A. George and J. W. H. Liu. An automatic nested dissection algorithm for irregular finite-element problems. SIAM J. Numer. Anal., 15:1053-1069, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... for structurally unsymmetric systems (Tarjan s algorithm) It is also possible to approach the problem of transforming the matrix to block upper triangular form starting from the standard techniques used to produce separator sets for structurally symmetric matrices, e.g. nested dissection [Geo73, GL78]. The ordering H2 starts with the construction of separator sets of the adjacency matrix of A A T as in the standard approaches. For the implementation of H2 we used a straight forward implementation of automatic nested dissection [GL81] However, other initial orderings could have been used ....

A. George and J. W. H. Liu. An automatic nested dissection algorithm for irregular finite-element problems. SIAM J. Numer. Anal., 15:1053-1069, 1978.

....degree algorithm selects a vertex with minimum degree in G k 1 . This vertex is numbered next in the ordering and is eliminated from G k 1 to form the graph G k . The whole selection elimination process is then repeated for G k . Another effective method for reducing fill is nested dissection [23, 24]. The method starts with computing a vertex separator S in G. All vertices in S are ordered after those in G(V S) The method is recursively applied to each component of G(V S) until a component This work is supported by the German Federal Department of Science and Technology (PARALOR project) ....

....that are based on function score AMF are proposed such as approximate 4 minimum mean local fill (AMMF) or approximate minimum increase in neighbor degree (AMIND) For more details consult [56] 2. 3 Top down orderings The most popular top down scheme is George s nested dissection (ND) algorithm [23, 24]. The basic idea of this approach is to find a subset of vertices S in G, whose removal partitions G in two subgraphs G(B) and G(W ) with V = S [ B [ W and jBj; jW j jV j for some 0 1. Such a partition of G is denoted by (S; B; W ) The set S is called vertex separator of G. If we order ....

A. George, J.W.H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., Vol. 15, No. 5, 1053--1069, 1978.

....Minnesota 55455 (teng cs.umn.edu) 1 to dividing the vertices of the graph into sets of equal size with few edges joining vertices in different sets. Graph partitioning has been an active field of research for several years, both theoretically [2, 9, 17, 21, 31, 32, 33] and experimentally [1, 14, 15, 16, 19, 27, 29, 37, 39, 42]. Optimal partitioning is an NP hard problem, and finding good graph partitions in practice can be very expensive. Graphs from large scale problems in scientific computing are often defined geometrically. They are meshes of elements in d dimensional Euclidean space (typically d = 2 or 3) This ....

....algorithm to partition the graph all the way down to fragments of 3 vertices or less; partial geometric uses the geometric algorithm down to fragments of 100 vertices and then uses minimum degree on the fragments. We also tabulate fill and height for Sparspak s nested dissection routine [19], for Matlab s minimum degree routine [22] and for nested dissection with separators from spectral partitioning as described by Pothen et al. 38] Sparspak s nested dissection routine uses a fast but fairly simple partitioning algorithm, which generally does not perform as well as the newer ....

Alan George and Joseph W. H. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM Journal on Numerical Analysis, 15:1053--1069, 1978.

....The multiple minimum degree algorithm (MMD) 8] has been shown to result in good orderings for a broad range of matrices. The Reverse Cuthill McKee algorithm [5] reduces the envelope of the matrix, and hence storage and computation requirements for band factorization. Nested dissection algorithms [3, 6, 7, 10, 16] usually result in an ordering with more parallelism, but often at the cost of increased fill in [13] For sparse factorization on distributed memory multiprocessors, performance critically depends on the cost of communication, which is affected by how the data and computation are mapped onto ....

A. George, M. Heath, and E.G.-Y. Ng. An Automatic Nested Dissection Algorithm for Irregular Finite Element Problems. SIAM Journal on Numerical Analysis, 15:1053--1069, 1978.

.... the higher communication cost due to fine granularity of the mapping (see [CT88] and Section 7) Another class of data clustering heuristics is based on reordering methods which have been developed to solve the fill in problem [GL81] These problems are NP Complete [Gil80] In [LS76] Geo73] GL78] and [GM78] several reordering heuristics have been introduced for its solution : Cuthill McKee, reverse Cuthill McKee, automatic nested dissection, and minimum degree. Generalizations of these algorithms appear in [Gil80] and [Liu89b] The Cuthill McKee ordering scheme and automatic nested ....

Alan George and Joseph W. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM Journal Numerical Analysis, 15(5):1053--1069, 1978.

