H.C. Elman, 1989. Relaxed and stabilized incomplete factorizations for non-selfadjoint linear systems. BIT, 29: 890--915.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of problems. Among all these preconditioners the Incomplete LU factorizations [69, 21] are the most popular ones, and attempts have been made to improve 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 ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for nonself -adjoint linear systems. BIT, 29:890--915, 1989.

....resulting linear system, like the one corresponding to the uniform grid, can be quite challenging for iterative methods if convection is strong. Again, a simple reordering of the coe#cient matrix can improve the situation dramatically. To illustrate this, we take the following example from Elman [26]. Consider the following partial di#erential equation in# (0, 1) #u 2P = g, 2) where P 0 and the right hand side g and the boundary conditions are determined by the solution u(x, y) 2P (1 x) 2Py . This function is nearly identically zero in# except for ....

....for boundary layers of width O(#) near x = 0 and y = 1, where # = 1 2P . A uniform coarse grid was used in the region where the solution is smooth, and a uniform fine grid was superimposed on the regions containing the boundary layers, so as to produce a stable and accurate approximation; see [26] for details. We performed experiments with P = 500 and P = 1000. These values are considerably larger than those used in [26] The resulting matrices are of order 5041 and 7921, with 24921 and 39249 nonzeros, respectively. The convergence criterion used was a reduction of the residual norm to ....

[Article contains additional citation context not shown here]

H. C. Elman, Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, BIT, 29 (1989), pp. 890--915.

....positive value for zero or negative pivots. While this trivially guarantees the existence of the factorisation, it is likely to lead to a large condition number for (M 1 A) Choosing the repair value is not trivial: too small a value will give unstable recurrences during the solution process [23, 24], so, even while the preconditioner is SPD, the iterative method may diverge. Overestimates of the optimal repair value will make the preconditioner too diagonally dominant, in e ect turning it into a Jacobi method. Kershaw s choice for the repair pivot is m kk = P j k jm kj j P j k jm jk ....

....1 in absolute value. For example, solving Lu = v for a matrix L with constant diagonals L = n ; 0; 0; 1 ; d) corresponds to the recurrence du i 1 u i 1 n u i n = v i , with a characteristic polynomial dx n 1 x n 1 n . The stability of this was analysed by Elman [23, 24], and found to be equivalent to the factors being diagonally dominant. For a short proof, consider the recurrence a 0 x i n X j=1 a j x i j = f i with characteristic solutions x i = i where is a solution of a 0 i n X j=1 a j i j = 0: Now suppose that the matrix is ....

Howard C. Elman. Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems. BIT, 29:890-915, 1989.

....Meijerink and van der Vorst (1977) and then was extended to H matrices and block H matrices, for which theoretical properties such as existence, stability and accuracy can be established. For details one can refer to Axelsson (1994, 1985) Concus, Golub and Meurant (1985) Donato and Chan (1992) Elman(1986, 1989), Gustafsson (1978) Manteuffel (1980) Saad (1996) and references therein. For general unsymmetric matrices, although a number of efficient incomplete LU factorization techniques have been presented (see Axelsson (1994) and Saad (1996) it is more difficult to give theoretical assurances about ....

Elman, H. (1989), `Relaxed and stabilized incomplete factorizations for non-selfadjoint linear systems', BIT 29, 890--915.

.... Since ILU and IC factorizations were the most popular preconditioners, at least in a sequential environment, many attempts have been made to improve them, for instance by including more fill [118] by 15 modifying the diagonal of the ILU factorization in order to force rowsum constraints [87, 7, 6, 122, 171, 61], or by changing the ordering of the matrix [173, 174] A set of experiments with respect to the effects of ordering is contained in [56] Saad [144] proposed a few variants on the incomplete LU approach for the matrix A, one of which is in fact an incomplete LQ decomposition. In this approach it ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems. BIT, 29:890--915, 1989.

