E. Chow and Y. Saad, Approximate inverse preconditioners for general sparse matrices, Tech. Rep. UMSI 94/101, Minnesota Supercomputer Institute, University of Minnesota, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....norm. Kolotilina and Yeremin [64] presented an algorithm in which the inverse was delivered in factored form, which has the advantage that singularity of M can be easily detected. In [22] an algortithm is presented which uses the 1 norm for the minimization. We also mention Chow and Saad [18], who use GMRES for the minimization of kAM Gamma IkF . This approach has the disadvantage that one does not have easy control over the amount of fill in in M , and drop tolerance strategies have to be applied. The approach has the advantage that it can be used to correct explicitly some given ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Technical Report Research Report UMSI 94/101, University of Minnesota Supercomputing Institute, Minneapolis, Minnesota, 1994.

....SPARSE APPROXIMATE INVERSE PRECONDITIONERS TONY F. CHAN , W. P. TANG y AND W. L. WAN Abstract. There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle [21] and Chow and Saad [11] also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices ....

....the application of the preconditioner is simply matrix vector multiply instead of backsolve and it can be done easily in parallel. On the other hand, due to its algebraic nature, it is applicable to both general and PDE problems. Moreover, recent studies of Grote and Huckle [21] and Chow and Saad [11] show that it is robust for matrices in Harwell Boeing collections. The main idea of sparse approximate inverse is described as follows. Dept. of Mathematics, Univ. of Calif. at Los Angeles, Los Angeles, CA 90095 1555. Email: chan math.ucla.edu, wlwan math.ucla.edu. Supported by grants from ONR: ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Colorado Conference on Iterative Methods, April 5-9, 1994. To appear in SIAM J. Sci. Comput.

....some sparse approximation for the full matrix A Gamma1 . This procedure turns out to be not robust if the pattern of nonzero elements is not optimally selected. An example for an incomplete inverse is to prescribe a pattern for P , e.g. that of A, and to determine P as Frobenius inverse of A [4]. 8. Beyond CG type methods. Many of the here dicussed methods have been already beyond the CG type. The reason for the large variety in iterative solvers is that methods should converge rapidly. Importantly, there is no single optimal method, but each method performs differently, depending on ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Research Report UMSI 94/101, University of Minnesota, Supercomputer Institute, 1200 Washington Avenue South, Minneapolis, Minnesota 55415, 1994.

....is singular, it may still be possible to choose a suitable preconditioner for a consistent system (1.1) by approximately minimizing some norm of AM Gamma I . There has been a lot of interest, recently, in such preconditioners (see, Cosgrove, Diaz and Griewank, 1992, Kolotilina and Yeremin, 1993, Chow and Saad, 1994, Huckle and Grote, 1994, de Doncker and Gupta, 1995, and Grote and Huckle, 1995, and the references contained therein) and there is some evidence that they are effective in practice. However, we have been somewhat surprised that there has been no proper assessment of their effectiveness in ....

....1 ftp to 130.246.8.32 and the directory pub harwell boeing for the Harwell Boeing matrices, and to ftp.cis.ufl.edu and the directory pub umfpack matrices for the Davis collection. 5 NUMERICAL EXPERIMENTS 10 sets were (with the exception of problem BP200) used by either Huckle and Grote (1994) or Chow and Saad (1994). All our numerical experiments were performed on each of the examples listed in Table 5.1. Since giving full results for each test problem would present the reader with an indigestible amount of data, we feel it is more helpful to only give comprehensive results for a subset of the main test set ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrics. Research Report UMSI 94/101, University of Minnesota Supercomputer Institute, Minneapolis, Minnesota, USA, 1994.

....approximate inverse, large sparse matrices, algebraic multilevel method. AMS subject classification: 65F05, 65F10, 65F50, 65Y05, 93B40. 1 Introduction For the solution of large sparse linear systems of the form Ax = b; A 2 GL (n; R) b 2 R n (1) sparse approximate inverse approximations [20, 19, 7, 15, 5, 21] have become popular as preconditioners for Krylov subspace [10, 25, 14] techniques. The main idea is to construct a matrix B that approximates A Gamma1 but B is still sparse. Several techniques have been developed, for example minimizing the norm of kAB Gamma Ik subject to some prescribed ....

