T.A. Manteuffel, 1980. An incomplete factorization technique for positive definite linear systems. Mathematics of Computation, 34: 473--497.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[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 ....

....of S in Algorithm 5 are positive and coincide with the entries of p = q in Algorithm 4. Proof. This follows immediately from Theorem 6 and Property 1. 2 It is well known that the ILU decomposition of an H matrix exists for any of the dropping strategies discussed in Section 2. 3, see, e.g. [20, 19]. It immediately follows that W and Z of Algorithm 4 exist for this case. Likewise for M matrices we know that the computed L and U are again M matrices. Consequently W and Z have to be nonnegative in this case. However this argument only applies for the theoretic way of dropping in p and q. A ....

T. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--490, 1980.

....generator matrices and their fast matrix vector multiplications. The conjugate gradient type methods [2, 4, 37] are possible choices. However, their convergence rates are slow in general. To speed up the convergence rate, we consider preconditioned conjugate gradient methods. We note that the MILU [31, 38] and MINV [25, 26] based preconditioners are not appropriate due to their expensive construction costs. One of the early applications of preconditioned conjugate gradient methods in solving queueing networks was done by Chan [9, 10] For Markovian overflow networks with traffic density close to 1, ....

T. Manteuffel, An Incomplete Factorization Techniques for Positive Definite Linear Systems, Math. Comp., 34 (1980), pp. 473--497.

....[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 [4, 5, 6]. See [7] and [8] for additional information and references. On the other hand, an incomplete factorization can fail for a general SPD matrix due to the occurrence of nonpositive pivots. This is commonly referred to as pivot breakdown. Matrices arising from FEM modelling of solids and structures ....

....used. The problem is that if a nonpositive pivot occurs, the loss of information about the true factors due to dropping has already been so great that no local trick can succeed in recovering a good preconditioner it s too late. A more effective strategy was suggested by Manteuffel in [4]. If an incomplete factorization of A fails due to pivot breakdown, a global diagonal shift is applied to A prior to reattempting the incomplete factorization. That is, the incomplete factorization is carried out on A = A ff diag(A) where ff 0 and diag(A) denotes the diagonal part of A. If ....

Manteuffel T. An incomplete factorization technique for positive definite linear systems. Mathematics of Computation 1980; 34:473--497.

....preconditioner. A possible remedy to both problems, is to shift the matrix B by a scalar a prior to computing its IC factorization. Thus, a preconditioner for B B t aI is constructed and applied to the original (nonshifted) matrix B. This approach was suggested by several authors, see, e.g. [13]. For large a, the preconditioner is obviously inaccurate. For smaller values of a on the other hand, the preconditioner, when it exists, does represent B more accurately but the norm of its inverse may become large, or very large. A compromise is therefore needed to obtain a shift a such that the ....

T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Mathematics of Computations, 34:473-497, 1980.

....A possible remedy to both problems, is to shift the matrix B by a scalar ff prior to computing its IC factorization. Thus, a preconditioner for B = B ffI is constructed and applied to the original (nonshifted) matrix B. This approach was suggested by several authors, see, e.g. [13]. For large ff, the preconditioner is obviously inaccurate. For smaller values of ff on the other hand, the preconditioner, when it exists, does represent B more accurately but the norm of its inverse may become large, or very large. A compromise is therefore needed to obtain a shift ff such that ....

T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Mathematics of Computations, 34:473--497, 1980.

....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 ....

....of factorized sparse approximate inverse in Algorithm 4 we can apply the whole theory available for incomplete ILU decompositions. As an example consider the stability of the ILU for H matrices. It has well known, that for any choice of , 7 the ILU decomposition of an H matrix exists. See [19, 18] for details. It immediately follows that Z, W of the Algorithm 4 exist for this case. Likewise for M matrices we know that the computed L, U are again M matrices. Consequently Z, W have to be nonnegative in this case. We obtain one immediate conclusion for the symmetric positive definite case. ....

T. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473-490, 1980.

....A s and this has been improved and extended by Elman [58] who also studied the stability properties of these methods. Notay [80] gave strategies for choosing = i;j dynamically in order to improve the robustness and performance for anisotropic problems. 3.3. Shifted ILU. Manteuffel [74] considered ILU factorizations of diagonally shifted matrices. He proved the following two results: 1. If A is symmetric positive definite, then there exists a constant ff 0, such that the ILU factorization of A ffI exists. This result shows that even though the ILU factorization of an SPD ....

T.A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--497, 1980.

....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 ....

Manteuffel T.A., "An Incomplete Factorization Technique for Positive Definite Linear Systems ", Math. Comp. 34 (1980), pp. 473-497.

