A. George. Computer implementation of the finite element method. Technical report, STAN-CS-208, Department of Computer Science, Standford University, San Francisco, Ca, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....all j, we obtain the lower bound on the envelope size. The upper bound is obtained by using the inequality c j (A) jffi(V j )j with the upper bound in Lemma 3.1. 2 Cuthill and McKee [3] proposed one of the earliest ordering algorithms for reducing the envelope size of a sparse matrix. George [12] discovered that reversing this ordering often leads to a significant reduction in envelope size and work. Since then the reverse CuthillMcKee (RCM) ordering has become one of the most popular envelope size reducing orderings. However, we do not know of any published quantitative results on the ....

A. George, Computer implementation of the finite element method, Tech. Rep. 208, Department of 20 Computer Science, Stanford University, Stanford, CA, 1971.

....of a linear system tends to be more efficient when the rows and columns of M are ordered so that the profile of the associated G is reduced. A number of heuristic algorithms for bandwidth and profile reduction are available. The reverse Cuthill McKee (RCM) algorithm, a modification by George [8] of the algorithm defined by Cuthill and McKee [4] is perhaps the most widely used. This algorithm was originally developed for bandwidth reduction. The Gibbs Poole Stockmeyer (GPS) algorithm [10] was developed to address both bandwidth and profile reduction, whereas profile reduction is the ....

A. GEORGE, Computer implementation of the finite element method, STANCS -71-208,Computer Science Dept., Stanford Univ.,Stanford, Calif., 1971.

....of minimal degree, levels of increasing distance from that vertex are first constructed. The enumeration is then performed level by level with increasing vertex degree (within each level) Several variations of this method have been suggested, the most popular being reverse Cuthill McKee (RCM) [6] where the level construction is restarted from a vertex of minimal degree in the final level. In many cases, it has been shown that 6 L. OLIKER, X. LI, P. HUSBANDS, AND R. BISWAS RCM improves the profile of the resulting matrix. The class of CM algorithms are fairly straightforward to implement ....

A. George, Computer implementation of the finite element method, Stanford University Technical Report STAN-CS-208, Stanford, CA, 1971.

....minmax or maxmin problems, where the first quantifier is over all triangulations of the point set, and the second is over all triangles in the triangulation. The problem of automatically generating optimal triangulations has been a subject for research since the 1960 s (see e.g. the discussion in [Geor71]) In spite of this attention, very little is known about constructing optimal triangulations in polynomial time. Exhaustive search can be ruled out since a set of n points has, in general, exponentially many triangulations. Greedy approaches (such as eliminating triangles from worst to best) are ....

J. A. George. Computer implementation of the finite element method. Techn. Rep. STAN-CS-71-208, Ph.D. Thesis, Comput. Sci. Dept., Stanford Univ., 1971.

....is Ework(A) # Esize(A) 2 2n . Proof. The proof follows from equations (2.1) and (2.2) by an application of the Cauchy Schwarz inequality. We omit the details. # Cuthill and McKee [3] proposed one of the earliest ordering algorithms for reducing the envelope size of a sparse matrix. George [14] discovered that reversing this ordering leads to a significant reduction in envelope size and work. The envelope parameters obtained from the RCM ordering are never larger than those obtained from CM [29] The RCM ordering has become one of the most popular envelope size reducing orderings. ....

A. George, Computer implementation of the Finite Element Method, Tech. report 208, Department of Computer Science, Stanford University, Stanford, CA, 1971.

....of ordering to minimize fill in is NP complete [24] There exist a number of ordering heuristics that try to minimize the amount of fill in. 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 ....

A. George. Computer Implementations of the Finite Element Method. PhD thesis, Stanford University, 1971.

....only in its sparsity pattern, the actual factorization may be more expensive and require more storage. Minimizing the profile of a matrix is known to be an NP complete problem [18] A number of heuristic algorithms have been proposed, including the Cuthill McKee [4] Reverse Cuthill McKee [6, 19], Gibbs King [8] Gibbs PooleStockmeyer [7] and Sloan [26] algorithms. More recently, spectral orderings based on the Fiedler vector of the Laplacian matrix associated with A have been developed [1, 20, 21] Kumfert and Pothen [17] propose combining the Sloan algorithm with the spectral ordering. ....

A. George. Computer implementation of the finite-element method. PhD thesis, Department of Computer Science, Stanford Univeristy, 1971.

....is over all triangles or edges of a triangulation. Two example criteria are maxmin area and maxmin inscribed circle (see [Schu87] The problem of automatically generating optimal triangulations for a given point set has been a subject for research since the 1960 s (see e.g. the discussion in [Geor71]) Exhaustive search can be ruled out since a set of n points has, in general, exponentially many triangulations. In spite of the attention these optimization problems have received, only very little is known about constructing optimal triangulations in polynomial time. An important negative ....

J. A. George. Computer implementation of the finite element method. Techn. Rep. STAN-CS-71-208, Ph.D. Thesis, Comput. Sci. Dept., Stanford Univ., 1971.

....matrix A is Ework(A) Esize(A) 2 2n : Proof. The proof follows from Equations 2.1 and 2.2, by an application of the CauchySchwarz inequality. We omit the details. 2 Cuthill and McKee [3] proposed one of the earliest ordering algorithms for reducing the envelope size of a sparse matrix. George [14] discovered that reversing this ordering leads to a significant reduction in envelope size and work. The envelope parameters obtained from the reverse Cuthill McKee (RCM) ordering are never larger than those obtained from CM [29] The RCM ordering has become one of the most popular envelope size ....