....the tasks on a shared memory machine and so, a large grain model can be used. For distributed memory machines, the first experiments on the Fan In algorithm [1] were carried out on matrices arising from nine point finite difference operators on rectangular grids. The nested dissection ordering [6] and the subtree to subcube mapping [7] are used, achieving low fill in and good load balance for the problems under consideration. Thus, the performances obtained are better than for the distributed Fan Out algorithm. Indeed, the latter is used to solve finite elements problems arising from ....

Alan GEORGE and Joseph LIU. An automatic nested dissection algorithm for irregular finite element problems. Siam journal of numerical analysis, 1053--1069, October 1978.

....introduces some nonzeros into L. These nonzeros are called ll in. By a symmetric reordering of the columns and rows of A we can often reduce the ll in. Finding the ordering that gives the least ll in is NP hard [29] Two frequently used heuristics are Minimum Degree [14] and Nested Dissection [12]. After a ll reducing ordering has been found, the next step in a sparse Cholesky factorization is the symbolic factorization. Its purpose is to nd the nonzero structure of L so that memory can be reserved for the nonzeros before their numerical values are computed. In parallel Cholesky ....

A. George and J. W. H. Liu, An automatic nested dissection algorithm for irregular nite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053 1069. 26

....the tasks on a shared memory machine and so, a large grain model can be used. For distributed memory machines, the first experiments on the Fan In algorithm [1] were carried out on matrices arising from nine point finite difference operators on rectangular grids. The nested dissection ordering [6] and the subtree to subcube mapping [7] are used, achieving low fill in and good load balance for the problems under consideration. Thus, the performances obtained are better than for the distributed Fan Out algorithm. Indeed, the latter is used to solve finite elements problems arising from ....

Alan GEORGE and Joseph LIU. An automatic nested dissection algorithm for irregular finite element problems. Siam journal of numerical analysis, 1053--1069, October 1978.

.... 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 ....

J. A. George and J. W-H. Liu. An automatic nested dissection algorithm for irregular nite element problems. #### ## ###### #####, 15:1053-1069, 1978.

....induced simple chordless cycle of length 4 or more. In the minimum chordal completion problem we are required to construct a chordal graph with the fewest edges. This bounds the space required to solve a sparse linear system. This problem has received considerable attention from researchers; see [9, 15, 17, 18, 19, 21, 22, 25, 29, 37, 36, 41, 42, 50, 54, 57], amongst many others. Most of these approaches use one of two methods, minimum degree heuristic (see [37, 42, 54] for example) or nested dissection (see [17, 19, 36] Hybrid algorithms (see [2, 27] are also used. The space requirement is not the only measure of interest in solving sparse ....

....system. This problem has received considerable attention from researchers; see [9, 15, 17, 18, 19, 21, 22, 25, 29, 37, 36, 41, 42, 50, 54, 57] amongst many others. Most of these approaches use one of two methods, minimum degree heuristic (see [37, 42, 54] for example) or nested dissection (see [17, 19, 36]) Hybrid algorithms (see [2, 27] are also used. The space requirement is not the only measure of interest in solving sparse linear systems. The total time taken, which is the number of arithmetic operations or the operation count is an useful measure if we have to solve the xed system for ....

J. A. George and J. W. Liu. \An automatic nested dissection algorithm for irregular nite element problems". In SIAM Journal on Numerical Analysis v15, pages 1053-1069, (1978).

....to individual processors. Thus, the algorithms we develop will take into account both size and balance in choosing separators. Nested dissection algorithms differ primarily in the heuristics used for choosing separators. A typical approach to automatic nested dissection for irregular graphs [4] involves first finding a peripheral vertex, generating a level structure based on the connectivity of the graph, and then choosing a middle level of vertices as the separator. Such an approach is difficult to implement efficiently on a distributed parallel computer for a number of reasons, ....

J. A. George and J. W.-H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

.... A linear partitioning splits the vertex set at a point of the order, and a tree partitioning algorithm splits a tree defined by the order and the dependency of the LU decomposition [40] Famous ordering methods are Cuthill McKee, reverse Cuthill McKee [39] GPS [22] and nested dissection [21]. Chang and Hajj [8] proposed a new method for vertex separation. It removes vertices one by one, and check for 1 vertex separators at each step. The criterions for the removed vertices are (1) the vertex with maximum degree, and (2) the vertex of the minimum radius (the radius of a vertex is the ....

