A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1--19, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....techniques for the ordering phase can be found in [3, 12, 16, 17, 18, 19, 24] The principal ordering technique used for reordering the general sparse matrices for regular LU factorization algorithms involves two stages. In the rst stage, a ll reducing ordering, such as minimum degree ordering [5], is applied to the coecient matrix A. This is followed by application of Liu s scheme of elimination tree rotation [18, 19] which causes a reduction in the height of the elimination tree without a ecting the amount of ll in in the upper triangular factor U . The resulting elimination tree is ....

....4. Numerical factorization : The parallel fan out BSF algorithm described in section 2. Substitution :The parallel BS algorithm described in section 3 of Part I. In the regular scheme, we implemented each of the four phases as follows. Ordering : The ll reducing minimum degree ordering [5] followed by Liu s elimination tree rotation scheme [18] Symbolic factorization : The sequential symbolic factorization algorithm presented in [7] Numerical factorization : The parallel fan out algorithm given in [2, 15] Substitution :The elimination tree based forward and back ....

A.George and J.W.H.Liu, The evolution of minimum degree ordering algorithm, SIAM Review, Vol. 31, No. 1, 1989, pp. 1-19.

....to two versions of the AMD algorithm discussed in an earlier paper [1] approximate minimum external degree, both with and without aggressive absorption) Details of the method used in AMD can be found in that paper. For a discussion of the long history of the minimum degree algorithm, see [2]. 2 Availability In addition to appearing as a Collected Algorithm of the ACM, AMD Version 1.0 is available at http: www.cise.ufl.edu research sparse. The Fortran version is available as the routine MC47 in HSL (formerly the Harwell Subroutine Library) 3] 3 Using AMD in MATLAB The simplest ....

A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1--19, 1989.

....at that stage. GP based heuristics regard the symmetric sparse matrix as the adjacency matrix of a graph and follow a divide and conquer strategy to label the nodes of the graph by partitioning it into smaller subgraphs. The initial success of MD based heuristics prompted intense research [10] to improve their run time and quality and they have been the methods of choice among practitioners. The Multiple Minimum Degree (MMD) algorithm by George and Liu [9, 10] and the Approximate Minimum Degree (AMD) algorithm by Davis, Amestoy, and Duff [5] represent the state of the art in MD based ....

....nodes of the graph by partitioning it into smaller subgraphs. The initial success of MD based heuristics prompted intense research [10] to improve their run time and quality and they have been the methods of choice among practitioners. The Multiple Minimum Degree (MMD) algorithm by George and Liu [9, 10] and the Approximate Minimum Degree (AMD) algorithm by Davis, Amestoy, and Duff [5] represent the state of the art in MD based heuristics. Until now, with the exception of Rothberg and Hendrickson [35] most researchers have focused on ordering sparse matrices arising in finite element ....

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Alan George and Joseph W.-H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1--19, March 1989.

....such as the conjugate gradient method [Saa96] can also be used. Furthermore, since every node in the roadmap is directly connected to a relatively small number of neighboring nodes, Q is a sparse matrix. Sparse matrix ordering algorithms greatly reduce the running time of iterative solvers [GL89, GMS92] 5 Computing the probability of folding In this and next section, we describe the application of SRS to compute two specific ensemble properties: the probability of folding in protein folding and the escape time in ligand protein binding. Protein folding is one of the most marvelous ....

A. George and J. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1-- 19, 1989.

....) ll edges. c) A non minimal triangulation of G with less ll. d) A minimum triangulation of G. MD has given rise to a large amount of research with respect to improving the running time of its practical implementations, and the number of papers written on this subject is in the hundreds [1, 11]. However, very little is known theoretically as to its quality. It has in fact been analyzed theoretically only to a limited extent, which makes it dicult to gain control over this heuristic in order to improve it yet further, although recent research has been done on algorithms for low ll ....

J. A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31:1-19, 1989.

....such as the conjugate gradient method [Saa96] can also be used. Furthermore, since every node in the roadmap is directly connected to a relatively small number of neighboring nodes, Q is a sparse matrix. Sparse matrix ordering algorithms greatly reduce the running time of iterative solvers [GL89, GMS92] 5 Computing the probability of folding In this and next section, we describe the application of SRS to compute two specific ensemble properties: the probability of folding in protein folding and the escape time in ligand protein binding. Protein folding is one of the most marvelous ....

