I. S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635--657.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....envelope schemes for sparse matrix factorization, these orderings have been used in the past few years in several other applications. The RCM ordering has been found to be an effective preordering in computing incomplete factorization preconditioners for preconditioned conjugate gradient methods [4, 6]. Envelope reducing orderings have been used in frontal methods for sparse matrix factorization [7] Such orderings have also been used in parallel matrix vector multiplication and tridiagonalization of sparse symmetric matrices. The wider applicability of envelope reducing orderings prompts us ....

I. S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635--657.

....to have algorithms that do not need, or depend less on, manual subdomain ordering and coloring. Extensive discussions on the effects of ordering and coloring of nodes or elements, in the context of iterative and direct sparse matrix computations, can be found in many research papers, see, e.g. [1, 9, 15, 17]. Some of the ideas and techniques can also be applied, with certain modifications, to the coloring and ordering of overlapping subdomains. We will not consider these techniques in this paper, since our interest is in automating the construction of the preconditioner. In this paper, we shall ....

I. S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, (1989), pp. 635--657.

.... them, for instance by including more fill [70] or by modifying the diagonal of the ILU factorization in order to force rowsum constraints [58, 6, 5, 73, 95, 34] or by changing the ordering of the matrix [96, 97] A collection of experiments with respect to the effects of ordering is contained in [30]. More recently, it was discovered that a multigrid inspired ordering can be very effective for discretized diffusion convection equations, leading in some cases to almost grid independent speeds of convergence [93, 11] see also [24] In these publications the ordering strategy is combined with a ....

....publications the ordering strategy is combined with a drop tolerance strategy for discarding small enough fill in elements. Red black ordering is an obvious approach to improve parallel properties for well structured problems, but it got a bad reputation from experiments, like those reported in [30]. If carefully done though, they can lead to significant gains in efficiency. Elman and Golub [35] suggested such an approach, in which Red Black ordering was combined with a reduced system technique. The idea is simple, eliminate the red points, and construct an ILU for the reduced system of ....

I. S. Duff and G. A. Meurant. The effect of ordering on preconditioned conjugate gradient. BIT, 29:635--657, 1989.

....can be found in Saad [87] and Donato [48] In the same paper, Saad also suggested an incomplete LU factorization with partial pivoting to improve the numerical stability. 3.5. Multilevel and Re ordered ILU. The performance of ILU and MILU is dependent on the ordering of the rows and columns of A [52]. Therefore various ordering strategies have been proposed to try to improve the performance. Some of these orderings are designed to improve the degree of parallelism in the application of the preconditioners and we shall discuss these in later sections. Here we describe several orderings which ....

....Orderings: Many other orderings have been proposed in the literature. For a comprehensive numerical study of the effect of the ordering on the convergence rate of incomplete factorization preconditioned CG methods for model elliptic problems, the reader is referred to the paper by Duff and Meurant [52]. Analysis of some of the orderings was given by Eijkhout [54] 16 Beauwens [22] defined a class of S P consistent orderings which share some similarities with the wavefront ordering. In addition to being attractive for parallel implementation, these orderings also enhance the convergence ....

I.S. Duff and G.A. Meurant. The effect of ordering on preconditioned conjugate gradients. BIT, 29:635--657, 1989.

....on the spectrum of ATA. Preconditioning techniques that accelerate the convergence of these methods have received extensive attention in the literature. Some examples of the literature of preconditioning symmetric positive definite linear systems and least squares problems are [25] 2] 39] 59] [21] [4] 61] 3] They can be summarized as follows: Column scaling. C = diag(di) where di are norms of columns of A. SSOR preconditioning [9] C = I L T, where A has been normalized so that ATA = L I L with L strictly lower triangular, and is a scalar parameter. Incomplete Cholesky ....

....the position set, we can first determine the positions in the border blocks and then the positions in all the diagonal blocks in parallel. We have not implemented a parallel version of this algorithm, but it is an area for future work. The quality of a IC preconditioner is related to the ordering [21] [23] 24] A good preconditioner must have not only potential for parallel processing but must also reduce the number of iterations so that overall computation time is reduced. How does this reordering and modification of the sparsity pattern affect the quality of the IC preconditioner At ....

I. S. Duff and G. A. Meurand. The effect of ordering on preconditioned conjugate gradients. BIT, 29:635 657, 1989.

....SSOR preconditioning with preconditioning based on Incomplete and Modified Incomplete Cholesky factorizations. The ordering of the variables in the domain affects both the convergence rate of the preconditioned problem and the implementation of the sparse triangular solvers. Duff and Meurant [40] show that the ordering of variables in incomplete Cholesky factorizations influences the spectrum of the preconditioned matrix and hence the convergence rate of iterative solvers. The ordering also influences the implementation of the sparse triangular solvers. When the sparsity pattern of the ....

