L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings i. theory. SIAM J. Matrix Anal. Appl., 14(1):45--58, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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, such that W AZ D, see e.g. 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 ....

Y. Kolotilina and Y. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

....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 diagonal D, such that Zn AW = D, see 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 ....

Y. Kolotilina and Y. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14:45-58, 1993.

....for convection dominated problems. Another approach that received attention is the concept of constructing an explicit approximation for the inverse of a given matrix A. The idea is to find a sparse matrix M such that kAM Gamma Ik is small for some convenient 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 ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

.... categorized first by their result: some algorithms produce a single sparse matrix approximating A e.g. SPAI[22] Chow and Saad s MR method[15, 16] Tang and Wan s local inverse[35] and others produce factored approximations (approximate inverses of the triangular factors) e.g. FSAI[25], AINV[4] The factored form has the advantages of guaranteed non singularity, extra sparsity from good orderings, and apparently more effect per nonzero thanks to its more implicit nature. However, the non factored form has the advantages of robustness with respect to orderings bad pivots ....

....an approximate inverse algorithm that works when the matrix is known only as a linear operator. Actually, a little more is known: the adjoint of the operator may be used in the algorithm as well. This rules out the Frobenius norm minimization 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 ....

L. Kolotilina and A. Yeremin, Factorized sparse approximate inverse preconditionings I. theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....approximating A is based on the construction of an incomplete Cholesky factorization. This incomplete factorization is often used as a preconditioner when solving linear systems with matrix A. In this paper we use another preconditioning technique, namely that of sparse approximate inverses (cf. [1, 7, 9, 11]) In Remark 3.10 we comment on the advantages of the use of sparse approximate inverse preconditoning for approximating d(A) Let A = LL T be the Cholesky decomposition of A. Then using techniques known from the literature a sparse approximate inverse GE of L, i.e. a lower triangular matrix GE ....

....= I . Note that d(A) d(L) 2 = Q n i=1 L 2=n ii . We will construct a sparse lower triangular approximation G of L Gamma1 and approximate d(A) by d(G) Gamma2 = Q n i=1 G Gamma2=n ii . The construction of a sparse approximate inverse that we use in this paper was introduced in [9, 10, 11] and can also be found in [1] Some of the results derived in this section are presented in [1] too. 4 A. REUSKEN We first introduce some notation. Let E ae f(i; j) j 1 i; j ng be a given sparsity pattern. By #E we denote the number of elements in E. Let SE be the set of n Theta n matrices ....

[Article contains additional citation context not shown here]

L. .Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditionings I : Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....pattern prescribed in advance for the approximate inverse factors, and sparsity is preserved during the process, by discarding elements having magnitude smaller than a given positive threshold. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ( 20] 21] 22] and [23]) This approach, known as FSAI , approximates A 1 by the factorization G T G, where G is a sparse lower triangular matrix approximating the inverse of the lower triangular Cholesky factor, L, of A. This technique has obtained good results on some di#cult problems and is suitable for ....

L.Yu Kolotilina, A.Yu. Yeremin, and A.A. Nikishin. Factorized sparse approximate inverse preconditionings. IV: Simple approaches to rising e#ciency. Technical report,

....a pattern prescribed in advance for the approximate inverse factors, and sparsity is preserved during the process, by discarding elements having magnitude smaller than a given positive threshold. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ( 20] 21] [22] and [23] This approach, known as FSAI , approximates A 1 by the factorization G T G, where G is a sparse lower triangular matrix approximating the inverse of the lower triangular Cholesky factor, L, of A. This technique has obtained good results on some di#cult problems and is ....

L.Yu Kolotilina, A.Yu. Yeremin, and A.A. Nikishin. Factorized sparse approximate inverse preconditionings. III: Iterative construction of preconditioners. Zap. Nauchn. Semin. POMI, 248:17--48, 1998.