....A similar strategy is to factor a shifted matrix A ffI , where ff is a positive scalar so that A ffI is well conditioned [30, 47] Such a strategy too obviously has a tradeoff between stable and accurate factorization. For more studies on the stability of ILU factorizations, we refer to [19, 32, 45, 13, 48]. It is also possible to reorder the rows of the matrix so that their diagonal dominance in a certain sense is in decreasing order. In this way, small pivots are in the last rows of the matrix and may not be used in an ILU factorization. This strategy also has some problems since the values of ....

H. C. Elman. Relaxed and stabilized incomplete factorization for nonselfadjoint linear systems. BIT, 29(4):890--915, 1989.

....preconditioner, variants of the incomplete factorizations with fill in are feasible they just have to fulfill the restrictions on the block structure of the matrices. Additionally, a lumping of the defect matrix entries to the main diagonal can be done in the factorization , e.g. MILU or RILU [2,3,21]. Although the acceleration effect of MILU RILU [21] was observed in the sequential code there was no decrease in the iteration count of the parallel calculations. This behavior seems to be connected with the unfavourable local numbering used in the parallel code (boundary nodes are numbered ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for non-selfadjoint linear systems. BIT, 29(4):890--915, 1989.

....Benzi, Daniel B. Szyld and Arno van Duin responding to the uniform grid, is very challenging for iterative methods if convection is strong. Again, a simple reordering of the coe#cient matrix can improve the situation dramatically. To illustrate this, we take the following example from Elman [22]. Consider the following partial di#erential equation in# = 0, 1) 0, 1) #u 2P #u #x 2P #u #y = g (2) where the right hand side g and the boundary conditions are determined by the solution u(x, y) e 2P (1 x) 1 e 2P 1 e 2Py 1 e 2P 1 . This function is ....

....except for boundary layers of width O(#) near x =0andy =1,where# =1 2P . A uniform coarse grid was used in the region where the solution is smooth, and a uniform fine grid was superimposed to the regions containing the boundary layers, so as to produce a stable and accurate approximation; see [22] for details. We performed experiments with P = 500 and P = 1000. These values are considerably larger than those used in [22] The resulting matrices are of order 5041 and 7921, with 24921 and 39249 nonzeros, respectively. When ILU(0) preconditioning was used, no iterative solver converged in ....

[Article contains additional citation context not shown here]

Howard C. Elman. Relaxed and stabilized incomplete factorizations for non-selfadjoint linear systems. BIT, 29:890--915, 1989.

....solvers. Since ILU and IC factorizations were the most popular preconditioners, at least in a sequential environment, many attempts have been made to improve them, for instance by including more fill [115] by modifying the diagonal of the ILU factorization in order to force rowsum constraints [82, 7, 6, 118, 164, 57], or by changing the ordering of the matrix [165, 166] A set of experiments with respect to the effects of ordering is contained in [52] Saad [140] proposed a few variants on the incomplete LU approach for the matrix A, one of which is in fact an incomplete LQ decomposition. In this approach it ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems. BIT, 29:890--915, 1989.

....incomplete factors L and U may be much worsely conditioned than the original matrix A. A coupled effect is that the long recurrences associated with solving with these factors are unstable [6, 16] A remedy is also to use diagonal perturbations, this time to make the factors diagonally dominant [27, 37, 17], but the perturbations in this case may need to be very large. Work supported in part by the National Science Foundation under grant NSF CCR 9618827 and in part by NASA under grant NAG2 904. Experimental Study of ILU 2 ILU preconditioners have also been applied successfully to indefinite ....

....to the regular, unmodified factorization, ILU Gamma1 corresponds to MILU, and ILU 1 corresponds to the modification of Jennings and Malik [22] For elliptic problems that are not M matrices, modification may also have a stabilizing effect if it increases the value on the diagonal. Elman [17] used this as part of his criteria to modify certain rows and not others, in his stabilized factorization based on RILU. Recently, the method of diagonal compensation [1] has been developed for preconditioning positive definite matrices with incomplete factorizations. Essentially, the SPD matrix ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems. BIT, 29:890--915, 1989.