....Unfortunately, the convergence rate obtained with this ordering is often worse than the convergence rate obtained with the natural ordering. The slower convergence rate of red black incomplete factorization preconditioners was observed experimentally by Ashcraft and Grimes [4] and Duff and Meurant [40]. Kuo and Chan [70] proved that for problems in square domains with Dirichlet boundary conditions naturally ordered preconditioners converge asymptotically faster than red black ordered preconditioners. Their results apply to conjugate gradient with SSOR, incomplete Cholesky, and modified ....

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Ian S. Duff and G erard Meurant. The effect of ordering on preconditioned conjugate gradient. BIT, 29(4):635--657, 1989.

....Some remarks on orderings Without preconditioning GMRES is independent of the ordering of the unknowns. If an ILU preconditioner is used then the convergence of preconditioned GMRES strongly depends on the ordering used. For symmetric problems the influence of the ordering is investigated in [10]. Duff and Meurant conclude that preconditioned CG converges fast for local orderings, which means that neighbouring nodes in the underlying mesh have numbers that are not too far apart. This is closely related to minimizing the bandwidth or profile of the matrix, and motivates us to order the ....

I.S. Duff and G.A. Meurant. The effect of ordering on preconditioned conjugate gradient. BIT, 29:635--657, 1989.

....with 1666 triangles, 5357 dof s, and a spectral condition number cond 2 (A) 2:4 Delta 10 10 . Again, it is clear that block Jacobi scaling significantly improves the quality of the preconditioning. The results above were obtained with a minimum degree ordering of the nodes. It is well known [24] that the numbering of unknowns may have significant influence on the performance of preconditioned iterations. For point versions of the AINV preconditioner, this has been investigated in [14] and [11] Here we consider nodal orderings, corresponding to orderings of the block partitioned matrix ....

I. S. Duff and G. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635--657.

.... We found a need for an additional aspect to handle the implicit permutation of vectors and matrix rows and columns, which occur frequently in numerical code, to address issues like: ordering for sparsity [6] parallel partitioning [7] block triangular form [8] and effective preconditioning [9]. An implicit permutation declaration, written with a view annotation, like view A through (p, allows implicit permutations to be dealt with in one place, rather than being smeared throughout the code. It says that all references to the specified arrays should act as if the arrays were ....

Duff, I.S. and G. Meurant, The effect of ordering on preconditioned conjugate gradients. BIT, 1989. 29: p. 685--657.

....equation can be solved by iterative methods. The speed of convergence depends very much on global properties (a local correction affects the whole solution) whereas for parallelism one wants to split the problem into smaller (almost) independent subproblems. These two requirements are in conflict [13]. A critical topical question in the use of incomplete factorization based preconditionings on parallel environments is how to overcome the above mentioned trade off between high level parallelism and rate of convergence [13,14] An answer to the above question requires to clearly identify why ....

....(almost) independent subproblems. These two requirements are in conflict [13] A critical topical question in the use of incomplete factorization based preconditionings on parallel environments is how to overcome the above mentioned trade off between high level parallelism and rate of convergence [13,14]. An answer to the above question requires to clearly identify why there is a trade off. To this end, Doi and Lichnewsky [8,9] relate this phenomenon to the number of incompatible nodes (any node i which is connected to at least two nodes j and k along the same direction (axis) such that j 6= k ....

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I.S. Duff and G.A. Meurant. The effect of ordering on preconditioned conjugate gradients. BIT 29, 635--657, 1989.

....preconditioners involve an interesting hybrid of combinatorial and algebraic techniques. GV 89] provides a short introduction to preconditioning, A 87, A 92, AL 86] provide 5 analysis of Preconditioned Conjugate Gradient, multilevel preconditioners are presented in [AV 89, BPX 90] and [E 81, DM 89, V 89, HVY 91] provide efficient implementations of preconditioned iterative methods. Preconditioners have been constructed using a wide variety of methods, including diagonal scaling, partial Gaussian elimination resulting in sparse partial LL T factorizations, and algebraic transforms. For ....

I.S. Duff and G.A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT 29, (1989), 635---657.

....by Simon [154] who considered large nonsymmetric linear systems. For the systems he considered he concluded that standard techniques used for sparse direct solvers were not too helpful for use in preconditioners based on level of fill. Immediately following this was a paper by Duff and Meurant [56] which concluded, similarly, that ICCG does not in general benefit in any significant manner form reordering. These studies were limited to certain types of reorderings and certain types of preconditioners. It is now known [18] that certain reorderings, such as Reverse Cuthill McKee are beneficial ....

