability evaluated. 1 Unsymmetric Pattern Multifrontal Method Multifrontal methods for the LU factorization of sparse matrices [1, 4, 5, 8] decompose the sparse matrix into a set of overlapping dense submatrices called frontal matrices. Each frontal matrix contains one or more pivots and is partially factorized according to these pivots. The updated entries in the unfactorized portion of a frontal matrix (called the contribution ....

T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for parallel sparse LU factorization, Tech. Rep. TR-93-018, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1993.

....memory access and that do not use dense matrix kernels. We discuss unifrontal methods in Section 2. We summarize the multifrontal method in Section 3, and in particular our earlier work on an unsymmetric pattern multifrontal method. We refer to this prior method as UMFPACK V1.1 (Davis 1995, Davis and Duff 1991, Davis and Duff 1997) The combination of unifrontal and multifrontal methods is discussed in Section 4. The combined algorithm is based on UMFPACK V1.1 and the new frontal matrix strategy discussed here. This combined algorithm is available in Release 12 of the Harwell Subroutine Library (HSL ....

....normally be factorized but only a few steps of Gaussian elimination are possible, after which the Schur complement F 22 GammaF 21 F Gamma1 11 F 12 (contribution block) needs to be summed (assembled) with other data at the parent node. In the unsymmetric pattern multifrontal method (Davis 1995, Davis and Duff 1991, Davis and Duff 1997) the ordering, 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. For example, consider the LU 7 5 4 7 6 ....

[Article contains additional citation context not shown here]

Davis, T. A. and Duff, I. S. (1991), Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Technical Report TR-91-023, CISE Dept., Univ. of Fl., Gainesville, FL.

....20] and Liu [23] More recently, several researchers have relaxed this heuristic by computing upper bounds on the degrees, rather than the exact degrees, and selecting a node of minimum upper bound on the degree. This work includes that of Gilbert, Moler, and Schreiber [22] and Davis and Duff [3, 4]. Davis and Duff use degree bounds in the unsymmetric pattern multifrontal method (UMFPACK) an unsymmetric Markowitz style algorithm. In this paper, we describe an approximate minimum degree ordering algorithm based on the symmetric analogue of the degree bounds used in UMFPACK. Section 2 ....

.... 9 10 7 4 3 5 6 8 9 10 7 1 4 3 5 6 8 9 10 2 7 1 4 3 5 6 8 9 10 2 7 1 4 3 5 6 8 9 10 2 7 1 4 3 5 6 8 9 10 2 7 7 4 3 5 6 8 9 10 2 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 1 (a) Elimination graph (c) Factors and active submatrix (b) Quotient graph [3, 4], d k i = min 8 : n Gamma k d k Gamma1 i jL p n V i j jA i n V i j jL p n V i j X e2E i nfpg jL e n L p j (4.1) where d k i d k i . Here we assume that p is the kth pivot, and that we compute the bounds only for variables i 2 L p . Algorithm 3 computes jL e n L p j for ....

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Report TR-91-023, CIS Dept., Univ. of Florida, Gainesville, FL, 1991. APPROXIMATE MINIMUM DEGREE 19

....typically the most costly part of a minimum degree algorithm, is not required in the RCM algorithm. However, for matrices with large profile, the frontal matrix can be large, and an unacceptable level of fill in can occur. 3. Multifrontal methods. In a multifrontal scheme for a symmetric matrix [2, 8, 9, 10, 17, 18, 28], it is normal to use an ordering such as minimum degree to reduce the fill in. Such an ordering tends to reduce fill in much more than profile reduction orderings. The ordering is combined with a symbolic analysis to generate an assembly or computational tree, where each node represents the ....

....possible, after which the remaining reduced matrix (the Schur complement (F 22 F 23 ) Gamma F 21 F Gamma1 11 (F 12 F 13 ) needs to be summed (assembled) with other data at the parent node. In the unsymmetric pattern multifrontal method, the tree is replaced by a directed acyclic graph (dag) [9], and a contribution block may be assembled into more than one subsequent frontal matrix. 4. Combining the two methods. Let us now consider an approach that combines some of the best features of the two methods. Assume we have chosen a pivot and determined a frontal matrix as in a normal ....

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Report TR-91-023, CIS Dept., Univ. of Florida, Gainesville, FL, 1991.

....that have irregular memory access and that do not use dense matrix kernels. We discuss unifrontal methods in Section 2. We summarize the multifrontal method in Section 3, and in particular our earlier work on an unsymmetric pattern multifrontal method. We refer to this prior method as UMFPACK V1.1 [8, 10, 11]. The combination of unifrontal and multifrontal methods is discussed in Section 4. The combined algorithm is based on UMFPACK V1.1 and the new frontal matrix strategy discussed here. This combined algorithm is available in Release 12 of the Harwell Subroutine Library [31] as the package MA38. In ....

....(1) cannot normally be factorized but only a few steps of Gaussian elimination are possible, after which the Schur complement F 22 Gamma F 21 F Gamma1 11 F 12 (contribution block) needs to be summed (assembled) with other data at the parent node. In the unsymmetric pattern multifrontal method [8, 10, 11], the ordering, 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. For example, consider the LU factors of a matrix from [11] L ....

[Article contains additional citation context not shown here]

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Report TR-91-023, CISE Dept., Univ. of Fl., Gainesville, FL, 1991.

....22] and Liu [25] More recently, several researchers have relaxed this heuristic by computing upper bounds on the degrees, rather than the exact degrees, and selecting a node of minimum upper bound on the degree. This work includes that of Gilbert, Moler, and Schreiber [24] and Davis and Duff [7, 8]. Davis and Duff use degree bounds in the unsymmetric pattern multifrontal method (UMFPACK) an unsymmetric Markowitz style algorithm. In this paper, we describe an approximate minimum degree ordering algorithm based on the symmetric analogue of the degree bounds used in UMFPACK. Section 2 ....

....approximation for the minimum degree and indicate its lower complexity. We assume that p is the kth pivot, and that we compute the bounds only for supervariables i 2 L p . Rather than computing the exact external degree, d i , our Approximate Minimum Degree algorithm (AMD) computes an upper bound [7, 8], d k i = min 8 : n Gamma k; d k Gamma1 i jL p n ij; jA i n ij jL p n ij X e2E i nfpg jL e n L p j 9 = 4.1) The first two terms (n Gamma k, the size of the active submatrix, and d k Gamma1 i jL p n ij, the worst case fill in) are usually not as tight as ....

T. A. Davis and I. S. Duff, Unsymmetric-pattern multifrontal methods for parallel sparse LU factorization, Tech. Report TR-91-023, CIS Dept., Univ. of Florida, Gainesville, FL, 1991.

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