Dutto, L.C., "The Effect of Ordering on Preconditioned GMRES Algorithm, for Solving the Compressible Navier-Stokes Equations," International Journal for Numerical Methods in Engineering, vol. 36, pp. 457--497, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and the strategy adopted for PROBE will follow in a later section. Ordering of unknowns Several authors have shown that the ordering of the unknowns plays an important role in the convergence of the iterative solver, especially when an incomplete factorization preconditioner is used [4, 31]. A number of ordering strategies have been compared in [32] The Reverse Cuthill McKee strategy [33] a well known bandwidth reduction algorithm, was shown to be the most efficient of those studied, which included two natural orderings, two domain decomposition orderings, and the minimum ....

Dutto, L.C., "The Effect of Ordering on Preconditioned GMRES Algorithm, for Solving the Compressible Navier-Stokes Equations," International Journal for Numerical Methods in Engineering, vol. 36, pp. 457--497, 1993.

....found this quasi Newton strategy to be competitive with explicit multigrid solvers, albeit with increased storage requirements. Other factors which can influence the efficiency of a Newton Krylov algorithm include the ordering of the un1 American Institute of Aeronautics and Astronautics knowns [9] and the type of preconditioner used. Possible reordering algorithms include reverse Cuthill McKee (RCM) nested dissection (ND) and quotient minimum degree (QMD) The choice of a preconditioner is dependent on the tradeoff between speed and memory. Many authors have used the incomplete ....

Dutto, L. C., "The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier--Stokes equations," International Journal for Numerical Methods in Engineering, vol. 36, pp. 457--497, 1993.

....the maximum number of elements per row and thus the memory usage. The ILU(p) and ILUT(p, factorizations are compared below, along with other aspects of the preconditioning. Ordering of unknowns The ordering of the unknowns plays an important role in the convergence of the iterative solver [4, 23]. It can greatly affect the quality of the incomplete factorization. As mentioned in [20] for some cases, numbering the nodes across the wakecut (designated or 3) produces a more efficient algorithm than using the natural ordering (designated or 1) The downside is that the bandwidth is twice as ....

Dutto, L.C., "The Effect of Ordering on Preconditioned GMRES Algorithm, for Solving the Compressible Navier-Stokes Equations," International Journal for Numerical Methods in Engineering, vol. 36, pp. 457--497, 1993.

....them. The most efficient strategy for our applications is to compute the preconditioner only once. Ordering of unknowns Since it can greatly affect the quality of the incomplete factorization, the ordering of the unknowns plays an important role in the convergence of the iterative solver [19, 35]. Several ordering strategies have been compared in [26] The Reverse CuthillMcKee strategy [36] a well known bandwidth reduction algorithm, was shown to be the most efficient of those studied, which included two natural orderings and the minimum neighbouring ordering. Start up For some flow ....

Dutto, L.C., "The Effect of Ordering on Preconditioned GMRES Algorithm, for Solving the Compressible Navier-Stokes Equations," International Journal for Numerical Methods in Engineering, vol. 36, pp. 457--497, 1993.

....preconditioners have been used in the finite element solution of elliptic p.d.e. s on unstructured meshes, such as simple diagonal scaling or incomplete Choleski factorization for example [12, 26] The second of these is generally superior, especially if a good ordering is chosen for the unknowns [8]. However, parallel methods based upon incomplete factorizations are usually difficult to implement efficiently due to the triangular solves that must be performed at each preconditioning step. From a parallel implementation point of view domain decomposition preconditioners are far more 3 ....

L. C. Dutto, The Effect of Ordering on Preconditioned GMRES Algorithm for Solving The Compressible Navier-Stokes Equations, Int. J. Num. Meth. in Eng. 36(1993) 457--497.

....et peu couteuse en m emoire se heurte a ces constats. 1. Il faut un bon pr econditionnement. Dans un cadre classique, le choix d une factorisation incompl ete de J(u i ) comme pr econditionneur coupl ee avec des techniques de renum erotation des noeuds du maillage donne de bons r esultats [Dut91] mais ici la matrice est inconnue. Et seul le r esidu fournit les informations sur le probl eme etudi e. 2. Un red emarrage est n ecessaire pour r eduire le cout m emoire mais une d egradation de la convergence est alors observ ee par rapport a GMRES classique. Depuis peu, des algorithmes de ....