....become popular as preconditioners for Krylov subspace [10, 25, 14] techniques. The main idea is to construct a matrix B that approximates A Gamma1 but B is still sparse. Several techniques have been developed, for example minimizing the norm of kAB Gamma Ik subject to some prescribed pattern [19, 7, 15] or biconjugate techniques ZAW D, where Z; W are lower triangular matrices and D is diagonal [5] 1.1 Sparse approximate inverses are smoothing While sparse approximate inverse matrices are quite powerful as preconditioners for a large class of matrices, there are cases when the sparse ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Research Report UMSI 94/101, University of Minnesota, Super Computing Institute, Minneapolis, Minnesota, 1994.

.... lies in determining a good sparsity structure for M, so that the solution of (3) yields an effective preconditioner, and a considerable amount of research has already been done in that direction (Yeremin et al. 6, 7, 8] Grote and Simon [9] Cosgrove, Diaz and Griewank [10] Chow and Saad [11], and Grote and Huckle [2] For the rest of this paper we shall restrict ourselves to SPAI, the method proposed by Grote and Huckle [2] and to spai 1.1, a parallel implementation of SPAI written by one us (Barnard [5] A closely related version of the parallel SPAI preconditioner is included in ....

E. Chow and Y. Saad, Approximate Inverse Preconditioners for General Sparse Matrices, in Proc. Colorado Conf. on Iterative Meth., 1994.

....polynom preconditioners, or incomplete LU decompositions of A [2] But these preconditioners either lead to unsatisfactorily convergence or are hard to parallelize. A very promising approach is the choice of sparse approximate inverses for preconditioning, M A Gamma1 and M sparse [10,4,3,7,6,8]. Then, in the basic iterative scheme only matrix vector multiplications with M appear and it is not necessary to solve a linear system in M like in the incomplete LU approach. Obviously, A Gamma1 is a full matrix in general, and hence not for every sparse matrix A there will exist a good ....

Chow,E., Saad,Y.: Approximate Inverse Preconditioners for general sparse matrices, Research Report UMSI 94/101, University of Minnesota Supercomputing Institute, Minneapolis, Minnesota, 1994.

....[11] 12] 13] but only consider fixed sparsity patterns. Simon and Grote solve (4) explicitly, but only allow for a banded sparsity pattern in M [9] 10] The approach of Cosgrove, Diaz, and Griewank [4] is similar to ours, but differs in the criteria used for augmenting M . Chow and Saad [3] use an iterative method to compute an approximate solution of (4) Their method automatically generates new entries, to which they apply a dropping strategy to remove the excessive fill in appearing in M . In section 2 we introduce the spai algorithm, which computes a sparse approximate inverse ....

....column m k of M , it is easy to single out the most difficult columns and concentrate on them to improve the convergence of the iterative solver. This may prove useful in connection with flexible preconditioning, where the preconditioner is adapted during the iterative process (see [17] 14] [3]) 3 Theoretical Properties of M We shall now derive rigorous bounds on the spectrum of the preconditioned matrix AM . Furthermore, we shall estimate the difference between M and A Gamma1 , and derive conditions that guarantee that M is nonsingular. Let M be an approximate inverse of A ....

E. Chow, Y. Saad. Approximate inverse preconditioners for general sparse matrices, Colorado Conf. on Iterative Meth., April 5--9, 1994.

....(i) iii) Incomplete LU decomposition of A, with A LU and defining M = LU) Gamma1 ; but like Gaussian Elimination this is not very effective in a parallel environment. A very promising approach is the choice of sparse approximate inverses for preconditioning, M A Gamma1 and M sparse [8,3,2,6,5,7]. Then, in the basic iterative scheme only matrix vector multiplications with M appear and it is not necessary to solve a linear system in M like in the incomplete LU approach. Obviously, A Gamma1 is a full matrix in general, and hence not for every sparse matrix A there will exist a good ....