A. George and J. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1-- 19, 1989.

....Degree algorithm [50] is a local heuristic that performs its pivot selection by selecting from the graph a node of minimum degree. The Minimum Degree algorithm is known to be a very fast and general purpose algorithm, and has received much attention over the last three decades (see for example [1, 13, 37]) However, the algorithm is intrinsically sequential, and very little can be theoretically proven about its eciency. 3.2.2 Nested Dissection The Nested Dissection algorithm [14] is a global heuristic recursive algorithm which computes a vertex set S that separates the graph into two parts A and ....

A. George and J. W.-H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31:1-19, 1989.

....here. Thanks to Haim Kaplan for referring us to Tarjan s paper on augmented union find data structures. 6 CHAPTER 1 Background This chapter provides background material for the next two chapters, which describe original research results. This chapter is based on material from [4] 5] 10] [12], and from Sivan Toledo s lecture notes on high performance computing. 1.1. Iterative Solvers The term iterative methods refers in this thesis to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system Ax = b at each step. These ....

A. George and J. W. H. Liu. The evolution of the minimum-degree ordering algorithm. SIAM Review, 31:1--19, 1989.

....the vertex with minimum degree in the corresponding graph as v k . A number of enhancements to the basic minimum degree algorithm have been developed, such as the use of a quotient graph representation, mass elimination, incomplete degree update, multiple elimination, and external degree. See [13] for a historical survey of the minimum degree algorithm. Many of these enhancements, although initially proposed for the minimum degree algorithm, can be applied to other greedy approaches as well. Other greedy approaches differ from minimum degree by the choice of minimization criteria for ....

....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, we use a quotient graph representation. Some particular details of our implementation are given below. Prototype The prototype for our ....

Alan George and Joseph W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1--19, March 1989.

....the number of iterations with SQMR, selecting the same density parameters as those used for the experiments reported in Table 9, but using di#erent orderings to permute the original pattern of MSym Frob . More precisely we consider the reverse Cuthil MacKee ordering [13] RCM) the minimum degree [18, 31] ordering (MD) the spectral nested dissection ordering [28] SND) and lastly we reorder the matrix by putting the denser rows and columns first (DF) It can be seen that MSym Frob is not too sensitive to the ordering and none of the tested orderings appears superior to the others. 16 Example 1 ....

J. George and J.W.H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31:1--19, 1989.

....to sparse symmetric factorization by Tinney and Walker [58] Over the years many enhancements have been proposed to the basic algorithm that have greatly improved its efficiency. In the following we briefly describe some of these enhancements. For a detailed survey the reader is referred to [27]. 3 Perhaps one of the most important enhancements is the concept of supernodes. Two vertices u; v of an elimination graph G k belong to the same supernode, if adj G k (u) fug = adj G k (v) fvg. In this context the vertices u; v are called indistinguishable. Indistinguishable vertices possess ....

....I is j adj G k (I)j. Instead of using true degrees, the vertex to be eliminated next is selected according to its external degree. The motivation is that the only edges added by the elimination of v 2 I are between vertices in adj G k (I) As a result, one obtains slightly better orderings [27]. A key feature of the minimum degree algorithm is that one vertex is eliminated in each step. Once the new elimination graph has been built, the degree of all vertices that were adjacent to the newly eliminated vertex have to be updated. The most time consuming part of the minimum degree ....

A. George, J.W.H. Liu, The evolution of the minimum degree ordering algorithm, SIAM Review, Vol. 31, No. 1, 1--19, 1989.

....and work than the partial pivoting numerical factorization requires. This challenge has been met for the class of reordering algorithms based on the minimum degree heuristic. Modern implementations of minimum degree heuristics use a clique cover to represent the graph GA of the matrix 1 A (see [7]) A clique cover represents the edges of the graph (the nonzeros in the matrix) as a union of cliques, or complete subgraphs. The clique cover representation allows us to simulate the elimination process with a data structure that only shrinks and never grows. There are two ways to initialize the ....

A. George and J. W. H. Liu. The evolution of the minimum-degree ordering algorithm. SIAM Review, 31:1--19, 1989.

....performance, we test two of the reordering algorithms typically used for unstructured grids. The Reverse Cuthill McKee strategy [24] is a well known bandwidth reduction algorithm. The minimum neighbouring algorithm [25] is a modification of the minimum degree reordering of George and Liu [26]. It was designed to minimize the lost information in a matrix incomplete factorization. Start up Different relaxation techniques have been suggested in order to get past the rather violent nonlinear startup, particularly in transonic flows. One way to relax the solution is to damp the Newton ....