L.C Dutto. The effect of ordering on preconditioned gmres algorithm applied to solve the compressible navier-stokes equations on unstructured grids. July 1991. Submitted to Int. J. for Numerical Methods in Engineering. 20 R.Choquet

....et peu couteuse en m emoire se heurte a ces constats. 1. Il faut un bon pr econditionnement. Dans un cadre classique, le choix d une factorisation incompl ete de J(u i ) comme pr econditionneur coupl ee avec des techniques de renum erotation des noeuds du maillage donne de bons r esultats [Dut91] mais ici la matrice est inconnue. Et seul le r esidu fournit les informations sur le probl eme etudi e. 2. Un red emarrage est n ecessaire pour r eduire le cout m emoire mais une d egradation de la convergence est alors observ ee par rapport a GMRES classique. Depuis peu, des algorithmes de ....

L.C Dutto. The effect of ordering on preconditioned gmres algorithm applied to solve the compressible navier-stokes equations on unstructured grids. July 1991. Submitted to Int. J. for Numerical Methods in Engineering.

....stability issues have also been discussed, such as systematically checking some estimate of the norm of (LU) Gamma1 , furthering some understanding of what is required for robust preconditioners. As for any complete or incomplete factorization, the ordering of the matrix plays an important role [10]. Whereas preordering the matrix does not cause any more difficulties with ILUS than with other ILU factorizations, one disadvantage of ILUS is that it cannot easily accommodate dynamic orderings, i.e. orderings that can be generated as the factorization progresses. For example, partial ....

L. C. Dutto. The effect of ordering on preconditioned GMRES algorithms, for solving the compressible Navier-Stokes equations, Int. J. Numer. Methods Engrg., 36 (1993), pp. 457-497.

....methods. Thus, for structured meshes, work has been done both on the development of new implicit schemes [LS88] Sen90] and on the use of new tools to solve large sparse non symmetric linear systems[WYY85] HTS93] Also for unstructured meshes, new schemes[HFM86] SF94] and techniques [JHF91] [Dut91], LF94] Lan94] have been developped. However, in most cases, each implicit step leads to one costly non linear system to be solved. When three dimensional space problems and or chemical context are considered, the size of the systems is usually large. Thus, the memory size of current computers ....

L.C Dutto. The effect of ordering on preconditioned gmres algorithm applied to solve the compressible navier-stokes equations on unstructured grids. July 1991. Submitted to Int. J. for Numerical Methods in Engineering.

....Engineering, Queen s University, Kingston, Ontario. 1 2 Weighted Graph Ordering for PCG Methods 1 Introduction Preconditioned conjugate gradient (PCG) methods have been proven to be robust and competitive techniques for the solution of matrices arising from PDE s in a number of applications [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] ....

....[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 ....

[Article contains additional citation context not shown here]

Laura C. Dutto. The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations. Research report, Universit 'e de Montr'eal, CRM, February 1992. to appear in International Journal for Numerical Methods in Engineering.

....two other measures of the effectiveness of the preconditioner are shown in the Table. First, the condition of the original and preconditioned matrix is shown in the second and third columns. In the fourth column is shown another measure of effectiveness of preconditioning suggested by Dutto[4] which uses the ratio of the Frobenius norm of the remainder matrix, defined as R = A Gamma M , to the Frobenius norm of original matrix. The Frobenius norm is defined as kAkF = s X i;j (A(i; j) 2 kAkF = s X diagonal A 0 A (6) As shown in the Table, both measures indicate the ....

L. C. Dutto, The Effect of Ordering on Preconditioned GMRES Algorithm, for Solving the Compressible Navier-Stokes Equations, International Journal of Numerical Methods in Engineering, 36 (1993), pp. 457--497.

....conjugate gradient method (PCG) with CGSTAB [20, 21] acceleration. An incomplete LU (ILU) type preconditioning is used [22] Iterative methods for N S equations 3 Poor results can be obtained with ILU preconditioning unless careful attention is paid to the ordering of the unknowns in the matrix [12, 3, 23, 24, 25], and even to the discretization used in the preconditioning matrix [3] which may be different, in general, from the discretization used in the actual Jacobian. Another level of sophistication is introduced in this article, by noting that an ILU factorization of the frozen coefficient matrix may ....