A. George, Computer implementation of the finite element method, Tech. Rep. 208, Department of Computer Science, Stanford University, Stanford, CA, 1971.

....and can be performed prior to numerical 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 ....

A. George, Computer Implementation of the Finite Element Method, PhD thesis, Dept. of Computer Science, Stanford University, 1971.

.... The multifrontal approach was first developed by Duff and Reid for symmetric, indefinite matrices [54] and then extended to unsymmetric matrices [55] However, the multifrontal approach has been used most extensively for the Cholesky factorization of symmetric, positive definite matrices [70, 65, 67]. Most recently, Davis and Duff have generalized the method to take advantage of unsymmetric pattern matrices [34, 35] One of the major advantages of a multifrontal method is that the regularity found in the dense matrix operations can be used to take advantage of advanced architectural features ....

A. George. Computer Implementation of the Finite-Element Method. PhD thesis, Report STAN CS-71-208, Department of Computer Science, Stanford University, Stanford, CA, 1971.

....In practice, the methods which have been successfully employed to discretise general computational domains are based upon either the advancing front concept or the Delaunay approach. Both methods start from a triangulation of the boundaries of the computational domain. The advancing front method [58, 59] discretises the volume by a marching approach in which points and elements are simultaneously created [60, 61] The key to success 25 is the implementation of an appropriate method of grid control. To this end, the grid is generated so as to meet, as closely as possible, the distribution of ....

A. J. George, Computer implementation of the finite element method, PhD Thesis STAN--CS--71-- 208, Stanford University, 1971.

....[7] and mesh partitioning [14, 21] Das et al. 6] also used a bandwidth reduction method for renumbering a grid to double the computational rate of an Euler solver on an iPSC 860 processor. There are a number of popular bandwidth reduction algorithms such as the Reverse Cuthill McKee [5, 9] and Gibbs Poole Stockmeyer [10] algorithms. These methods can be viewed physically as splitting the mesh into a number of layers (surfaces in three dimensions) The members within each layer are then numbered contiguously in memory. Fig. 6(b) shows the sparsity pattern of a 4913 vertex matrix ....

....been stored and operated on in double precision for the Jacobi and electromagnetic solvers, and in single precision for the Euler solver. The following numbering schemes have been analysed: ffl original numbering from grid generator; ffl bandwidth reduction using Reverse Cuthill McKee (RCM) 5] [9]; ffl RCM with random numbering within each level (RCMLR) ffl Greedy method with various block sizes; ffl completely random numbering; ffl vector colouring for the Euler solver. The first four renumbering schemes have been applied to the first set and all other sets have been consistently ....

A. George, 1971. Computer implementation of the finite element method. Technical Report STAN--CS--71--208, Computer Science Department, Stanford University, California.

....planes. Several reorderings have been considered in the literature as alternatives to the natural order. Among these are Minimum Degree (MD) Multiple Minimum Degree (MMD) Reverse Cuthill McKee (RCM) Nested Dissection (ND) and Multicoloring (MCL) For a description of these reorderings see [6] [11], 12] 16] 21] MMD, MD, and ND reorderings attempt to minimize the fill in the factors, while RCM reduces the bandwidth of the matrix. The degree of parallelism (DOP) of a preconditioning algorithm is the number of processors that can work simultaneously on constructing or applying the ....

A. GEORGE, Computer Implementation of the Finite Element Method, Report STAN-CS-208, Stanford University, Department of Computer Science, 1971.

....the profile is an NP complete problem[17] but due to its practical importance a number of heuristics have been proposed. Efforts in the 70 s and early 80 s developed techniques based upon a breadth first search in the graph of the matrix yielding methods known as reverse Cuthill McKee (RCM) [4, 7], Gibbs King (GK) 10] Gibbs Poole Stockmeyer (GPS) 9] and Sloan [19] More recently, several researchers have independently suggested ordering the vertices based upon their value in the Fiedler vector of the graph of the matrix [1, 18] This spectral approach generally produces better orderings ....

A. George. Computer implementation of the finite element method. Technical Report STANCS -208, Stanford University, Dept. of Computer Science, 1971.

....in this work. The Cuthill McKee algorithm [1] is a well known bandwidth and profile reduction algorithm. It generates the level structure rooted at each vertex of low degree, and it numbers these of minimal width. The algorithm selects the numbering producing the smallest bandwidth. George [6] has shown that the ordering obtained reversing the previous one gives, in general, a more suitable ordering. The Reverse Cuthill McKee algorithm (noted rcm henceforth) never increases the profile [11] and hence has no effect on bandwidth. The minimum neighboring algorithm (noted mineig ....

A. George. Computer Implementation of the Finite Element Method. PhD thesis, Departement of Computer Science, Stanford University, Stanford, USA, 1971. Also as Research Report STAN CS-71-208, Stanford University, Stanford, USA.

No context found.

A. George. Computer implementation of the finite element method. Technical report, STAN-CS-208, Department of Computer Science, Standford University, San Francisco, Ca, 1971.

No context found.

George, J.A,"Computer Implementation of the Finite Element Method", Techical Report No. STAN-CS-71-208, Computer Science Dept., Stanford University, 1971.

No context found.

A. George, Computer implementation of the finite element method, Tech. Report 208, Department of Computer Science, Stanford University, Stanford, CA, 1971.

No context found.

A. George, Computer implementation of the finite element method, Tech. Report 208, Department of Computer Science, Stanford University, Stanford, CA, 1971.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC