M. Benzi and M. Tuma. A comparative study of sparse approximate inverse preconditioners. Appl. Numer. Math., 30(2-3):305--340, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....algorithms such as SPAI[22] and FSAI[25] as well as Tang and Wan s local inverse method[35] since they all require the ability to access submatrices of M # . Chow and Saad s MR method[15, 16] is a possibility as it only uses the matrix as an operator. However, the impressive performance[5] of the incomplete inverse 33 factorization algorithms makes them the most attractive choice. I chose to adapt the AINV[4] algorithm. The original form of AINV is a column oriented, left looking, dot product based algorithm that constructs a factored approximate inverse via biconjugation, shown ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, to appear in Appl. Numer. Math., 30 (1999).

....AINV approach by Benzi, Meyer, and Tuma [5] Once computed, the approximate inverse M is applied as a preconditioner to the linear system (1) for use with a Krylov subspace iterative method. For a comparative study of various sparse approximate inverse preconditioners we refer to Benzi and Tuma [6]. By choosing an a priori sparsity pattern for M , the cost of computing M can be greatly reduced. Possible choices include powers of A or A A, as suggested by Huckle [16] and Chow [11] Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Num. Math. 30, 1999, pp. 305-340.

....easy to parallelize and cheap to evaluate, because M is sparse. Recently, various algorithms have been proposed, all of which attempt to compute directly a sparse approximate inverse of A [5,9,17] For a comparative study of various approximate inverse preconditioners we refer to Benzi and Tuma [6]. Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First introduced by Benson and Frederickson [3,4] they were shown to be e ective on various dicult elliptic problems on unstructured grids by Tang and Wan [25] Advantages of sparse approximate ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Num. Math. 30, 1999, pp. 305-340.

....such as SPAI[22] and FSAI[25] as well as Tang and Wan s local inverse method[35] since they all require the ability to access submatrices of M T # AM 1 # . Chow and Saad s MR method[15, 16] is a possibility as it only uses the matrix as an operator. However, the impressive performance[5] of the incomplete inverse 33 factorization algorithms makes them the most attractive choice. I chose to adapt the AINV[4] algorithm. The original form of AINV is a column oriented, left looking, dot product based algorithm that constructs a factored approximate inverse via biconjugation, shown ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, to appear in Appl. Numer. Math., 30 (1999).

....T J in H red . Whereas this is doubtlessly cheap, it is also guaranteed to fail as fi 0. For small values of fi 0 we may therefore expect to do better with choices such as a bidiagonalization of J = GammaQBG, or an automatic sparse approximate inverse construct for approximating J in M red [5]. We have experimented with such variants and found that, whereas they do indeed improve robustness for small values of fi, the overall performance for the examples presented in x4 is not improved by much. ....

M. Benzi and M. Tuma. A comparative study of sparse approximate inverse preconditioners. Trans. Applied Numerical Math., 30:305, 1999.

....easy to parallelize, and cheap to evaluate because M is sparse. Recently, various algorithms have been proposed, all of which attempt to compute directly a sparse approximate inverse of A [5,9,17] For a comparative study of various approximate inverse preconditioners we refer to Benzi and Tuma [6]. Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First introduced by Benson and Frederickson [3,4] they were shown to be e ective on various dicult elliptic problems on unstructured grids by Tang and Wan [25] Advantages of sparse approximate ....

M. Benzi and M. Tuma, A Comparative Study of Sparse Approximate Inverse Preconditioners, Appl. Num. Math. 30, 1999, pp. 305-340.

....if M is sparse. Recently, various alternatives have been proposed, all of which attempt to compute directly a sparse approximate inverse of A. See for instance Grote and Huckle [1] 2] Benzi and Tuma [3] or Chow and Saad [4] For a comparative study of these different approaches we refer to [5]. Approximate inverse techniques are also gaining in importance as robust and parallel smoothers for Multi grid methods see Benson [6] and more recently Tang and Wan [7] 2 Description of the Block SPAI Algorithm The original SPAI algorithm computes the preconditioner M explicitly by ....

M. Benzi and M. Tuma, A Comparative Study of Sparse Approximate Inverse Preconditioners, Los Alamos National Lab., Tech. Report LA-UR-98-0024, January 1998 A Block Version of SPAI 9

....so, a symmetric partitioning scheme is likely more appropriate. Furthermore, a symmetric reordering keeps the eigenvalues intact. Generally, iterative methods involve preconditioning. Suppose we have an explicit preconditioner such as an approximate inverse M # A 1 . See Benzi and Tuma [3] for a survey of approximate inverse preconditioners. In that case, we need to find P and Q such that both PAQ and Q T MP T # (PAQ) 1 are well partitioned. By well partitioned we mean that (1) the communication costs are low, 2) the block rows of PAQ are balanced (i.e. have approximately ....

M. Benzi and M. T uma, A Comparative Study of Sparse Approximate Inverse Preconditioners, Tech. Rep. LA-UR-98-0024, Los Alamos National Laboratory, Los Alamos, NM, 1998.

....in [BT98b] Since we restrict our comparison of the preconditioners to scalar machines, we cannot exploit the inherent parallelism in the construction of such preconditioners. Thus, we can t expect them to prove efficient in our test scenario. For a comparison of the parallel version we refer to [BT98a]. Finally we introduce a new physically motivated improvement of splitting based preconditioners. From the theory of characteristics for hyperbolic partial differential equations, we know the the directions of the propagation of information in advance. By performing an appropriate numbering of the ....

M. Benzi and M. Tuma. A comparative study of Sparse Approximate Inverse Preconditioners. Los Alamos National Laboratory Technical Report, (LA-UR-98-24), 1998.

....(like dot products) If A is structurally symmetric or nearly so, a symmetric partitioning scheme is likely more appropriate. Generally, iterative methods involve preconditioning. Suppose we have an explicit preconditioner such as an approximate inverse M A Gamma1 . See Benzi and Tuma [3] for a survey of approximate inverse preconditioners. In that case, we need to find P and Q such that both PAQ and Q T MP T (PAQ) Gamma1 are well partitioned. By well partitioned, we mean that (1) the communication costs are low, 2) the block rows of PAQ are balanced (i.e. have ....

M. Benzi and M. T uma, A comparative study of sparse approximate inverse preconditioners, Tech. Rep. LA-UR-98-0024, Los Alamos Natl. Lab., 1998.

....(like dot products) If A is structurally symmetric or nearly so, a symmetric partitioning scheme is likely more appropriate. Generally, iterative methods involve preconditioning. Suppose we have an explicit preconditioner such as an approximate inverse M A Gamma1 . See Benzi and Tuma [3] for a survey of approximate inverse preconditioners. In that case, we need to find P and Q such that both PAQ and Q T MP T (PAQ) Gamma1 are well partitioned. By well partitioned, we mean that (1) the communication costs are low, 2) the block rows of PAQ are balanced (i.e. have ....

M. Benzi and M. T uma, A comparative study of sparse approximate inverse preconditioners, Tech. Rep. LA-UR-98-0024, Los Alamos Natl. Lab., 1998.

....type of preconditioners may solve certain problems that are not suitable for ILU type preconditioners [10] and they offer an alternative for the traditional ILU type preconditioners, There exist several techniques to construct SAI preconditioners. They can be roughly categorized into three groups [2], i.e. sparse approximate inverses based on Frobenius norm minimization [8, 10, 16, 17] factorized sparse approximate inverses [3, 20] sparse approximate inverses computed from an ILU factorization [27] Each of these groups contains a variety of different constructions and each of them has its ....

....and each of them has its own merits and drawbacks. In other words, none of them is absolutely better than others in all comparison metrics (construction cost, application cost, robustness, efficiency, etc. A comprehensive survey and comparison of several existing SAI techniques can be found in [2]. In this paper, we only discuss in detail the one that we will use with our multi level block ILU preconditioning (BILUM) technique and refer interested readers to the original papers for detailed discussions of other SAI techniques. Our choice of this SAI is based on the ready availability of ....

[Article contains additional citation context not shown here]

M. Benzi and M. Tuma. A comparative study of sparse approximate inverse preconditioners. Appl. Numer. Math. to appear.

....as a preconditioner for a given sparse matrix A. The preconditioning operation is thus reduced to a matrix vector operation with M . In other words, these preconditioners have the property of affording maximum potential parallelism. They have been shown to be efficient for certain type of problems [7, 15, 21]. In some cases, the construction of a sparse approximate inverse is possible even if the matrix does not have a stable ILU factorization. One drawback of many sparse approximate inverse techniques is their high construction cost, unless the computation can be done efficiently on parallel ....

....matrices. The resulting sparse approximate inverse has a factored form M = LDU , where L is a lower triangular matrix, D is a diagonal matrix, and U is a upper triangular matrix. In some cases, such form of sparse approximate inverses is also called factorized sparse approximate inverse [7]. The main advantages of the new preconditioners are its robustness and low construction cost. This paper is organized as following. Section 2 outlines several existing sparse approximate inverse techniques. Section 3 discusses in detail our new technique and its efficient construction and ....

[Article contains additional citation context not shown here]

M. Benzi and M. Tuma. A comparative study of sparse approximate inverse preconditioners. Appl. Numer. Math. to appear.

....so, a symmetric partitioning scheme is likely more appropriate. Furthermore, a symmetric reordering keeps the eigenvalues intact. Generally, iterative methods involve preconditioning. Suppose we have an explicit preconditioner such as an approximate inverse M A Gamma1 . See Benzi and Tuma [3] for a survey of approximate inverse preconditioners. In that case, we need to find P and Q such that both PAQ and Q T MP T (PAQ) Gamma1 are well partitioned. By well partitioned, we mean that (1) the communication costs are low, 2) the block rows of PAQ are balanced (i.e. have ....

M. Benzi and M. T uma, A comparative study of sparse approximate inverse preconditioners, Tech. Rep. LA-UR-98-0024, Los Alamos National Laboratory, Los Alamos, New Mexico, 1998.

....put in recent years into developing alternative preconditioning strategies that have natural parallelism while being comparable to ILU methods in terms of robustness and convergence rates. This work has resulted in several new techniques known as sparse approximate inverse preconditioners; see [7] for a recent survey and extensive references. Sparse approximate inverse preconditioners are based on directly approximating the inverse of the coefficient matrix A with a sparse matrix G A Gamma1 . The application of the preconditioner only requires matrix vector products, which are easily ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999), pp. 305--340.

.... the preconditioned conjugate gradient method [2] In the last few years there has been considerable interest in explicit preconditioning techniques based on directly approximating A Gamma1 with a sparse matrix M ; see, e.g. 6] 7] 14] 16] 21] 22] 25] 28] and the recent survey [9]. Sparse approximate inverses have been shown to result in good rates of convergence of the preconditioned iteration (comparable to those obtained with incomplete factorization methods) while being well suited for implementation on vector and parallel architectures; see, e.g. 5] 8] 11] ....

....industrial problems. Also, IC and SSOR preconditioners are difficult to implement in parallel, particularly for unstructured problems. With regards to polynomial preconditioners, which offer good potential for parallelization, our own experience is that they perform well only on easy problems [9]. We are interested in algebraic preconditioners that are robust, parallelizable, and effective at reducing the number of iterations. Based on our experience, the FSAI method developed by Kolotilina and Yeremin in [28] is one of the very few naturally parallel techniques that are also quite ....

[Article contains additional citation context not shown here]

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999), pp. 305--340.

....These are general purpose, algebraic preconditioners that have been used successfully to solve a wide range of problems, particularly from PDE s. For a detailed treatment of ILU preconditioning we refer the reader to [47] For a recent survey of sparse approximate inverse preconditioners, see [11]. 2.1. Incomplete factorization techniques. Incomplete factorization methods compute sparse approximations L, U to the exact triangular factors L and U of A. Notice that this already assumes that A has an LU factorization. The matrix M = L U A is then used as the preconditioner. ....

....to the FSAI preconditioner [37] there is no need to prescribe the sparsity pattern of the approximate inverse factors in advance. It should be emphasized that the approximate inverse factors can be calculated directly from A, and that no knowledge of the factors L or U is needed. In [10] [11] it was found that this preconditioner tends to deliver better convergence rates than SPAI, while being much cheaper to compute. A drawback of approximate inverse preconditioners in factored form (compared to SPAI) is that the preconditioner may not be defined for general sparse matrices. ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999), pp. 305--340.

....M which directly approximates A Gamma1 . The preconditioning step with an approximate inverse preconditioner M only requires matrix vector products, and is easily implemented on vector and parallel architectures. There exist several proposals for computing sparse approximate inverses; see [5] [7] and the references 22 Michele Benzi and Gene H. Golub therein. These techniques can outperform standard incomplete factorization methods when implemented on high performance computers; see [6] 7] The explicit bounds in Section 3 could be used to compute approximate inverse preconditioners M ....

....architectures. There exist several proposals for computing sparse approximate inverses; see [5] 7] and the references 22 Michele Benzi and Gene H. Golub therein. These techniques can outperform standard incomplete factorization methods when implemented on high performance computers; see [6] [7]. The explicit bounds in Section 3 could be used to compute approximate inverse preconditioners M A Gamma1 for use in conjugate gradient (CG) calculations. For instance, we can form a matrix M whose entries m ij are defined as the arithmetic mean of the lower and upper bounds from the ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Los Alamos National Laboratory Technical Report LA-UR-98-0024, Los Alamos, NM, 1998.

....for which ILU type methods fail, and therefore they can be useful even on sequential computers. A comprehensive survey of sparse approximate inverse preconditioners, together with the results of extensive numerical tests aimed at assessing the performance of the various methods, can be found in [5]. One of the conclusions of that study was that factorized forms, in which the approximate inverse is the product of two sparse triangular matrices, tend to perform better than nonfactorized ones, in the sense that they often deliver better convergence rates for the same amount of nonzeros in the ....

....better than nonfactorized ones, in the sense that they often deliver better convergence rates for the same amount of nonzeros in the preconditioner. Factorized approximate inverses are also much less expensive to compute than other forms, at least in a sequential environment. As mentioned in [5], another potential advantage of the factorized approach is the fact that such preconditioners are sensitive to the ordering of the equations and unknowns. Indeed, the amount of fill in in the inverse triangular factors of a sparse matrix A depends very strongly on the ordering. In contrast, the ....

[Article contains additional citation context not shown here]

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., to appear.

....[6] However, in this paper we will not make use of such techniques. There exist several other sparse approximate inverse techniques, but in this paper we limit ourselves to a comparison between SPAI and AINV. A more complete comparative study of these and other techniques is forthcoming [5]. 4 Numerical Experiments. The numerical experiments were performed on a set of twenty nonsymmetric sparse matrices (mostly drawn from the Harwell Boeing collection [10] a few from other sources) These matrices originate from a variety of applications such as oil reservoir simulation, circuit ....

....method, in some cases requiring more than 30 times longer than AINV, but it must be kept in mind that SPAI was designed for use on massively parallel architectures. In a parallel implementation SPAI would certainly look much better, although parallel implementations of AINV are possible as well [5]. Furthermore, we observe that whereas the computation of the ILU(0) and AINV preconditioners did not vectorize, there was some benefit from vectorization in the computation of SPAI. This is due to the use of high level BLAS in the SPAI algorithm. Finally, we notice that both methods can be ....