....A similar strategy is to factor a shifted matrix A ffI , where ff is a positive scalar so that A ffI is well conditioned [27, 44] Such a strategy too obviously has a tradeoff between stable and accurate factorization. For more studies on the stability of ILU factorizations, we refer to [19, 29, 42, 13, 45]. It is also possible to reorder the rows of the matrix so that their diagonal dominance in a certain sense is in decreasing order. In this way, small pivots are in the last rows of the matrix and may not be used in an ILU factorization. This strategy also has some problems since the values of the ....

H. C. Elman. Relaxed and stabilized incomplete factorization for nonselfadjoint linear systems. BIT, 29(4):890--915, 1989.

....Libre de Bruxelles, Service de M etrologie Nucl eaire (CP 165) 50 av. F.D. Roosevelt, B 1050 Brussels, Belgium. ynotay ulb.ac.be. Research supported by the Fonds National de la Recherche Scientifique , Belgium. 2 M.M. MAGOLU AND Y. NOTAY 1. it does not suffer a lot from existence problem [13, 17, 18]; 2. it is robust with respect to discontinuities and anisotropy [4, 34] Two major inconveniences are that the optimal value of the relaxation parameter strongly varies from a problem to another and the behavior could be very sensitive to variations of around the observed opt [12, ....

H.C. Elman, Relaxed and stabilized incomplete factorizations for nonself-adjoint linear systems, BIT, 29 (1989), pp. 890--915.

....are unable to resolve fine features of the solution, such as boundary layers. A possible solution is to use centered differences, but with a local mesh refinement over regions where the solution is expected to exhibit strong variations. To illustrate this, we take the following example from Elman [15]. Consider the following partial differential equation in Omega = 0; 1) Theta (0; 1) Gamma Deltau Gamma 2P u x 2P u y = g (5.5) where P 0, and the right hand side g and the boundary conditions are determined by the solution u(x; y) e 2P (1 Gammax) Gamma 1 e 2P Gamma 1 ....

....x = 0 and y = 1, where ffi = 1=2P . A uniform coarse grid was used in the region where 14 Michele Benzi and Miroslav Tuma the solution is smooth, and a uniform fine grid was superimposed to the regions containing the boundary layers, so as to produce a stable and accurate approximation; see [15] for details. We performed experiments with P = 500 and P = 1000. These values are considerably larger than those used in [15] The resulting matrices are of order 5041 and 7921, with 24921 and 39249 nonzeros, respectively. The convergence criterion used was a reduction of the initial residual ....

[Article contains additional citation context not shown here]

H. C. Elman, Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, BIT, 29 (1989), pp. 890--915.

....5 point finite difference discretization of Deltau fi(u x u y ) f , then for fi sufficiently large, the incomplete LU factors may be very ill conditioned even though A has a very modest condition number. Remedies for reducing the condition numbers of e L and e U have been discussed by Elman (1989) and van der Vorst (1981) 2.4 Some General Comments on ILU The use of incomplete factorizations as preconditioners for symmetric systems has a long pedigree (Meijerink and van der Vorst 1977) and good results have been obtained for a wide range of problems. An incomplete Cholesky factorization ....

Elman, H. C. (1989), `Relaxed and stabilized incomplete factorizations for non-selfadjoint linear systems', BIT 29, 890--915.

....found in [37, 40, 41, 42, 44] Although different fourth order compact schemes may have different formula, their performances appear to be similar. These three discretization schemes are easy to implement and the resulting linear systems are frequently used as test examples for iterative methods [5, 10, 12, 24, 28, 33]. However, their relative advantages and disadvantages, especially for the fourth order compact schemes, in terms of the quality of computed solution, the algebraic properties of the coefficient matrix and the performance of iterative methods, are not completely clear. We mention that developing ....

....Exact solution of these two problems are not related to the Reynolds number and were chosen to model the non smooth boundary conditions. 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Problem 2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Problem 1 Figure 1: Characteristic directions of Problems 1 and 2. Problem 3 was used by Elman [12] to test the relaxed and stabilized incomplete factorizations for solving non self adjoint linear systems. Although the convection coefficients are constant, the solution is related to the Reynolds number. Even for moderate Re, the solution is nearly identically zero in Omega except for boundary ....

