J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Mathematics of Computation, 31 (1977), pp. 148--162.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in recent years which compute approximate solutions of (1) via an approximate inverse of A, instead of a factorization. One of the main motivations for using preconditioners of this type is parallelism. Another important reason is that ILU preconditioners, which have been developed for M matrices [20], often fail for indefinite matrices. A few of the approximate inverse techniques are based on minimizing kI Gamma AMk in some appropriate norm [18, 16, 14, 9] Others compute the approximate inverse in factored form by seeking two sparse unit upper triangular matrices W and Z, and a diagonal D, ....

....[3, 5, 2, 17, 23] As it turns out, the latter class of preconditioners show an algebraic behavior that is similar to that of the well known incomplete LU decompositions. For example, they are stable for M and H matrices, in perfect analogy with known results on incomplete LU decompositions in [20, 19]. It is worth mentioning that there has been some work on methods for inverting triangular matrices which are computed from a standard LU factorization, based on the same motivations, see [11] However, our paper does not consider these methods. The purpose of this paper is to take an in depth ....

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J. Meijerink and H. A. V. der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M--matrix. Math. Comp., 31:148--162, 1977.

....we reduce each semi linear subproblem to sequences of linear (sub)problems. The linearized problems are then approximated by the finite difference method proposed by Bakhvalov [1] The resulting finite dimensional problems are solved by the incomplete Choleski conjugate gradient (ICCG) method [16]. In the following, we describe these constituents of our numerical approach in more detail. 3.1.1. Newton s method. Let us consider the following semi linear problem Lw(P ) f(P; w) P where Omega Omega with Omega as well as L and f being defined in Section 1. The Newton s ....

....P M s6=r a r;s ; 1 r M; where r = i (j Gamma 1)N , 1 i N , 1 j N , and M = N . The right hand side vector g = g r ) 2 R M consists of the following components: g r = Gamma j Gamma1 ) F i;j : We apply the so called incomplete Choleski conjugate gradient (ICCG) method [16] to the linear system (10) Its pseudocode is given by: Algorithm 2. Set initial guess w and stopping tolerance ffl 3 . r = g Gamma Aw p = q oe = r; q) DO iteration = 1; m 11 ff = oe= p; Ap) w = w ffp IF max Omega jffiwj ffl 3 THEN STOP r = r Gamma ffAp = r; q) fi = ....

J. Meijerink and H. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31 (1977), pp. 148--162.

....impose on the local solvers to guarantee convergence of the additive Schwarz methods are the following O and (32) i = 1, p. 33) We note that condition (32) is satisfied automatically if A i is an H matrix. This occurs, e.g. if A i is an incomplete factorization of A i [9], 14] Condition (33) is equivalent to having the splitting A i = A i ( A i A i ) be H compatible. Note also that under the conditions (32) 33) since we have O, we conclude that # is a regular splitting. These conditions also provide us with the counterpart to ....

Meijerink, J. A. and van der Vorst, H. (1977), An iterative solution method for linear systems of which the coe#cient matrix is a symmetric M-matrix, Mathematics of Computation 31 pp. 148--162.

....) Figure 3: Sensitivity to the restart parameter of GMRES. GRE1107 test problem. 3.2 Symmetric positive definite linear systems In this section we illustrate, on the set of SPD matrices listed in Table 5, the SPD variant of the update presented in Proposition 5. As a preconditioner we use IC(t) [17]. We observe a similar improvement for SPD linear systems to what was seen in the previous section. This is illustrated in Table 6 where we show the number of CG iterations as we vary the dimension of the positive semi definite update. To show that the improvement of the update is not too closely ....

J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coe#cient matrix is a symmetric M-matrix. Mathematics of Computation, 31:148-- 162, 1977. 16

....a general SPD matrix is a delicate issue which must be addressed if one wishes to design reliable preconditioners. As is well known, the existence of an incomplete factorization A LL (or A LDL in the square root free case) has been established for certain classes of matrices. Already in [3] the existence of the incomplete Cholesky factorization was proved, for arbitrary choices of the sparsity pattern, for the class of M matrices. This existence result was extended shortly thereafter to a somewhat larger class (that of H matrices with positive diagonal entries) by several authors ....

Meijerink JA, van der Vorst HA. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation 1977; 31:148--162.

....systems, Preconditioned conjugate gradient; Variable prcconditioncrs. AMS(MOS) subject classifications. 65F10 1. Introduction. Krylov subspace techniques have increasingly been viewed as general purpose iterative methods, especially since the popularization of preconditioning techniques [2] in the mid 70 s. Although iterative methods lack the robustness of direct methods, they are effective for the large class of problems arising from partial differential equations of the elliptic type. An important gap in the literature concerns the development of truly general purpose iterative ....