Chow,E., Saad,Y.: Approximate Inverse Preconditioners for general sparse matrices, Research Report UMSI 94/101, University of Minnesota Supercomputing Institute, Minneapolis, Minnesota, 1994.

.... by the standard (dense) QR factorizations (Cosgrove et al. 1992, Gould and Scott 1998, Grote and Huckle 1997) In a further attempt to increase sparsity and reduce computational costs in the solutions of the subproblems, it has been suggested to use a few steps of GMRES to solve the subsystems (Chow and Saad 1994). A recent study indicates that the computed approximate inverse may be a good alternative for ILU (Gould and Scott 1998) but it is much more expensive to compute both in terms of time and storage, at least if computed sequentially. This means that it is normally only attractive to use this ....

Chow, E. and Saad, Y. (1994), Approximate inverse preconditioners for general sparse matrices, Technical Report Research Report UMSI 94/101, University of Minnesota Supercomputing Institute, Minneapolis, Minnesota.

....of the ILU type is the possibility of breakdowns during the incomplete factorization process, due to the occurrence of zero or exceedingly small pivots. This situation typically arises when dealing with matrices which are strongly unsymmetric and or indefinite, even if pivoting is applied (see [11], 49] and in general it may even occur for definite problems unless A exhibits some degree of diagonal dominance. Of course, it is always possible to safeguard the incomplete factorization process so that it always runs to completion, producing a nonsingular preconditioner, but there is also no ....

....has been devoted to explicit preconditioning based on the following approach: the sparse approximate inverse is computed as the matrix G which minimizes kI Gamma GAk (or kI Gamma AGk for right preconditioning) subject to some sparsity constraint (see [4] Ch. 8 of [2] 16] 43] 44] 32] 31] [11], 30] Here the matrix norm is usually the Frobenius norm or a weighted variant of it, for computational reasons. With this choice, the constrained minimization problem decouples into n independent linear least squares problems (one for each row, or 6 Michele Benzi and Miroslav Tuma column of ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Research Report UMSI 94/101, University of Minnesota Supercomputer Institute, Minneapolis, MN, USA, 1994.

....preconditioner. Yeremin et al. compute a factorized sparse approximate inverse [13, 12, 14] but only consider fixed sparsity patterns. Simon and Grote [9] solve (4) explicitly, but only allow for a banded sparsity pattern in M . The approach of Cosgrove, Diaz, and Griewank [5] Chow and Saad [4], and Grote and Huckle [11, 10] all suggest methods which capture the sparsity pattern of the main entries of A Gamma1 automatically and at a reasonable cost, but stop short of an actual parallel implementation. Gould and Scott [8] present results of a simulated parallel implementation based on ....

E. Chow and Y. Saad. Approximate Inverse Preconditioners for General Sparse Matrices. In Proc. Colorado Conf. on Iterative Meth., 1994.

....preconditioner. Yeremin et al. compute a factorized sparse approximate inverse [15, 14, 16] but only consider fixed sparsity patterns. Simon and Grote [10] solve (4) explicitly, but only allow for a banded sparsity pattern in M . The approach of Cosgrove, Diaz, and Griewank [5] Chow and Saad [4], and Grote and Huckle [12, 11] all suggest methods which capture the sparsity pattern of the main entries of A Gamma1 automatically and at a reasonable cost, but stop short of an actual parallel implementation. Gould and Scott [9] present results of a simulated parallel implementation based ....

E. Chow and Y. Saad. Approximate Inverse Preconditioners for General Sparse Matrices. In Proc. Colorado Conf. on Iterative Meth., 1994.

....norm. Kolotilina and Yeremin [64] presented an algorithm in which the inverse was delivered in factored form, which has the advantage that singularity of M can be easily detected. In [22] an algortithm is presented which uses the 1 norm for the minimization. We also mention Chow and Saad [18], who use GMRES for the minimization of kAM Gamma IkF . This approach has the disadvantage that one does not have easy control over the amount of fill in in M , and drop tolerance strategies have to be applied. The approach has the advantage that it can be used to correct explicitly some given ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Technical Report Research Report UMSI 94/101, University of Minnesota Supercomputing Institute, Minneapolis, Minnesota, 1994.