....8 3.637 0.048 0.369 1.886 y5.986 Table 11: Timing breakdown for CGS and OMN(10) 22 poorly conditioned. In these cases the robustness of IC preconditioning is not needed. For problems not studied here, we have seen that variants of IC preconditioning like the shifted IC preconditioner of [12] are quite robust and solve problems which cannot be solve by diagonal or polynomial preconditioning. One of these variants will be installed in future versions of CrayPCG. In comparing CrayPCG to the multifrontal solver, CrayPCG is competitive for moderately well to well conditioned problems. ....

T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--497, 1980.

....of incomplete LU (in this case Cholesky) factorizations for such matrices can be so poor that they become ineffective. It is tempting to simply shift the matrix A by a scalar ff and extract the preconditioning for A ffI which is then used for preconditioning the original matrix, see, e.g. [7]. This by itself does not work well enough in general. A modification of this idea whichinvolves more work, will lead to a more effective technique. This modification consists of exploiting a rational approximation to A ;1 based on an expansion in terms of the form (A ffI) i . Because the ....

T. A. Manteuffel, An incomplete Factorization technique for positive definite linear systems, Math. of Comp., 34 (1980), 473-497.

....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 andH 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 whichcanbe viewed as approximations of the inverses of ....

....of factorized sparse approximate inverse in Algorithm 4 we can apply the whole theory available for incomplete ILU decompositions. As an example consider the stability of the ILU for H matrices. It has well known, that for any choice of C, R the ILU decomposition of an H matrix exists. See [19, 18] for details. It immediately follows that Z# W of the Algorithm 4 exist for this case. Likewise for M matrices we know that the computed L# U are again M matrices. Consequently Z# W have to be nonnegative in this case. We obtain one immediate conclusion for the symmetric positive definite ....

T. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--490, 1980.

....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 the feasibility and efficiency of ....

Manteuffel, T. (1980), `An incomplete factorization technique for positive definite linear systems', Mathematics of Computation 34, 473--497.

....see Axelsson [3] and Saad [37] Section 3 extends the incomplete Cholesky factorization of section 2 to indefinite matrices. Two main issues arise: scaling the matrix and modifying the matrix so that the factorization is possible. We use a symmetric scaling of the matrix and Manteu#el s [28] shifted approach. We prove that the incomplete Cholesky factorization exists for H matrices with positive diagonal elements, and we establish bounds for the number of iterations required to compute the factorization. We also study how the shift depends on the scaling of the matrix. Modifications ....

....are retained. Thus, the advantage of having predictable storage requirements is lost. There have been several proposed modifications to the incomplete Cholesky factorization that are applicable to general positive definite matrices. The shifted incomplete Cholesky factorization of Manteu#el [27, 28] for the scaled matrix # A = D 1 2 AD 1 2 , D = diag(a ii ) 3.1) requires the computation of a suitable # # 0 for which the incomplete Cholesky factorization of # A #I succeeds. Manteu#el used a fixed fill factorization and showed that if # A #I is an H matrix, then the incomplete ....

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T. A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Math. Comp., 34 (1980), pp. 307--327.

.... of a larger pattern P can lead to better convergence rates [45] There are several modifications and extensions to define incomplete LU decompositions, mainly with the aim to derive stable versions of the solution algorithm [23, 38] Concerning the existence of ILU decompositions we refer to [19, 31, 33]. 6 A linear multigrid method Another class of solution methods for linear (and also nonlinear) systems are multigrid methods. For an introduction see [18, 32, 44] The first multigrid method (MGM) was formulated by Fedorenko in 1964 [9] This was a multigrid algorithm for the standard 5 point ....

T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comput. 34 (1980) 473--479.

....PILU preconditioners. The existence of preconditioners computed from the PILU algorithm can be proven for some classes of problems. Meijerink and van der Vorst [27] proved that if A is an M matrix, then the incomplete LU factorization exists for any pre determined sparsity pattern, and Manteu el [26] extended this result to H matrices with positive diagonal elements. These results immediately show that PILU preconditioners with sparsity patterns based on level values exist for these classes of matrices. This is true even when di erent level values are used for the various subdomains and ....

T. A. Manteuffel, An incomplete factorization technique for positive denite linear systems, Math. Comput., 34 (1980), pp. 307-327.

....processor such as the Intel i860. In this paper we present several possible graph reductions that can be employed to greatly improve the performance of an implementation on high performance RISC processors. Consider an implementation of any of the standard general purpose iterative methods [7, 15]: consistently ordered SOR, SSOR accelerated by conjugate gradients (CG) or CG preconditioned with an incomplete matrix factorization. It is evident that the major obstacle to a scalable implementation [6] is the inversion of sparse triangular systems with a structure based on the structure of ....

