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.

....with problem size, and this could easily explain the faster than expected growth in factofization time. It should also be noted as coincMental that the measured computation time growth of ILUCG for the grid circuit problem matches the theoretical result for a 2 D discretized Laplacian (see [ 18]) The conductance and capacitance to ground in the circuit implies that the linear system at each timestep is more diagonally dominant than the 2 D discretized Laplacian, and therefore ILUCG converges faster for these examples. That this accelerated convergence doesn t reflect itself in reduced ....

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., vol. 31, pp. 148-162, 1977.

....consider some common preconditioning methods: 1. The diagonal preconditioner, MD = diag(A) is the simplest preconditioner. 2. The SSOR preconditioner is given by M ot = D wL)D i(D wU) where A = L D U, and co is a design parameter. 3. Incomplete Cholesky factorization preconditioners [115] which are based on approxi mate factorizations of A in order to limit the amount of fill ins. Although preconditioners based on incomplete factorization are quite effective in reducing the number of iterations, significant computational effort is required to initially construct them. Moreover, ....

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, 31:148162,

....case includes many standard splittings such as the diagonal, tridiagonal, or triangular part, as well as block versions of them. The other notable example is incomplete factorizations :i Li Ui where the nonzeros of the factors are in the locations of the nonzeros of Ai, and in particular ILU(0) [30]. In these cases, the inequality (3.10) holds, or equivalently, we have (weak) regular splittings [30,41] 4. Comparison theorem This section can be read independently of the rest of the paper. Theorem 4.1 Let A O. Let A iV] iV M N be two weak regular splittings such that (4.1) M M . ....

....as block versions of them. The other notable example is incomplete factorizations :i Li Ui where the nonzeros of the factors are in the locations of the nonzeros of Ai, and in particular ILU(0) 30] In these cases, the inequality (3. 10) holds, or equivalently, we have (weak) regular splittings [30,41]. 4. Comparison theorem This section can be read independently of the rest of the paper. Theorem 4.1 Let A O. Let A iV] iV M N be two weak regular splittings such that (4.1) M M . Let w 0 be such that w A e for some e O. Then, 4.2) ivIl ;Vll. If the inequality in (4.1) is ....

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

....this reason we do not show these results in the tables. We report the results of experiments with the following accelerators: Bi CGSTAB, TFQMR, and GMRES with restart parameter m = 20. The preconditioners used were standard incomplete factorizations based on levels of fill (ILU(0) and ILU(1) see [35]) and Saad s dual threshold ILUT; see [41] 42] For ILUT, we used two di#erent sets of parameters, 10 2 , 5) and (10 3 , 10) The latter results in a very powerful but expensive preconditioner, containing up to five times the number of nonzeros in A. Right preconditioning was used in all ....

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.

....a good parallel preconditioner not only decreases the total number of floating point operations, but also possesses a high degree of parallelism. A good preconditioner depends both on the system matrix and the parallel computer. For instance, the incomplete Cholesky factorization preconditioner [29] is very successful on sequential computers, but performs not as good on vector and parallel computers. Polynomial preconditioners [30] are very well suited for parallel computers [31, 32] and experiments have shown that, implemented on a distributed memory computer, they can be much more ....

....accumulate operation introduces the calculations that should be accounted for by tpva. calc. After every communication step the received vectors of [ n pl 2 complex words are added to the partially accumulated vector in the diagonal processor. A total ofp 1 2 vectors must be added, therefore [29] cylinder = 2(X 1) rcalc tpva.calc The complex accumulate is straightforward, the partial inner products are first accumulated in the horizontal direction, followed by an accumulation in the vertical direction. In both sweeps pl 2 complex number are added. This leads to the following ....

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) 148-162.

....SIMD or vector processors) Numerical examples show that, in both cases, the overhead in the number of iterations required for convergence of the preconditioned iteration is small relatively to the speed up gained. 1 Introduction The Incomplete LU (ILU) decomposition of a sparse matrix [15] [16] [24] is considered as one of the most powerful and robust methods for the solution of sparse linear systems of equations. The idea is to construct sparse triangular matrices L and U such that LU approximates the coefficient matrix in some sense. Then LU serves as a preconditioner in a ....

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

....methods is to reject those fill in entries outside a priori or adaptively (dynamically) chosen sparsity pattern. An incomplete factorization A = LL 0 R is called the incomplete Cholesky decomposition, whose existence and uniqueness has been proved for M matrix by Meijerink and van der Vorst [37]. Because LL is s.p.d. LL is also s.p.d. thus the incomplete Cholesky factorization can be used as the preconditioner of the CG method. The CG method with the incomplete Cholesky decomposition preconditioner is called the ICCG method [37, 31] When a sparsity pattern S is given a ....

....for M matrix by Meijerink and van der Vorst [37] Because LL is s.p.d. LL is also s.p.d. thus the incomplete Cholesky factorization can be used as the preconditioner of the CG method. The CG method with the incomplete Cholesky decomposition preconditioner is called the ICCG method [37, 31]. When a sparsity pattern S is given a priori, the incomplete Cholesky decomposition A = LL 0 R, which is called factorization by position, is described by Figure 4.6. If (i; j) 2 S, the fill in entry is accepted. On the other hand, the sparsity set can be given dynamically during the ....

Meijerink, J. A. and H. A. van der Vorst, "An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix," Math. Comput., vol. 31, pp. 148--152, 1977.

....many standard splittings such as the diagonal, tridiagonal, or triangular part, as well as block versions of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) [26]. In these cases, the inequality (3.10) holds, or equivalently, we have (weak) regular splittings [26] 36] 4. Comparison Theorem. This section can be read independently of the rest of the paper. N = M Gamma N be two weak regular splittings such that : 4.1) Let w 0 be such that w = ....

....versions of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) 26] In these cases, the inequality (3. 10) holds, or equivalently, we have (weak) regular splittings [26], 36] 4. Comparison Theorem. This section can be read independently of the rest of the paper. N = M Gamma N be two weak regular splittings such that : 4.1) Let w 0 be such that w = A e for some e 0. Then, k M Nkw kM Nkw : 4.2) If the inequality in (4.1) is strict, ....

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

....using METIS s ordering. Hence, we only use the natural ordering for MICC. We use both orderings to unmodified ICC. We implemented a sparse Cholesky factorization algorithm specifically for this project. The code can perform complete, no fill incomplete (sometimes known as ICC(0) or ICCG(0) [20]) and droptolerance incomplete factorization. When performing incomplete factorizations, the code can modify the diagonal to ensure that the row sums of LL T are equal to those of A [16] The code implements a sparse left looking algorithm. The code is e#cient in the sense that its running time ....

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. Mathemathics of Computation, 31:148--162, 1977.

.... but is widely used to solve indefinite or nearly semi definite symmetric problems. The key to the success of these iterative methods is often the use of efficient preconditioners 3 aimed at accelerating their convergence and considerable work has also been published on these aspects (see [2, 7, 9, 21, 22] among many others) Extensions have been made on variable or flexible preconditioners allowing for preconditioners to be fine tuned through the course of the resolution [30] and even across separate closely coupled resolutions [6] For the case of the matrices arising in the GCV application ....

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--152, 1977. 20

....The other code is a column oriented left looking sparse Cholesky code. The code is e#cient in the sense that its running time is proportional to the number of floating point operations that it performs. It can perform complete, no fill incomplete (sometimes known as IC(0) or ICCG(0) [20]) and drop tolerance incomplete factorization. When performing incomplete factorizations, the code can modify the diagonal to ensure that the row sums of LL T are equal to those of A [14] or it can use a relaxed modification, which is more robust [2, 4, 5, 27] The performance of this code is ....

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, Mathemathics of Computation, 31 (1977), pp. 148--162.

....this research, we call so. briefly PBEs) The PBEs means the matrix elements corresponding to the periodic boundary conditions under discreted system. To solve these systems, various preconditioned iterative methods are applied to. For example, the preconditioned conjugate gradient (PCG) method [9][12] is used for symmetric positive definite systems. Here, preconditioners should be chosen to attain fast convergence. The preconditioning means the transformation of a linear system to an equivalent system which is easier to solve than the given one [3] 7] When deciding on which preconditioner K ....

....in [10] the converging behavior is distinctive between the SC and the block incomplete factorization. The convergence rate of the CG method depends on spectral properties of the given matrix, such as the eigenvalue distribution and the condition number, and on the given right hand side [3] 4] 5][12][13] In this paper, the spectral properties by using the SC and block incomplete Cholesky factorization are evaluated. In section II, physical model, the linear system and typical solver for this system is presented. In section III, the SC preconditioner is detailed, and the e#ect of the SC and ....