....reduce the value of (8.1) Cosgrove, Diaz and Griewank 1992) and so, in principle, ensure convergence of the preconditioned iterative method. Cosgrove et al. 1992) Huckle and Grote (1994) and Gould and Scott (1995) use a (dense) QR factorization to solve the small least squares problems while Chow and Saad (1994) use GMRES. Gould and Scott (1995) show that this technique gives almost as good a preconditioner as ILU but is much more expensive to compute both in terms of time and storage, at least if computed sequentially. One problem with these approaches is that, although the residual for the ....

Chow, E. and Saad, Y. (1994), Approximate inverse preconditioners for general sparse matrices, Technical Report UMSI 94/101, University of Minnesota Supercomputer Institute.

....Simon and Grote solve (4) explicitly, but only allow for a banded sparsity pattern in M [7] 8] In [6] the sparsity of the approximate inverse is selected dynamically, but the criteria for choosing new elements are oriented towards reducing the 1 norm. A recent approach by Chow and Saad [2] uses an iterative method to compute an approximate solution of (4) Although their method generates new entries in M automatically at each iteration, they must apply a drop strategy to remove the excessive fill in appearing in M . In section 2 we introduce the SPAI algorithm to compute a sparse ....

....column m k of M , one could easily single out the toughest columns and concentrate on them to improve the convergence of the iterative solver. This may prove useful in connection with flexible preconditioning, where the preconditioner is adapted during the iterative process (see [12] and [2]) 3 Theoretical Properties We shall now derive rigorous bounds on the spectrum of the preconditioned matrix AM . Furthermore, we shall estimate the difference between M and A Gamma1 , and derive conditions that guarantee that M is nonsingular. Let M be an approximate inverse of A obtained ....

E. Chow, Y. Saad. Approximate Inverse Preconditioners for General Sparse Matrices, Colorado Conference on Iterative Methods, April 5-9, 1994.

No context found.

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Technical Report UMSI 94-101, University of Minnesota Supercomputer Institute, Minneapolis, MN 55415, May 1994.

....j Gamma Am j k 2 2 ; j = 1; 2; n; where e j and m j are the jth columns of the identity matrix and a matrix Q 2 S, respectively. Note that each of the n columns can be computed independently. Different strategies for selecting a nonzero structure of the approximate inverse are proposed in [4] and [9] In [9] the initial sparsity pattern is taken to be diagonal with further fill in allowed depending on the improvement in the minimization. The work [4] suggests controlling the sparsity of the approximate inverse by dropping certain nonzero entries in the solution or search directions of ....

....columns can be computed independently. Different strategies for selecting a nonzero structure of the approximate inverse are proposed in [4] and [9] In [9] the initial sparsity pattern is taken to be diagonal with further fill in allowed depending on the improvement in the minimization. The work [4] suggests controlling the sparsity of the approximate inverse by dropping certain nonzero entries in the solution or search directions of a suitable iterative method (e.g. GMRES) This iterative method solves the system Am j = e j such that minm j ke j Gamma Am j k 2 2 ; for j = 1; 2; n: ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. SIAM Journal on Scientific Computing, --, 1997. To appear.

....dropping elements in x with small magnitudes. If the right handside b and the initial guess for x are sparse, this is a very economical method for computing a sparse approximate solution. We have used this technique to construct preconditioners based on approximating the inverse of A directly [6]. This paper is organized as follows. In Section 2 we describe the sparse approximate inverse algorithm and some techniques for finding sparse approximate solutions with the Schur complement. Section 3 describes how block partitioned factorizations may be used as preconditioners. The most ....