....of reorderings and certain types of preconditioners. It is now known [18] that certain reorderings, such as Reverse Cuthill McKee are beneficial in preconditioning methods, in particular with some form of dropping strategy. The beneficial impact of well chosen fill ins was already demonstrated in [56] for some orderings. What seems to be also clear is that the best approaches for direct solvers (such as Nested Dissection and minimal degree ordering) are not the best for iterative solvers. Since ILU and IC factorizations were the most popular preconditioners, at least in a sequential ....

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I. S. Duff and G. A. Meurant. The effect of ordering on preconditioned conjugate gradients. BIT, 29:635--657, 1989.

....as a Preconditioner M. A. DeLong and J. M. Ortega July 3, 1995 1 Introduction It is well known (see, e.g. [2] and [5] that the use of red black or multicolor orderings to parallelize SSOR or ILU preconditioning may seriously degrade the rate of convergence of the conjugate gradient method, as compared with the natural ordering. The SOR iteration itself, however, does not suffer this degradation. ....

I. Duff and G. Meurant, The Effect of Ordering on Preconditioned Conjugate Gradients, BIT, 29, pp.635-657, 1989.

....2.2: Incomplete Cholesky factorization In our discussion we have ignored the role played by the ordering in the matrix. This is an important issue since the ordering of the matrix affects the fill in the matrix, and thus the incomplete Cholesky factorization. In particular, Duff and Meurant [14] and Eijkhout [15] have shown that the number of conjugate gradient iterations can double if the minimum degree ordering is used to reorder the matrix. However, we note that these studies were not done with limited memory preconditioners, and thus it is not clear that the same conclusion holds for ....

....In particular, Figures 4.2 and 4.3 look similar when oe is smaller. The improvement is most noticeable for bcsstk09, the easiest problem in the test set. The decrease in computational time for an incomplete Cholesky factorization is not guaranteed. For example, the results of Duff and Meurant [14] comparing a level 1 factorization with a level 0 incomplete Cholesky factorization on grid problem with five point and nine point stencils showed that the extra work in the computation of the conjugate gradient iterates was not always offset by the work saved from the reduction (if any) in the ....

I. S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635--657.

....This gives rise to fronts of uncoupled nodes and hence the preferable structure of the matrix. In order for high degree of parallelity and equal processor load this idea is limited to simple domains or isotropic problems. Other papers where anisotropic problems are studied are for instance [6] and [13] However, the aspect of parallel solution is not considered in these papers. In [6] the effect of various node ordering strategies on the preconditionedconjugate gradient method is experimentally studied. In [13] an optimal order PCG algorithm for problems with strong anisotropy and with ....

....matrix. In order for high degree of parallelity and equal processor load this idea is limited to simple domains or isotropic problems. Other papers where anisotropic problems are studied are for instance [6] and [13] However, the aspect of parallel solution is not considered in these papers. In [6] the effect of various node ordering strategies on the preconditionedconjugate gradient method is experimentally studied. In [13] an optimal order PCG algorithm for problems with strong anisotropy and with varying dominance directions is presented. With the new stategy presented in this paper, ....

S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29(4), 635-657, 1989.

....the other hand, in such a way that the part of the computation related to the interior nodes reduces to local triangular solves which may be carried out independently. Some degradation is expected in the convergence rate but, as will be seen, limited compared with the gain in parallelism, see also [9, 10, 20]. Here, we consider the use of this principle within the framework of the results of [19] In the latter paper, a parallelization technique is proposed which is based on the equivalence between any iterative scheme applied to the global system with the same 1 iterative scheme applied to an ....

....for discrete problems of the type (1. 1) solved with the conjugate gradient algorithm, see [18, 21] Moreover, from our tests, it appears that this number of iterations do generally not increases when #proc: increases from 1 to 4 (in accordance with the theoretical and experimental results in [9, 20]) Then, it increases slightly less then O( #proc: 1=4 ) and, for 256 processors, remains not greater than about twice that required by the single or four processor version, which represents a quite reasonable cost for such a massively parallel method. 1 indeed, for the choice U d = P d = D ....

I. S. Duff and G. A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635--657.

....incomplete factorizations as preconditioning techniques are often efficient [31,32] Their major drawback is that they are not easy to parallelize without seriously affecting the convergence. Several attempts have been reported in the literature, including reordering strategies, see, e.g. [1,4,8,11,16,20,21,30,38,45 47,49], domain decomposition type approaches [9,10,22,25,27,36,41,42] and truncated Neumann series approaches, 44,3,48] This reflects the difficulty of the task. Recent surveys of techniques for achieving parallelism may be found in [13,17] 1 Research supported by the Commission of the European ....

....the application of the preconditioner. A special treatment of the interface gridpoints allows to alleviate the significant decrease of the convergence rate that is characteristic for DD methods and for most of the orderings that have been suggested for general parallel computations (see, e.g. [16,14]) Our exposition is organized as follows. In Section 2, we give a brief overview of our terminology and notation. Section 3 consists of background material, including a description of the preconditioned conjugate gradient (PCG) method, and a description of the generalized incomplete factorization ....