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, in preparation.

....proposed in the literature. Some of Iterative Methods in Scientific Computation II David R. Kincaid et al. eds. pp. 1 11. Copyright c fl1999 by IMACS All rights of reproduction in any form reserved. ISBN 0 123 45678 9 2 Benzi, Mar in, and Tuma the main references include [1] 2] [4], 9] 10] 11] 12] 14] 15] 16] 18] In particular, 4] provides a fairly complete survey of the existing algorithms, together with a comparison on a broad range of test problems. In this contribution we focus on the AINV preconditioner [1] 2] This method, which is based on ....

....Computation II David R. Kincaid et al. eds. pp. 1 11. Copyright c fl1999 by IMACS All rights of reproduction in any form reserved. ISBN 0 123 45678 9 2 Benzi, Mar in, and Tuma the main references include [1] 2] 4] 9] 10] 11] 12] 14] 15] 16] 18] In particular, [4] provides a fairly complete survey of the existing algorithms, together with a comparison on a broad range of test problems. In this contribution we focus on the AINV preconditioner [1] 2] This method, which is based on incomplete (bi)conjugation, is fairly robust (comparable to ILU type ....

[Article contains additional citation context not shown here]

Benzi, M. and Tuma, M., A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math. (1999), to appear.

....for which ILU type methods fail, and therefore they can be useful even on sequential computers. A comprehensive survey of sparse approximate inverse preconditioners, together with the results of extensive numerical tests aimed at assessing the performance of the various methods, can be found in [6]. One of the conclusions of that study was that factorized forms, in which the approximate inverse is the product of two sparse triangular matrices, tend to perform better than nonfactorized ones, in the sense that they often deliver better convergence rates for the same amount of nonzeros in the ....

....better than nonfactorized ones, in the sense that they often deliver better convergence rates for the same amount of nonzeros in the preconditioner. Factorized approximate inverses are also much less expensive to compute than other forms, at least in a sequential environment. As mentioned in [6], another potential advantage of the factorized approach is the fact that such preconditioners are sensitive to the ordering of the equations and unknowns. Indeed, for a sparse matrix A the amount of inverse fill, which is defined as the number of structurally nonzero entries in the inverse ....

[Article contains additional citation context not shown here]

M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., to appear.

.... factors Z Z and D D , and a factorized approximate inverse is obtained as M Gamma1 = Z D Gamma1 Z T : The stability of this procedure for certain classes of matrices, including diagonally dominant ones, was proved in [4] In addition, numerical experiments in [4] and [8] [9] showed that this approach performs well on linear systems arising from various applications, such as the discretization by finite differences of elliptic partial differential equations and the finite element analysis of simple structures. In particular, the experiments in [8] 9] showed that on ....

....in [4] and [8] 9] showed that this approach performs well on linear systems arising from various applications, such as the discretization by finite differences of elliptic partial differential equations and the finite element analysis of simple structures. In particular, the experiments in [8] [9] showed that on vector computers, this technique can be superior to IC methods because of good vectorization properties. To our knowledge, until now there have been no reports of the application of explicit preconditioners to shell problems. Numerical Experiments In this section, the results of ....

[Article contains additional citation context not shown here]

M. Benzi and M. Tuma. A Comparative Study of Sparse Approximate Inverse Preconditioners. Technical Report LA-UR-98-0024, Los Alamos National Laboratory, Los Alamos, NM. To appear in Applied Numerical Mathematics.

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M. Benzi and M. Tuma. A comparative study of sparse approximate inverse preconditioners. Appl. Numer. Math., 30(2-3):305--340, 1999.

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M. Benzi and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Applied Numerical Mathematics, 30 (1999), pp. 305--340.

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