A. George and J. W. H. Liu, "An Automatic Nested Dissection Algorithm for Irregular Finite Element Problems," SIAM J. Num. Anal., Vol. 15, No. 5, Oct. 1978, pp. 1053--1069.

....x k in G k 1 . The value j adj G k 1(x k )j is called the degree of x k in G k 1 and will be denoted by deg G k 1(x k ) This work is supported by the German Federal Department of Science and Technology (PARALOR project) Another effective method for reducing fill is nested dissection [17, 18] which is based on a divideand conquer paradigm. The method starts with computing a vertex separator S A in G A . All vertices in S A are ordered after those in G(X A S A ) The method is recursively applied to each component of G(X A S A ) until a component consists of a single ....

....strategies that are based on this function are proposed such as approximate minimum mean local fill (AMMF) or approximate minimum increase in neighbor degree (AMIND) For further details consult [44] 2. 3 Top down methods The most popular top down scheme is George s nested dissection algorithm [17, 18]. The basic idea of this approach is to find a set of vertices S in a graph G = X; E) whose removal partitions G in two subgraphs G(W ) and G(B) with X = W [ B [ S and jW j; jBj jXj for some 0 1. If S is ordered after W and B, no fill edge can occur between vertices in W and B. ....

A. George, J.W.H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., Vol. 15, No. 5, 1053--1069, 1978.

....clearly by this dissection process. The fill in relates to the size of the separators and in broad terms, the smaller the separator, the lower the fill. Nested dissection algorithms differ primarily in the heuristics used for choosing separators. An approach called Automatic Nested Dissection [8] involves first finding a peripheral vertex, generating a level structure based on the connectivity of the graph, and then choosing a middle level of vertices as the separator. More recent heuristics for computing graph separators include spectral methods [20] methods based on geometric ....

....leader of the group of processors. This is indeed feasible because the size of the graph is small enough. Next, the leader processor can apply any sequential algorithm to compute a separator in G k . In our current implementation, we use the inexpensive level search based AND heuristic of Sparspak [8, 5]. Now each vertex in G k is marked to be either in part A k , B k , or the separator S k . This partition in G k is projected to the graph at level 0, i.e. G 0 . Recall that each vertex v in G k represents a group of vertices in G 0 ; if v 2 A k then all vertices in the group are assigned to part ....

[Article contains additional citation context not shown here]

J. A. George and J. W.-H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

....factorization. The ideal choice is an ordering that introduces the least fill, but the problem of computing such an ordering is NP complete [22] Consequently, almost all ordering algorithms are heuristic in nature. Examples include reverse CuthillMcKee [5, 6, 8] automatic nested dissection [9], and minimum degree [18] A greedy ordering heuristic numbers columns successively by selecting at each step a column with the optimal value of a metric. In the minimum degree algorithm of Tinney and Walker [21] the metric is the number of nonzero entries (and hence operations) in the rank 1 ....

A. George and J. W.-H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

....nested dissection. The fill in relates to the size of the separators and in broad terms, the smaller the separator, the lower the fill. Nested dissection algorithms differ primarily in the heuristics used for choosing separators. An early approach called Automatic Nested Dissection [11] (AND) involves first finding a pseudoperipheral vertex, generating a level search structure from this pseudoperipheral vertex, and then choosing a middle level of vertices as the separator. Typically, AND incurs significantly higher fill in than MMD. Other heuristics for computing graph ....

....is small enough, i.e. approximately O(N=P ) 2. Recall that vertex weights are associated with this graph. The leader processor can apply any sequential algorithm to compute a separator in GK . In our current implementation, we use the inexpensive level search based AND heuristic of SPARSPAK [7, 11] extended in the natural manner to a graph with vertex weights. We modify the AND heuristic to select as the middle level the level such that the total weight to the left and right of the level is roughly half the original number of vertices. This middle level is chosen as the vertex separator ....

J. A. George and J. W.-H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

.... disadvantage is the higher communication cost due to fine granularity of the mapping (see [CT88] and Section 7) Another class of data clustering heuristics is based on reordering methods developed to solve the fillin problem [GL81] These problems are NP Complete [Gil80] See [LS76] Geo73] GL78] and [GM78] for the following methods : Cuthill McKee, reverse Cuthill McKee, automatic nested dissection, and minimum degree. Generalizations of these algorithms appear in [Gil80] and [Liu89b] which can be used to partition a mesh into P connected submeshes (P way partition) A generalization ....

Alan George and Joseph W. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM Journal Numerical Analysis, 15(5):1053--1069, 1978.