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I.S. Duff and G.A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT 29 (1989) 635--657.

....in reducing the iteration count. This problem is a 2 dimensional mesh with an aspect ratio of 10 Gamma5 . In the case of poor aspect ratios, a weighted Laplacian should be more appropriate for computing the spectral ordering, but we defer this topic for future research. Duff and Meurant [8] indicate that ordering becomes more significant when the problem becomes more difficult (discontinuous coefficients, anisotropy, etc. Another problem from the Harwell Boeing collection BCSSTK17 did not converge quickly for levels of fill below two, indicating that it is a difficult problem. ....

I. Duff and G. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635--657.

....[10, 4, 16, 2, 13, 5, 19, 32, 30] The successful application of PCG methods depends to a great extent on the formation of a rapidly convergent preconditioner. A number of studies have examined the effect of matrix ordering on the quality of preconditioners based on incomplete factorization [6, 7, 8, 12, 13, 25, 11]. In [6, 7, 8] evidence was presented to demonstrate how matrix ordering can have a profound effect on the quality of preconditioners, and a heuristic noted that was shown to produce good matrix ordering. This study examines the use of efficient algorithms from combinatorial graph theory which ....

....the matrix rows represent vertices, and the graph edges are encoded in the off diagonals, the magnitude of the off diagonal coefficients providing the strength of the connections. The reader may wish to review [26, 28, 17, 6] for relevant information on this view of matrices. Duff and Meurant [12] studied a large number of preconditioner orderings for matrices arising from isotropic and anisotropic PDE s discretized on a regular grid. Their study considered orderings based solely on the sparsity pattern of the matrix, and concluded that Reverse Cuthill McKee (RCM) ordering [17] was, in ....

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I. S. Duff and G. A. Meurant. The effect of ordering on preconditioned conjugate gradients. BIT: Nordisk Tijdskrift for Informationbehandling, 29:635-- 657, 1989.

....example is the case of (2d 1) point central finite differences on a regular FULLY PARALLEL M ILU PRECONDITIONING 5 rectangular grid of d dimensions for a second order partial differential equation with Dirichlet boundary conditions. The two dimensional d = 2 case was studied in some detail in [5] in terms of the impact of orderings on convergence. When a natural lexicographical ordering is applied to the grid and the no fill ILU preconditioning is computed, the induced wavefront pattern is as denoted in Figure 1. If the ordering starts at the southwest corner, then each point in the ....

....trick [6] may be applied to reduce the number of computations required in the linear solve. Here, the matrix vector product and the preconditioning are fused together, reducing the required number of floating point operations. Second, a 3 D analog of the van der Vorst ordering may be applied [5]. This technique is based in sweeping inward from four corners into the center of the grid, rather than sweeping fully from one corner of the grid to the opposite corner. Numerical experiments for some problems have indicated that any loss of convergence due to this technique is minor. This ....

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Duff, I. S. and G. A. Meurant, "The Effect of Ordering on Preconditioned Conjugate Gradient," BIT, 29:635--657, 1989.

....down [42, 45, 51] Also, factorisations are inherently recursive, and coupled with the sparseness of the incomplete factorisation, this gives very limited parallelism in the algorithm using a natural ordering of the unknowns. Different orderings may be more parallel, but take more iterations [25, 27, 43]. 8.4.3 Analytically inspired preconditioners In recent years, a number of preconditioners have gained in popularity that are more directly inspired by the continuous problem. First of all, for a matrix from an elliptic PDE, one can use a so called fast solver as preconditioner [12, 28, 63] A ....

I.S. Duff and G.A. Meurant. The effect of ordering on preconditioned conjugate gradients. BIT, 29:635--657, 1989.

....but not on each other, they can all be updated in parallel. Thus the degree of parallelism is n 2 =2, a substantial increase from O(n) for the natural ordering. However, since the data dependence are completely local and there is no global sharing of information, the convergence rate is poor [30]. In fact, it can be shown that the condition number of the preconditioned system in the red black ordering is only about 1=4 that of the unpreconditioned system for ILU, MILU and SSOR, with no asymptotic improvement as h tends to zero [49] One way to strike a better balance between parallelism ....

I. S. Duff and G. A. Meurant. The effect of ordering on preconditioned conjugate gradient. BIT, 29:635--657, 1989. 51

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I.S. Duff and G.A. Meurant, The effect of ordering on preconditioned conjugate gradients, BIT 29, 635-657, (19S9).

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I. Duff and G. Meurant, The Effect of Ordering on Preconditioned Conjugate Gradients, BIT 29, 635-657 (1989).

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Duff, I. S. and G. A. Meurant, "The Effect of Ordering on Preconditioned Conjugate Gradient, " BIT, 29:635--657, 1989.

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