J.A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp., 31(137):148 162, 1977.

....L(u) in (8.1) Left: 100. Right: 1000. Example 8.2. Our last set of experiments involves two matrices from the Harwell Boeing collection [7] 23] In all tests, the right hand side was chosen to be the vector of all ones. Additional ( xed) incomplete LU preconditioning was applied [24], 29] In our experiments, we used the Matlab function luinc with tolerance tol to build the preconditioning matrix P . As mentioned earlier in this section, in the exible algorithm this amounts to run the inner solver with the preconditioned matrix AP . We display the convergence history of ....

J.A. Meijerink and H. van der Vorst. An iterative solution method for linear systems of which the coecient matrix is a symmetric M-matrix. Mathematics of Computation, 31:148-162, 1977.

....their extensive use. Computer Science Department, Vanderbilt University, Nashville, TN. tComputer Science Department, University of Minnesota Preconditioning has been recognized as a powerful technique for improving the conver gence of iterative methods for solving linear systems of equations [11, 22]. The original matrix A is multiplied by the inverse of a preconditioning matrix M which is close to A in some sense. This has the effect of bringing the condition number of the preconditioned matrix closer to 1, thereby increasing the convergence rate [9, 14] Applying precondition ing to the ....

....With SOR(k) k SOR iterations may be performed in each preconditioning step. Each iteration consists of the solution of two triangular systems of equations, which is as expensive as a matrix vector multiplication in serial computers. ILUT(p,r) is an extension of the Incomplete LU factorization [11] that uses a two parameter strategy for dropping elements [20] The first parameter controls the allowable fill in in the sparse matrix. The second parameter controls the magnitude of the remaining elements in the factorization. When constructing the current row of L and U, elements that are ....

J.A. Meijerink and H.A. Van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp. 31 (1977) 148.

....techniques recently appeared which compute approximate solutions of (1) via an approximate inverse of A, instead of a factorization. One motivation for using such preconditioning techniques is parallelism. Another motivation is that the ILU preconditioners, which have been developed for M matrices [19], often fail for indefinite matrices. A few of these approximate inverse techniques are based on minimizing the norm III AMll in some appropriate norm [15, 13, 11, 7] Others compute the approximate inverse in factored form by seeking two sparse unit upper triangular matrices W, Z, and a ....

....e.g. 21, 4, 5, 3, 14] The latter class of preconditioners turns out to have an algebraic behavior that is similar to the well known case of the incomplete LU decompositions, e.g. they are stable for M and H matrices. This is the perfect analogy to the result on incomplete LU decompositions in [19, 18]. The purpose of this paper is to take an in depth look at the relationships between these different preconditionings, using the incomplete LU decomposition as a reference point. In particular, we will show that these methods generate factors which can be viewed as approximations of the inverses ....

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J. Meijerink and H. A. V. der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric m-matrix. Math. Comp., 31:148-162, 1977.

....A k in (3.3) by the symmetric part of F (i.e. F F ) 2) and estimate the spectral radius of the symmetric part of F . In this way, the smoothing of the prolongator maintains a sense of energy minimization. We have found that this procedure is quite e ective when an incomplete LU factorization [23] is used as a smoother. For the Reynolds numbers that we have considered, the resulting multigrid procedure is quite ecient. Our numerical results (Section 5) demonstrate convergence for the F solve within about 25 multigrid iterations. However, for highly convective ows it should be possible to ....

J.A.Meijerink and H. V. der Vorst, An iterative solution method for linear systems of which the coecient matrix is a symmetric m-matrix, Math. of Comp., 31 (1977), pp. 148-162.

....perturbations to an M matrix that yields another M matrix. matrix that arises by zeroing out the off diagonal elements in the first column of A via one step of Gaussian elimination. If A is an M matrix then A ( exists and is also an M matrix. 21 Theorem 5 ( due to Meijerink and van der Worst [50]) Let A = aii) be an n x n M matrix. If the matrix B = bii) is such that aij bij 0 for i j and 0 aii bii then B is also an M matrix. We also need the well known fact that MGS applied to A produces a upper triangular factor that is identical to the matrix L T produced by the Cholesky ....