Laura C. Dutto. The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations. Research report, Universit'e de Montr'eal, CRM, February 1992. to appear in International Journal for Numerical Methods in Engineering.

....preconditioners have been used in the finite element solution of elliptic p.d.e. s on unstructured meshes, such as simple diagonal scaling or incomplete Choleski factorization for example [12, 26] The second of these is generally superior, especially if a good ordering is chosen for the unknowns [8]. However, parallel methods based upon incomplete factorizations are usually difficult to implement efficiently due to the triangular solves that must be performed at each preconditioning step. From a parallel implementation point of view domain decomposition preconditioners are far more ....

Dutto, L.C., (1993), "The Effect of Ordering on Preconditioned GMRES Algorithm for Solving The Compressible Navier-Stokes Equations". Int. J. Num. Meth. in Eng., 36, 457--497.

....matrix, Peter Brown, had found that reordering this matrix with reverse Cuthill McKee (RCM) ordering makes ILU more effective. The solution, keeping all other parameters the same, was found in 37 steps in this case. In general, reordering has a large effect on the accuracy of ILU preconditioners [14, 15]. Consider now LNS3937. If no pivoting is used, the problem is small pivots. We thresholded the pivots for ILU(30) This decreases condest and helps the residual be reduced further, but there is still no convergence. By increasing amount of fill in with the thresholding also does not help ....

L. C. Dutto. The effect of ordering on preconditioned GMRES algorithms, for solving the compressible Navier-Stokes equations. Int. J. Numer. Methods Engrg., 36:457--497, 1993.

....means that the blocks will have to be inverted. If the block size is small, then inversion will not be too slow, however some type of approximate inverse may be needed for larger sized blocks. Further details on such block preconditioners are found in Axelsson [3] and Concus et al. 31] 49 Dutto [40] has looked at the importance of how the unknowns are ordered in the case of block ILU preconditioning for the Navier Stokes equations and has shown that certain ordering algorithms, such as reverse Cuthill Mcgee, will improve the preconditioner and accelerate convergence of GMRES. Ordering of ....

....have been implemented here. Further investigations could be carried out on other algorithms such as element by element techniques or a more advanced incomplete LU factorization involving higher levels of fill in. Re ordering the unknowns may also lead to convergence acceleration (see Dutto [40] for details) We now see how the size of Krylov dimension affects the rate of convergence when using GMRES with ILU(0) preconditioning. Table 4.5 shows the CPU times for convergence of case A2 on mesh 2 (with a fixed algorithmic Courant number of 50 used in the local timestepping) for different ....

L.C. Dutto. The effect of ordering on preconditioned GMRES algorithm for solving the compressible Navier-Stokes equations. International Journal for Numerical Methods in Engineering, 36:457--497, 1993.

....by Jones and Plassmann (1994) The situation for unsymmetric systems is, however, much less clear. Although there have been many experiments on using incomplete factorizations and there have been studies of the effect of orderings on the number of iterations (Benzi, Szyld and van Duin 1997, Dutto 1993), there is very little theory governing the behavior for general systems and indeed the performance of ILU preconditioners is very unpredictable. Allowing high levels of fill in can help but again there is no guarantee, as we have argued in Section 1. 3 Some Other Forms of Preconditioning 3.1 ....

Dutto, L. (1993), `The effect of ordering on preconditioned GMRES algorithm for solving the Navier-Stokes equations', Int J. Numerical Methods in Engineering 36, 457--497.

....of x and y coordinates) were also tested in the solution of the compressible Navier Stokes equations in [133] The RCM ordering gave slightly better convergence rates over a wide range of problems. RCM is also more efficient in that it creates fewer wavefronts, thus producing longer vectors. Dutto [39] has carried out a systematic study of the effect of various orderings on convergence and has reported similar results. 4.4 Newton Krylov methods All the iterative methods discussed so far sacrifice convergence properties by making a lower order approximation on the left hand side of Eqn. 12) ....