....[2, 3, 6] Liu [9] developed an efficient implementation of this algorithm that reduces the number of degree updates that need to be performed. It is referred to as the multiple minimum degree (MMD) algorithm. Another popular heuristic for ordering sparse matrices is the nested dissection [5] algorithm, which permutes the matrix into a bordered block diagonal form, such that the zero off diagonal blocks suffer no fill in during the factorization process. George [4] showed that for complete nested dissection of regular n Theta n finite element grids, the fill in introduced by nested ....

A. George, M. Heath, and E.G.-Y. Ng. An Automatic Nested Dissection Algorithm for Irregular Finite Element Problems. SIAM J. Numer. Anal., 15:1053--1069, 1978.

.... used heuristics are for j = 1 to n do for i j where l ji 6= 0 do cmod(j, i) cdiv(j) end do Figure 1: Sparse column Cholesky for i = 1 to n do cdiv(i) for j i where l ji 6= 0 do cmod(j, i) end do Figure 2: Sparse submatrix Cholesky Minimum Degree [14] and Nested Dissection [12]. After a fill reducing ordering has been found, the next step in a sparse Cholesky factorization is the symbolic factorization. Its purpose is to find the nonzero structure of L so that memory can be reserved for the nonzeros before their numerical values are computed. In parallel Cholesky ....

A. George and J. W. H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

....only a small number of edges join the two sets. Graph bisection has long been studied from both theoretical and heuristic points of view, motivated by such applications as VLSI layout, divide and conquer algorithms, solution of sparse linear systems, and load balancing for parallel computation [9, 15, 20]. Theoretical work includes separator theorems for various special classes of graphs [1, 10, 12, 21, 23] and approximation algorithms that guarantee to produce bisections within a polylog factor [19] or, recently, a constant factor [6] of optimal. It remains to be seen whether these approximation ....

Alan George and Joseph W. H. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM Journal on Numerical Analysis, 15:1053--1069, 1978.

....matters worse, both the problem of minimizing fill and the problem of finding the lowest possible etree are known to be NP hard [21, 26] There are methods for finding orderings that give low etrees and few fill edges. Nested dissection is a method for ordering G that was developed to reduce fill [5, 6] and has also been shown to produce low etrees [13] Another approach is first to compute a fill reducing ordering P and then to find an equivalent ordering (giving the same fill edges) of the adjacency graph G of L L T that produces a low etree. The main motivation behind this approach is ....

A. George and J. W. H. Liu, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

....toute paire de sommets dans le graphe G. Factorisation de Cholesky 7 Un probl eme non r esolu actuellement est de trouver de bons s eparateurs. En effet, le parall elisme mis en evidence est tributaire de la taille de ces s eparateurs : plus ils sont petits, meilleur est le parall elisme [2]. 1.2 Factorisation symbolique Au cours de cette phase, la structure de la matrice L est construite. Cette structure qui servira a l allocation de l espace m emoire, accueillera aussi bien la matrice A que L. Ce traitement all ege par cons equent la factorisation num erique. 1.2.1 Quelques ....

Alain GEORGE and Joseph LIU. An automatic nested dissection algorithm for irregular finite element problems. Siam journal of numerical analysis, 1978.

.... for structurally unsymmetric systems (Tarjan s algorithm) It is also possible to approach the problem of transforming the matrix to block upper triangular form starting from the standard techniques used to produce separator sets for structurally symmetric matrices, e.g. nested dissection [17, 20]. The ordering H2 starts with the construction of separator sets of the adjacency matrix of A A T as in the standard approaches. For the implementation of H2 we used a straight forward implementation of automatic nested dissection [19] However, other initial orderings could have been used such ....

A. George and J. W. H. Liu, An automatic nested dissection algorithm for irregular finite-element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053--1069.

....at the very end. The components resulting from removing the separator are recursively ordered, one after the other, and placed before the separator in the elimination ordering. George [12] first proposed this method for eliminating nodes in a mesh, and later generalized it in a paper with Liu [13] for eliminating the nodes in an arbitrary graph. Bounds on the fill produced by nested dissection orderings are known for planar graphs and arbitrary graphs with bounded degree [1, 14, 21] The minimum degree heuristic repeatedly finds a vertex of minimum degree and eliminates it. This heuristic ....

J. A. George and J. W. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM Journal of Numerical Analysis, 15:1053--1069, 1978.

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A. George and J.-W.H. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM J. Numer. Anal., 15:1053--1069, 1978.

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