Esmond G. Ng amd Padma Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM Journal on Matrix Analysis and Applications, To appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....algorithms in wide use are based on a greedy approach such that the ordering is chosen to minimize some quantity at each step of a simulated n step symmetric Gaussian elimination process. The algorithms using such an approach are typically distinguished by their greedy minimization criteria [11]. 4.1 Graph Models In 1961, Parter introduced the graph model of symmetric Gaussian elimination [12] A sequence of elimination graphs represent a sequence of Gaussian elimination steps. The initial Elimination graph is the original graph for matrix A. The elimination graph of k s step is ....

....Degree (AMD) algorithm uses an estimate of the degree (or external degree) of a vertex [14] The Minimum Deficiency class of algorithms instead choose the vertex that would create the minimum number of fill in elements. A nice comparison of many of these different approaches can be found in [11]. 7 5 Implementation Our GGCL based implementation of MMD closely follows the algorithmic descriptions of MMD given, e.g. in [15, 13] The implementation presently includes the enhancements for mass elimination, incomplete degree update, multiple elimination, and external degree. In addition, ....

Esmond G. Ng amd Padma Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM Journal on Matrix Analysis and Applications, To appear.

....These solvers are generally complex pieces of software with many years of development effort invested in them. MFACT is another example for sparse, symmetric, positive definite systems, written by Raghavan and Ng [10] It uses an ordering heuristic called modified multiple minimum degree (MMMD) [11] which offers significant improvements in factorisation speed. We used the public domain serial version of this library as a base and re engineered it for PZFlex. Section 5 offers a performance comparison of these two solver approaches for a range of piezoelectric models. 3 A rapid ....

Ng, E.G. and Raghavan, P., "Performance of Greedy Ordering Heuristics for Sparse Cholesky Factorization", SIAM Journal of Matrix Analysis and Applications, Vol 20, No 4, pp 902--914, 1999.

....algorithms in wide use are based on a greedy approach such that the ordering is chosen to minimize some quantity at each step of a simulated n step symmetric Gaussian elimination process. The algorithms using such an approach are typically distinguished by their greedy minimization criteria [20]. In graph terms, the basic ordering process used by most greedy algorithms is as follows: 1. Start: Construct undirected graph G 0 corresponding to matrix A 2. Iterate: For k =1; 2; until G k = do: # Choose a vertex v k from G k according to some criterion 43 template #class ....

E. G. Ng and P. Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM Journal on Matrix Analysis and Applications,To appear.

.... Approximate Minimum Degree Amestoy, Davis and Du# [1] AMF Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and Raghavan [6] MMMD Modified Multiple Minimum Degree Ng and Raghavan [6] connecting vertices i and j in G exists if and only if a ij is nonzero. By symmetry, a j,i is also nonzero. The graph model of symmetric Gaussian elimination was introduced by Parter [7] A sequence of elimination graphs, G k , ....

.... Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and Raghavan [6] MMMD Modified Multiple Minimum Degree Ng and Raghavan [6] connecting vertices i and j in G exists if and only if a ij is nonzero. By symmetry, a j,i is also nonzero. The graph model of symmetric Gaussian elimination was introduced by Parter [7] A sequence of elimination graphs, G k , represent the fill created in each step of the factorization. The ....

Esmond G. Ng and Padma Raghavan, Performance of greedy ordering heuristics for sparse Cholesky factorization, submitted to SIAM J. Mat. Anal. Appl. (1997).

.... factorization and triangular solution using static data structures for the Cholesky factor (of the reordered matrix) obtained from step (ii) The numeric factorization step is the most expensive; the symbolic steps typically require a small fraction of the time required for numeric factorization [1, 7, 6, 8, 9, 13, 15, 18]. A simple column by column implementation with columns in # stored using a standard sparse storage scheme is very inecient on modern computers with deep cachehierarchies. The ineciency stems from indirect addressing and disregard for data locality leading to both a larger number of cache misses ....

E. G. Ng and P. Raghavan. The performance of greedy ordering heuristics for sparse Cholesky factorization. #### ## ###### ##### #####, 20(4):902-914, 1999.

....ordering, P 1 , reduces fill and the second ordering, P 2 , is a 1. Designing Object Oriented Sparse Direct Solvers 9 Abbreviation. Algorithm Name MMD Multiple Minimum Degree [LIU85] AMD Approximate Minimum Degree [ADD96] AMF Approximate Minimum Fill [ROT96] MMDF Modified Minimum Deficiency [NR97] MMMD Modified Multiple Minimum Degree [NR97] AMMF Approximate Minimum Mean Local Fill [RE98] AMIND Approximate Minimum Increase in Neighbor Degree [RE98] TABLE 1.1. Some of the algorithms in the Minimum Priority family. modification of P 1 to preserve numerical stability. In general, any ....