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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. Comput., 31, pp. 148--162, 1977.

....introduce some strategies for building symmetric preconditioners based on Frobenius norm minimization. In the later sections, we briefly present more classical techniques like a factorized approximate inverse preconditioner namely AINV [6, 7, 9] and FSAI [20] and incomplete Cholesky factorization [24]. In Section 3, we study the numerical behaviour of those preconditioners on a set of model problems representative of real calculations in electromagnetics applications. In particular, we give some clues to explain the poor behaviour of some of them. We conclude this paper with some remarks in ....

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.

.... employed on a wide range of symmetric problems, providing a good balance between computational costs and reduction of the number of iterations (see [11] and [18] Well known theoretical results on the 4 existence and stability of the factorization can be proved for the class of M matrices [37], and recent studies involve more general symmetric matrices, both structured and unstructured. On these problems, however, the occurrence of a small or negative pivot during the incomplete process may require some diagonal modi cations of the matrix to improve stability. Another important ....

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. Mathematics of Computation, 31:148-162, 1977.

....parameter) but it is likely to be more accurate than the positional dropping strategy. 2.1.2 Pivot repair A further algorithmic issue to consider is how the method deals with zero or negative pivots, should they occur. The existence question of incomplete LU factorisations was fully solved in [34] for the case of M matrices 2 . This paper and subsequent generalisations such as [1] established that for M matrices ll in can be totally or partially ignored, while the M matrix property is preserved for the remaining submatrix. As a result, all pivots are guaranteed to be positive and no ....

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 Comp, 31:148-162, 1977. 14

....# # =0 (B 1 i,# C i,# ) # B 1 i,# = A 1 i,# . Since A i,# is an M matrix, many of the standard iterative methods indeed represent regular splittings (and thus weak nonnegative splittings of either type) Jacobi, Gauss Seidel and their block variants, and also several ILU splittings; see [28, 39]. With these observations it should be obvious that we can now establish a theory for inexact restricted additive Schwarz methods by following the lines of the previous 476 ANDREAS FROMMER AND DANIEL B. SZYLD sections. In particular, analogous to Theorem 4.4 we get that if A is an M matrix and ....

J. A. Meijerink and H. 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.

....A i; In this section, we consider the case were the subdomain problems are solved approximatively or, in other words, inexactly. We represent this fact by using an approximation A i; of the matrix A i; In practice one uses for example an incomplete factorization of A i; see, e.g. [18], 25] As in [14] suppose that the inexact solves are such that the splittings A i; A i; A i; A i; are weak regular splittings (21) for i = 1; p; or that A i; is an M matrix and A i; A i; i = 1; p: 22) Note that (22) implies (21) The incomplete factorizations ....

.... [14] suppose that the inexact solves are such that the splittings A i; A i; A i; A i; are weak regular splittings (21) for i = 1; p; or that A i; is an M matrix and A i; A i; i = 1; p: 22) Note that (22) implies (21) The incomplete factorizations satisfy (21) [18]. The restricted multiplicative Schwarz iteration with inexact solves on the subdomains is then given by TRMS; 1 Y i=p (I R i; A 1 i; R i; A) In a way similar to (10) we construct matrices M i; T i A i; O O D:i; i ; 23) such that R i; A 1 i; R i; A = ....

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.

....using METIS s ordering. Hence, we only use the natural ordering for MICC. We use both orderings to unmodified ICC. We implemented a sparse Cholesky factorization algorithm specifically for this project. The code can perform complete, no fill incomplete (sometimes known as ICC(0) or ICCG(0) [12]) and droptolerance incomplete factorization. When performing incomplete factorizations, the code can modify the diagonal to ensure that the row sums of LL T are equal to those of A [8] The code implements a sparse left looking algorithm. The code is e#cient in the sense that its running time ....

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. Mathemathics of Computation, 31:148--162, 1977.

....the choice of . However, for 50, also the preconditioned methods are non convergent. For that reason we have limited ourselves in the numerical experiments to the natural scaling ( 1) Preconditioners We now consider two incomplete LU decompositions: the classic ILU decomposition and ILUD [17]. Both preconditioners are combined with Krylov subspace methods to solve the system (16) For the explanation of the incomplete LU decompositions we define the set P , which consists of pairs (i; j) such that the i th component of the vector of unknowns is connected to the j th component. It ....