....ATA is an M matrix, then A0)TA0) is also an M matrix. Proof: This results directly from Theorem 4 and Lemma 1. The results above can be generalized to IC and IMGS. The generalization to IC appears in the literature and can be stated as Lemma 3. Lemma 3 ( due to Meijerink and van der Worst [50]) Let A: aij) be a symmetric positive definite matrix and let AO) a) be the matrix that arises by from one step of lC. If A is an M matrix then AO) exists and is also an M matrix. A similar property can be derived for IMGS. Lemma 4 For a given A R x and a set P C P, Let be the matrix ....

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J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation, 31(137):148 162, 1977.

....Table 3: Single solve of the Bratu Problem, u 0 = 0; 6:8. of the form F (u s ; s) 0; u s ; s) s) 0; where , a scalar valued function, is chosen such that s is some arc length on the solution branch and u s is the solution of (13) for = s) We preconditioned GMRES by ILU(0) [18] of the discretized Laplace operator Delta. The first table Tab. 2 shows the results after a full continuation run: starting from the smallest solution (u; with = 1 the solution branch is followed along the (discretized) arc with s n = s 0 n Deltas for step size Deltas = 1 and n = 1; 2; ....

J. A. Meijerink and H. A. Van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148--162.

....Through work of Reid [79] it became clear that these methods, if used as iterative techniques, could be used to advantage for many classes of linear systems. With preconditioning [21] these methods could be made even more efficient. Among the first popular preconditioned methods was ICCG [69, 63]. In the period after 1975, we have seen that symmetric positive definite sparse systems were usually solved by preconditioned CG methods, when very large, and by sparse direct solvers if moderately large. For indefinite symmetric sparse systems special variants were proposed, like MINRES, and ....

....of iterative methods, one applies usually some form of preconditioning. Many different preconditioners have been suggested over the years, each of these preconditioners more or less successful for restricted classes 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 ....

J. A. Meijerink and H. A. Van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math.Comp., 31:148--162, 1977.

....diffusion equations arising from nuclear reactor simulations. He also introduced very useful matrix notations for the analysis and further developments along this line were made by Beauwens [19] Axelsson [2] considered these methods as generalized relaxation methods. Meijerink and Van der Vorst [75] considered these methods as incomplete factorizations and they proved the existence of ILU preconditioners for M matrices. Gustafsson [63] proposed a modified version of the ILU preconditioner with improved spectral properties. Finally, the paper of Kershaw [69] provided convincing numerical ....

....It is easy to see that a sufficient condition is that Algorithm ILU and Algorithm MILU given earlier in this section do not encounter a zero pivot (i.e. a r;r = 0. It turns out that this can only be guaranteed for certain special classes of matrices. Specifically, Meijerink and Van der Vorst [75] proved that the ILU factorization for arbitrary fill patterns exists for Mmatrices (i.e. matrices A with a i;j 0; i 6= j; and A 0 componentwise) Moreover, they proved that the splitting A = M Gamma N is regular, i.e. ae(M N) 1, which implies that the fixed point iteration x i 1 = ....

J.A. Meijerink and H.A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math Comp, 31:148--162, 1977.

....or BiCGstab( 19] The rate of convergence and the efficiency of Krylov subspace methods is often improved by preconditioning. The identification of an effective preconditioner may be a problem. For instance, for interior eigenvalues the construction of an effective incomplete LU factorization [10], 8] for A Gamma e I may require much fill in , which makes the construction expensive. As we will argue in x2.7, it may be a good strategy to compute a preconditioner K for A Gamma I for a fixed value of only, and to use K j (I Gamma e q e q ) 13) as the preconditioner for various ....

J. A. Meijerink and H. A. Van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148--162.

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J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Mathematics of Computation, 31 (1977), pp. 148--162.

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J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coe#cient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148--162.

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J.A. Meijerink and H.A. van der Vorst, 1977. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation, 31: 148--162.

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J.A. Meijerink and H.A. Van der Vorst, An Iterative Solution Method for Linear Systems of which the Coecient Matrix is a Symmetric M-Matrix, Math. Comput. 31 (1977), pp. 148-162.

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J. Meijerink and H. Van der Vorst, An iterative solution method for linear systems of which the coecient matrix is a symmetric M-matrix, Math. Comput., 31 (1977), pp. 148-162.

No context found.

J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coe#cient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148--162.

No context found.

J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp., 31:148--162, 1977.

No context found.

J.A. Meijerink and H. van der Vorst, An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric Mmatrix, Math. Comp. 31, pp. 148-162, 1977.

No context found.

J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coe#cient matrix is a symmetric Mmatrix. Math. Comp., 31:148--162, 1977.

No context found.

J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a sysmmetric M-matrix. Math. Comp., 31:148--162, 1977.

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