....matrices [7, 19] For these algorithms to be practical, they must provide approximations that are sparse. Block Approximate Inverse Techniques 4 A number of techniques have recently been developed to construct a sparse approximate inverse of a matrix, to be used as a preconditioner [5, 6, 8, 10, 15, 17, 18]. Many of these techniques approximate each row or column independently, focusing on (in the column oriented case) the individual minimizations min x ke j Gamma Axk 2 ; j = 1; 2; n (4) where e j is the j th column of the identity matrix. Such a preconditioner is distinctly easier than ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices, Technical Report UMSI 94/101, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota, 1994.

....generally fail on the preconditioned system, which may well have a condition number that is much worse than the un preconditioned system. An easily calculated indicator of instability of the ILU factors is the infinity norm condition estimate k(LU) Gamma1 ek1 , where e is a vector of all ones [3, 4]. In GMRES the residual norm is calculated as the iteration proceeds (see [11] section 3.2) Iteration is terminated if the calculated residual norm is reduced by more than a convergence tolerance ffl. The calculation is not always correct, due to roundoff errors, and this can lead to early ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Technical Report UMSI 94-101, University of Minnesota Supercomputer Institute, Minneapolis,MN 55415, May 1994.

....the entire matrix A, then an alternative parallel implementation is to compute the n individual columns of M simultaneously, with no communication required. We have found that this approximate inverse preconditioner is advantageous in many cases where the matrix A is nonsymmetric or indefinite [1]. 4 Conclusion We have described some implementations and algorithms for basic sparse matrix computations which may be used as part of a library, or because of its simple and open design, may be used as templates for the development of research or application codes. We have argued that a ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices, UMSI 94/101, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota, 1994.

No context found.

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices. Technical Report UMSI 94-101, University of Minnesota Supercomputer Institute, Minneapolis, MN 55415, May 1994.

....dropping elements in x with small magnitudes. If the right handside b and the initial guess for x are sparse, this is a very economical method for computing a sparse approximate solution. We have used this technique to construct preconditioners based on approximating the inverse of A directly [6]. This paper is organized as follows. In Section 2 we describe the sparse approximate inverse algorithm and some techniques for finding sparse approximate solutions with the Schur complement. Section 3 describes how block partitioned factorizations may be used as preconditioners. The most ....

....case for block preconditioners for block tridiagonal matrices [7, 19] For these algorithms to be practical, they must provide approximations that are sparse. A number of techniques have recently been developed to construct a sparse approximate inverse of a matrix, to be used as a preconditioner [5, 6, 8, 10, 15, 17, 18]. Many of these techniques approximate each row or column independently, focusing on (in the column oriented case) the individual minimizations min x ke j Gamma Axk 2 ; j = 1; 2; n (4) where e j is the j th column of the identity matrix. Such a preconditioner is distinctly easier than ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices, Technical Report UMSI 94/101, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota, 1994.

....Yet another possible strategy will be mentioned in Section 2.4. 2.2.2 Approximate inverse techniques A second, much cheaper approximation for (1) comes from approximate inverse techniques. Their most common application has been to independently approximate all the rows or columns of an inverse [2, 5, 7, 15, 12] or its factors [1, 16] and use it as a preconditioner. In the column case, for example, this can be done by minimizing the 2 norm of the residual, min x j ke j Gamma Ax j k 2 (5) for each column j of the matrix, where e j is the j th coordinate vector, and where x j is somehow constrained to ....

....vector, and where x j is somehow constrained to be sparse. The minimization may be done in many ways, most obviously by using a QR factorization. However, since the exact minimum is not required, it may be cheaper to use a few steps of a descent type method starting with a sparse initial guess [5]. It is useful in many circumstances to regard (5) as a general method to find a sparse approximate solution to a linear system [6] In our context, to solve Lz = v approximately, we focus on the minimization problem min z kv Gamma Lzk 2 (6) with respect to all sparse z. By constraining the ....

E. Chow and Y. Saad. Approximate inverse preconditioners for general sparse matrices, Technical Report UMSI 94/101, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota, 1994.

No context found.

E. Chow and Y. Saad, Approximate inverse preconditioners for general sparse matrices, Tech. Rep. UMSI 94/101, Minnesota Supercomputer Institute, University of Minnesota, 1994.

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