....of the matrix was scaled to be one. If the incomplete factorization fails (a negative diagonal element created during the factorization) a small multiple of the identity is added to diagonal, and the factorization is restarted. This process is repeated until a successful factorization is obtained [15]. The average number of conjugate gradient iterations required to solve one nonlinear iteration of the thermal equilibrium problem for the crystal model to a relative accuracy of 10 Gamma7 is approximately 700. The average number of conjugate gradient iterations required per nonlinear iteration ....

T. A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Mathematics of Computation, 34 (1980), pp. 473--497.

....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 and Main Results The Incomplete LU (ILU) decomposition of a sparse matrix [14] [15] 21] 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 ....

Manteuffel T.A., "An Incomplete Factorization Technique for Positive Definite Linear Systems ", Math. Comp. 34 (1980), pp. 473-497.

....arrays of SIMD or vector processors) Numerical examples show that, in both cases, the overhead in the number of preconditioned iterations required for convergence is small in comparison with the speed up gained. 1 Introduction The Incomplete LU (ILU) decomposition of a sparse matrix [17] [18] 28] is one of the most powerful tools for the solution of sparse linear systems of equations. The idea is to construct sparse triangular matrices L and U such that LU is a suitable approximation of the coefficient matrix A. LU serves as a preconditioner in a Krylov space acceleration ....

Manteuffel T.A., "An Incomplete Factorization Technique for Positive Definite Linear Systems", Math. Comp. 34 (1980), pp. 473-497.

....of incomplete LU (in this case Cholesky) factorizations for such matrices can be so poor that they become ineffective. It is tempting to simply shift the matrix A by a scalar ff and extract the preconditioning for A ffI which is then used for preconditioning the original matrix, see, e.g. [7]. This by itself does not work well enough in general. A modification of this idea which involves more work, will lead to a more effective technique. This modification consists of exploiting a rational approximation to A Gamma1 based on an expansion in terms of the form (A ffI) Gammai . ....

T. A. Manteuffel, An incomplete Factorization technique for positive definite linear systems, Math. of Comp., 34 (1980), 473-497.

....[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 ....

....of factorized sparse approximate inverse in Algorithm 4 we can apply the whole theory available for incomplete ILU decompositions. As an example consider the stability of the ILU for H matrices. It has well known, that for any choice of C, R the ILU decomposition of an H matrix exists. See [19, 18] for details. It immediately follows that Z; W of the Algorithm 4 exist for this case. Likewise for M matrices we know that the computed L; U are again M matrices. Consequently Z; W have to be nonnegative in this case. We obtain one immediate conclusion for the symmetric positive definite ....

T. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--490, 1980.

....enough to the full factorisation. Incomplete factorisations can be very effective, but there are a few practical problems. For the class of M matrices, these methods are well defined [52] but for other, even fairly common classes of matrices, there is a possibility that the algorithm breaks down [42, 45, 51]. Also, factorisations are inherently recursive, and coupled with the sparseness of the incomplete factorisation, this gives very limited parallelism in the algorithm using a natural ordering of the unknowns. Different orderings may be more parallel, but take more iterations [25, 27, 43] 8.4.3 ....

T.A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--497, 1980.

....is well understood and several implementations are available [20, 23] The main unresolved issue is that of finding robust preconditioners to accelerate the convergence of CG. One of the most popular and broadly applicable preconditioners is obtained from incomplete Cholesky (IC) factorization [3, 9, 16, 17]. An incomplete Cholesky factor L of A is obtained by discarding some or all of the fill produced in the course of Cholesky factorization. The amount of fill retained in L can be limited by using a drop threshold scheme [17, 24] Incomplete Cholesky with a drop threshold strategy is similar to ....

T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comput., 34:473--497, 1980. Blocked Incomplete Cholesky 11

....preconditioner is required to be hpd. An approach that often is used in these cases is to shift each block, D i , by a factor, oe i such that D i oe i is hpd, and therefore the resulting modification of M is hpd. If M is this modification, then M Gamma1 is the preconditioning matrix. See [11] for a discussion of this technique in the finite element case. It should be noted that the shifting technique is not generally successful in the case of an indefinite hermitian matrix A; however, the indefinite case is a difficult problem and no satisfactory general approach to preconditioning is ....

T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--497, 1980.

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T.A. Manteuffel, 1980. An incomplete factorization technique for positive definite linear systems. Mathematics of Computation, 34: 473--497.

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T. A. Manteuffel. An incomplete factorization technique for positive definite linear systems. Math. Comp., 34:473--497, 1980.

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