George, A., and Liu, J.W.H., "The evolution of the minimum degree ordering algorithm," SIAM Rev., no. 31, pp. 1--19, 1989.

....orderings that delay long dipaths in the triangular part of A (i.e. paths in A with monotonically increasing node indices) as these cause lots of I F fill, or alternatively orderings which give short and bushy elimination trees. Another useful characterization of I F fill using notions from [9] allows us to calculate the number of nonzeros in each column of the inverse factors very cheaply during a symbolic factorization: Theorem 4.3. Z ij #= 0 if and only if j is reachable from i strictly through nodes eliminated previous to i or in terms of the quotient graph model, if i is ....

A. George and J. Liu, The evolution of the minimum degree ordering algorithm, SIAM Review, 31 (1989), pp. 1--19.

....another vertex, eliminates it, and so on until all vertices are ordered. Since the development of the original symmetric minimum degree algorithm over 30 years ago, numerous changes have been suggested, each change improving the speed of the algorithm or the quality of the ordering found (see [6] for a survey) Important ideas incorporated in our code include the clique cover representation, supernodes, multiple eliminations, and approximate degree updates. The clique cover representation of the graph uses a set of cliques to cover all the edges in the graph. An elimination step in this ....

A. George and J. W. H. Liu, The evolution of the minimum degree ordering algorithm, SIAM Review, 31(1) (Mar. 1989), pp 1-19.

....T A takes# (n 2 ) work, but factoring it takes only O(n) work. This challenge has been met for the class of reordering algorithms based on the minimum degree heuristic. Modern implementations of minimum degree heuristics use a clique cover to represent the graph GA of the matrix 1 A (see [5]) A clique cover represents the edges of the graph (the nonzeros in the matrix) as a union of cliques, or complete subgraphs. The clique cover representation allows us to simulate the elimination process with a data structure that only shrinks and never grows. There are two ways to initialize the ....

A. George and J. W. H. Liu. The evolution of the minimum-degree ordering algorithm. SIAM Review, 31:1--19, 1989.

....A T A takes # n 2 ) work, but factoring it takes only O(n) work. This challenge has been met for the class of reordering algorithms based on the minimum degree heuristic. Modern implementations of minimum degree heuristics use a clique cover to represent the graph GA of the matrix 1 A (see [6]) A clique cover represents the edges of the graph (the nonzeros in the matrix) as a union of cliques, or complete subgraphs. The clique cover representation allows us to simulate the elimination process with a data structure that only shrinks and never grows. There are two ways to initialize the ....

A. George and J. W. H. Liu. The evolution of the minimum-degree ordering algorithm. SIAM Review, 31:1--19, 1989.

.... trapezoidal BDF2 method is also a possible choice and offers some advantages, see [26] All linear systems arising during the solution process have been solved by LU factorization and the minimum degree algorithm has been used to minimize the amount of fill in during the factorization process [27]. As shown in Figs. 2 5 for steady state solutions of the laser device of Fig. 1, as well as for the unstationary solutions x x x JMSM, Vol. 1, No. 1, Pages 6 8, 1999. 6 computed, the numerical solutions do not show spurious oscillations and this in spite of the fact the carrier densities are ....

A. George, J.W.H. Liu, The evolution of the minimum degree ordering algorithm, SIAM Review, Vol. 31, No. 1, pp. 1-19, 1989.

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A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1--19, 1989.

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Alan George and Joseph W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM REVIEW, 31(1):1--19, March 1989.

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A George and J. W. H . Liu, The evolution of the minimum degree ordering algorithm, siamrev 31 (1989), 1-19.

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A.George and J.W.H.Liu, The evolution of minimum degree ordering algorithm, SIAM Review, Vol. 31, No. 1, 1989, pp. 1-19. 103

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A. George and J. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31(1):1--19, 1989.

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J. A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31:1--19, 1989.

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J.A. GEORGE AND J.W.H. LIU, 1987. The Evolution of the Minimum Degree Ordering Algorithm, Technical Report CS-87-06, Department of Computer Science, York University, North York, Canada.

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