L. Dutto, The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations, Intl. J. for Numer. Meth. in Engrg., 36 (1993), pp. 457--497.

....methods. Thus, for structured meshes, work has been done both on the development of new implicit schemes [LS88] Sen90] and on the use of new tools to solve large sparse non symmetric linear systems[WYY85] HTS93] Also for unstructured meshes, new schemes[HFM86] SF94] and techniques [JHF91] [Dut91], LF94] Lan94] have been developped. However, in most cases, each implicit step leads to one costly non linear system to be solved. When three dimensional space problems and or chemical context are considered, the size of the systems is usually large. Thus, the memory size of current computers ....

L.C Dutto. The effect of ordering on preconditioned gmres algorithm applied to solve the compressible navier-stokes equations on unstructured grids. July 1991. Submitted to Int. J. for Numerical Methods in Engineering.

....achieved in the latter paper. The situation for unsymmetric systems is, however, much less clear. Although there have been many experiments on using incomplete factorizations and there have been studies of the effect of orderings on the number of iterations (D Azevedo, Forsyth and Tang 1992, Dutto 1993) that show similar behaviour to the symmetric case, there is very little theory governing the behaviour for general systems and indeed the performance of ILU preconditioners is very unpredictable. Allowing high levels of fill in can help but again there is no guarantee. In fact, a major problem is ....

Dutto, L. C. (1993), `The effect of ordering on preconditioned GMRES algorithm, for solving the NavierStokes equations', Int J. Numerical Methods in Engineering 36(3), 457--497.

....stability issues have also discussed, such as systematically checking some estimate of the norm of (LU) Gamma1 , furthering some understanding of what is required for robust preconditioners. As for any complete or incomplete factorization, the ordering of the matrix plays an important role [8]. Whereas preordering the matrix does not cause any more difficulties with ILUS than with other ILU factorizations, one disadvantage of ILUS is that it cannot easily accomodate dynamic orderings, i.e. orderings that can be generated as the factorization progresses. For example, partial column ....

L. C. Dutto. The effect of ordering on preconditioned GMRES algorithms, for solving the compressible Navier-Stokes equations, Int. J. Numer. Methods Engrg., 36 (1993), pp. 457-497.

....the coefficient matrix before performing the incomplete factorization can have the effect of producing stable triangular factors, and hence more effective preconditioners. The effects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems have been considered in [5, 8, 9, 10, 11, 13, 20, 26]. Some authors have concluded that the reorderings designed for sparse direct solvers are not recommended for use with preconditioned iterative methods; see, e.g. 8] 20] 26] Of course, there have been cases in which such preorderings have been tried for nonsymmetric problems. Simon [26] ....

....problems, and found essentially no improvement over the original ordering. Similar conclusions were reached by Langtangen [20] who applied a Minimum Degree reordering (MD) with ILU(0) preconditioning of matrices arising from a Petrov Galerkin formulation for convectiondiffusion equations. Dutto [13], in the context of a specific application (solving the compressible NavierStokes equations with finite elements on unstructured grids) was possibly the first to observe that MD and other direct solver reorderings can have a positive effect on the convergence of GMRES with ILU(0) preconditioning. ....

L.C. Dutto. The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations. Internat. J. Numer. Methods Engrg, 36:457--497, 1993.

....moderate number of available processors, with q N . After a global reordering of the unknowns, these are distributed into q groups using a partitioning strategy, and if desired, they are reordered locally to obtain diagonal blocks having a zero nonzero structure adapted to ILU(0) factorizations [2]. The block i (i = 1; q) has associated m i unknowns, where m i N=q and P N i=1 m i = N . After permuting rows and columns, the system matrix A is seen as a block structured one: A = 0 B B B B A 11 A 12 Delta Delta Delta A 1q A 21 A 22 Delta Delta Delta A 2q . ....

....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 henceforth) is a modification of the minimum degree algorithm [7] proposed by Martin [12,2], which reject the new relationships produced by the elimination of a vertex in the structure of the graph associated with the matrix. It was developed to save as much matrix information as possible for an incomplete Gaussian factorization when the factors are constrained to have the same ....

L. C. Dutto. The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations. International Journal for Numerical Methods in Engineering, 36(3):457--497, 1993.

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