....ordering, P 2 , is a 1. Designing Object Oriented Sparse Direct Solvers 9 Abbreviation. Algorithm Name MMD Multiple Minimum Degree [LIU85] AMD Approximate Minimum Degree [ADD96] AMF Approximate Minimum Fill [ROT96] MMDF Modified Minimum Deficiency [NR97] MMMD Modified Multiple Minimum Degree [NR97] AMMF Approximate Minimum Mean Local Fill [RE98] AMIND Approximate Minimum Increase in Neighbor Degree [RE98] TABLE 1.1. Some of the algorithms in the Minimum Priority family. modification of P 1 to preserve numerical stability. In general, any modification of P 1 increases fill and, hence, the ....

E. G. Y. Ng and P. Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. Technical Report, Computer Science Department, University of Tennessee, Knoxville, 1997.

.... AMD Approximate Minimum Degree Amestoy, Davis and Duff [1] AMF Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and Raghavan [6] MMMD Modified Multiple Minimum Degree Ng and Raghavan [6] Table 1. Algorithms that fit into the Minimum Priority family. the vertex with the minimum degree in the current elimination graph, hence reducing fill by controlling this worst case bound. In Multiple Minimum Degree (MMD) a maximal ....

.... AMF Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and Raghavan [6] MMMD Modified Multiple Minimum Degree Ng and Raghavan [6] Table 1. Algorithms that fit into the Minimum Priority family. the vertex with the minimum degree in the current elimination graph, hence reducing fill by controlling this worst case bound. In Multiple Minimum Degree (MMD) a maximal independent set of vertices of low degree are eliminated in ....

Esmond G. Ng and Padma Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. Submitted to SIAM J. Mat. Anal. Appl., 1997.

....pivot is selected; selecting the pivot with least de ciency leads to a minimum de ciency ordering algorithm. Exact de ciency is very costly to compute. Approximate minimum de ciency ordering algorithms have been successfully used for symmetric matrices, in the context of Cholesky factorization [40, 42]. In an nonsymmetric context, the de ciency of column c can be bounded by kC c kkR r k 0 X i2Cc j[i]j(kR i k j[c]j) 1 A : Any new nonzeros are limited to the kC c k by kR r k Householder update. Each super row i 2 C c , however, is contained in this submatrix and thus reduces the ....

E. G. Ng and P. Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM J. Matrix Anal. Applic., 20(4):902{ 914, 1999.

....of A. Consequently, the first step in sparse Cholesky factorization is that of ordering to reduce fill. The second step is that of symbolic factorization; this step determines the zero nonzero structure of L. A variety of efficient, graph theoretic algorithms exist for the ordering step [1, 13, 19] and the symbolic factorization steps [7, 14] The third and the fourth steps are numeric, namely, numeric factorization and triangular solution using the precomputed data structures. Consider the zero nonzero structure of L for a given ordering of A. Columns of L can naturally be grouped into ....

E. G. Ng and P. Raghavan. The performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM J. Matrix Anal. Appl., 1998. To appear.

....[7] First, the minimum deficiency algorithm is typically much more expensive than the minimum degree algorithm. Second, it has been believed that the quality of minimum deficiency orderings is not much better than that of minimum degree orderings [7] Results by Rothberg [19] and also by us [17]) demonstrate that minimum deficiency leads to significantly better orderings than minimum degree. However, current implementations of the minimum deficiency algorithm require substantially more time than MMD. In this paper, we develop two greedy heuristics that are less expensive to compute than ....

....on the performance of MMDF and MMMD. Section 5 contains some concluding remarks. The remaining part of this section describes recent related work. Related work. Rothberg has investigated metrics for greedy ordering schemes based on approximations to the deficiency [19] His work and our work [17] were done independently of each other. 1 Rothberg [19] ffl shows that the minimum deficiency algorithm is significantly superior to MMD in terms of the number of operations required to compute the Cholesky factor, ffl develops three approximate minimum fill (AMF) heuristics based on ....

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E. G.-Y. Ng and P. Raghavan, Performance of greedy ordering heuristics for sparse cholesky factorization, Tech. Rep. CS-97-XX, University of Tennessee, Knoxville, TN, 1997.

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Esmond G. Ng amd Padma Raghavan. Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM Journal on Matrix Analysis and Applications, To appear.

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