H. C. Elman, Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, BIT 29, 890--915 (1989).

....unstable and nonphysical solutions may result when the conditions on =h are violated, particularly when boundary layers are present in the solution. In this case, the computed solutions are still useful in that they allow to determine the existence and location of such boundary layers; see, e.g. [9]. In the remainder of the paper, we will not address the issue of whether the discrete solution is a good approximation to the continuous solution. We have employed three Krylov subspace accelerators as follows: Conjugate Gradient (CG) 14] for Problem 1, Bi CGSTAB [25] for Problem 2, and ....

H. C. ELMAN, Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, BIT, 29 (1989), pp. 890--915.

....of problems. Among all these preconditioners the Incomplete LU factorizations [69, 21] are the most popular ones, and attempts have been made to improve 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 ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for nonself -adjoint linear systems. BIT, 29:890--915, 1989.

....are unable to resolve fine features of the solution, such as boundary layers. A possible solution is to use centered differences, but with a local mesh refinement over regions where the solution is expected to exhibit strong variations. To illustrate this, we take the following example from Elman [16]. Consider the following partial differential equation in Omega = 0; 1) Theta (0; 1) Gamma Deltau Gamma 2P u x 2P u y = g (5.5) where P 0, and the right hand side g and the boundary conditions are determined by the solution u(x; y) e 2P (1 Gammax) Gamma 1 e 2P Gamma 1 ....

....for boundary layers of width O(ffi) near x = 0 and y = 1, where ffi = 1=2P . A uniform coarse grid was used in the region where the solution is smooth, and a uniform fine grid was superimposed to the regions containing the boundary layers, so as to produce a stable and accurate approximation; see [16] for details. We performed experiments with P = 500 and P = 1000; see Table 6. These values are considerably larger than those used in [16] The resulting matrices are of order 5041 and 7921, with 24921 and 39249 nonzeros, respectively. The convergence criterion used was a reduction of the initial ....

[Article contains additional citation context not shown here]

H. C. Elman, Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, BIT, 29 (1989), pp. 890--915.

....the resulting linear system, like the one corresponding to the uniform grid, is very challenging for iterative methods if convection is strong. Again, a simple reordering of the coefficient matrix can improve the situation dramatically. To illustrate this, we take the following example from Elman [16]. Consider the following partial differential equation in Omega = 0; 1) Theta (0; 1) Gamma Deltau Gamma 2P u x 2P u y = g (2) where the right hand side g and the boundary condi Table 2: Number of iterations for different orderings, several preconditioners Bi CGSTAB GMRES(20) ....

....for boundary layers of width O(ffi) near x = 0 and y = 1, where ffi = 1=2P . A uniform coarse grid was used in the region where the solution is smooth, and a uniform fine grid was superimposed to the regions containing the boundary layers, so as to produce a stable and accurate approximation; see [16] for details. We performed experiments with P = 500 and P = 1000. These values are considerably larger than those used in [16] The resulting matrices are of order 5041 and 7921, with 24921 and 39249 nonzeros, respectively. When ILU(0) preconditioning was used, no iterative solver converged in ....

[Article contains additional citation context not shown here]

H.C. Elman. Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems. BIT, 29:890--915, 1989.

....a set of d independent discrete convection di usion operators. Although the convection di usion equation is a more dicult problem than the Poisson equation (in particular, the analysis of solution algorithms is far less well developed) there are e ective solvers available for it, see for example [2, 9, 24, 42]. The Schur complement system is less straightforward. An operator Q S that is spectrally equivalent to the pressure mass matrix, as discussed above, is easy to implement and has also been shown to lead to (essentially) mesh independent rates of convergence for (3.1) 3.2) 8, 19] However, ....

H. C. Elman. Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems. BIT, 29:890-915, 1989.

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H.C. Elman, 1989. Relaxed and stabilized incomplete factorizations for non-selfadjoint linear systems. BIT, 29: 890--915.

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