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.

....grid in Figure 5 consists of 10800 grid cells. For the Poisseuille problem a Cartesian grid of 80 Theta 80 cells is used. The subdomain problems are solved using GMRES with RILU(ff) preconditioning [50, 48] and a relative stopping criterion. For ff = 0 we get the standard ILU preconditioner [36] and for ff = 1 we get the Modified ILU preconditioner [27] RILU(ff) 1] lies in between these two. With RILUD(ff) we mean RILU(ff) restricted to the diagonal. The momentum equations are solved using a RILUD(0:95) preconditioner and the pressure equations using a RILU(0:975) As a short hand, we ....

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.

....it provides in convergence rate. A final desirable property in a preconditioner is that it should parallelize well, especially on distributed memory computers. Probably the most effective preconditioning strategy in common use is to take K = LL T to be an incomplete Cholesky factorization of A [16]. For discretizations of second order PDEs in two dimensions, defined on a grid with spacing h, we have with incomplete Cholesky factorization, h Gamma2 ; with a modified IC factorization[9, 1] h Gamma1 ; and with a multigrid cycle, 1. Preconditioners such as multigrid and some domain ....

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, 31:148-- 162, 1977.

....Residual norm 8 smallest eigenpairs Complete Jacobi Davidson Linear system Figure 1. Residual norm against MVM for ILU and MILU preconditioning (h 1 = 180) For K , we considered three di erent preconditioners computed from A : the standard incomplete Cholesky factorization without ll in (ILU) [28, 30, 32, 33, 34], the modi ed version for which K and A have same row sum (MILU) 28, 30, 32, 35, 36, 37] and the algebraic multilevel preconditioner from [38] AML) which leads to a condition number independent of h although it is only twice as expensive per iteration than the incomplete factorization ....

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 1977; 31:148-162.

....in a few iterations at each time step. We implemented one of the simplest preconditioners based on the incomplete Cholesky (IC) factorization of A. So M is constructed with a Cholesky factorization of A, but only the entries corresponding to the nonzero positions of A are computed and stored [13]. Hence, M is as sparse as A and contains 7 diagonals. The incomplete Cholesky algorithm we used is a generic one for general sparse matrices taken from [9] For reference convenience, it is reproduced in Algorithm 4.2, where we use the notation M = m i;j ) Algorithm 4.2 Procedure for ....

....1;1 = p a 1;1 . 12 2. For i = 2 to n 3. For j = 1 to i Gamma 1 4. If a i;j = 0 then m i;j = 0 else 5. m i;j = i a i;j Gamma P j Gamma1 k=1 m i;k m j;k j =m j;j 6. m i;i = q a i;i Gamma P i Gamma1 k=1 m 2 i;k Since A is a Stieltjes matrix, Algorithm 4. 2 is guaranteed to finish [13]. Algorithm 4.2 actually computes a lower triangular matrix ML . The preconditioner is then taken as M = MLM T L . Note that even if the matrix entries are constant in our case, the preconditioning matrix M is no longer constant. So this explicitly constructed preconditioner ML has to be stored. ....

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.

....[26] Also in this class are the popular incomplete LU (ILU) factorization preconditioners. ILU type methods have been successfully applied to Markov chain problems by Saad [25] in a sequential environment. The existence of incomplete factorizations for nonsingular M matrices was already proved in [20]; an investigation of the existence of ILU factorizations for singular M matrices can be found in [11] Incomplete factorization methods work quite well on a wide range of problems, but they are not easily implemented on parallel computers. For this and other reasons, much effort has been put in ....

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

....using METIS s ordering. Hence, we only use the natural ordering for MICC. We use both orderings to unmodified ICC. We implemented a sparse Cholesky factorization algorithm specifically for this project. The code can perform complete, no fill incomplete (sometimes known as ICC(0) or ICCG(0) [12]) and drop tolerance incomplete factorization. When performing incomplete factorizations, the code can modify the diagonal to ensure that the row sums of LL T are equal to those of A [8] The code implements a sparse left looking algorithm. The code is e#cient in the sense that its running time ....

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. Mathemathics of Computation, 31:148--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.

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, 1977. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation, 31: 148--162.

No context found.

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.

No context found.

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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