....not require a pattern prescribed in advance for the approximate inverse factors, and sparsity is preserved during the process, by discarding elements having magnitude smaller than a given positive threshold. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ( 20] [21], 22] and [23] This approach, known as FSAI , approximates A 1 by the factorization G T G, where G is a sparse lower triangular matrix approximating the inverse of the lower triangular Cholesky factor, L, of A. This technique has obtained good results on some di#cult problems and is ....

L.Yu Kolotilina and A.Yu. Yeremin. Factorized sparse approximate inverse preconditionings. II: Solution of 3D FE systems on massively parallel computers. Int J. High Speed Computing, 7:191--215, 1995.

....More precisely in Section 2.1 we 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 ....

....does not require a pattern prescribed in advance for the approximate inverse factors, and sparsity is preserved during the process, by discarding elements having magnitude smaller than a given positive threshold. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ([20], 21] 22] and [23] This approach, known as FSAI , approximates A 1 by the factorization G T G, where G is a sparse lower triangular matrix approximating the inverse of the lower triangular Cholesky factor, L, of A. This technique has obtained good results on some di#cult problems and ....

L.Yu Kolotilina and A.Yu. Yeremin. Factorized sparse approximate inverse preconditionings. I: Theory. SIAM J. Matrix Analysis and Applications, 14:45--58, 1993.

....implementation, computes the inverse factors of A 1 using relations (3) and preserves a sparsity pattern for the factors L and U discarding entries with small modulus in the algorithm. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ( 33] 34] 35] and [36]) This approach, known as FSAI , has provided good results on some dicult problems and is suitable for a parallel implementation, but it requires the prescribing in advance a sparsity pattern for the approximate factors, and this can limit its use on very general problems, especially in the ....

L.Yu Kolotilina, A.Yu. Yeremin, and A.A. Nikishin. Factorized sparse approximate inverse preconditionings. IV: Simple approaches to rising eciency. Technical report,

....for a parallel implementation, computes the inverse factors of A 1 using relations (3) and preserves a sparsity pattern for the factors L and U discarding entries with small modulus in the algorithm. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ( 33] 34] [35] and [36] This approach, known as FSAI , has provided good results on some dicult problems and is suitable for a parallel implementation, but it requires the prescribing in advance a sparsity pattern for the approximate factors, and this can limit its use on very general problems, especially in ....

L.Yu Kolotilina, A.Yu. Yeremin, and A.A. Nikishin. Factorized sparse approximate inverse preconditionings. III: Iterative construction of preconditioners. Zap. Nauchn. Semin. POMI, 248:17-48, 1998.

....for a parallel implementation, computes the inverse factors of A 1 using relations (3) and preserves a sparsity pattern for the factors L and U discarding entries with small modulus in the algorithm. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ( 33] [34], 35] and [36] This approach, known as FSAI , has provided good results on some dicult problems and is suitable for a parallel implementation, but it requires the prescribing in advance a sparsity pattern for the approximate factors, and this can limit its use on very general problems, ....

L.Yu Kolotilina and A.Yu. Yeremin. Factorized sparse approximate inverse preconditionings. II: Solution of 3D FE systems on massively parallel computers. Int J. High Speed Computing, 7:191-215, 1995.

....suitable for a parallel implementation, computes the inverse factors of A 1 using relations (3) and preserves a sparsity pattern for the factors L and U discarding entries with small modulus in the algorithm. An alternative approach was proposed by Kolotilina and Yeremin in a series of papers ([33], 34] 35] and [36] This approach, known as FSAI , has provided good results on some dicult problems and is suitable for a parallel implementation, but it requires the prescribing in advance a sparsity pattern for the approximate factors, and this can limit its use on very general problems, ....

L.Yu Kolotilina and A.Yu. Yeremin. Factorized sparse approximate inverse preconditionings. I: Theory. SIAM J. Matrix Analysis and Applications, 14:45-58, 1993.

....additional M v matrix vector multiply; thus, it is 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. Examples are the FSAI approach by Kolotilina and Yeremin [17], the MR algorithm by Chow and Saad [10] and the 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 ....

L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditionings: I. Theory, SIAM J. Matrix Anal. Appl. 14, 1993, pp. 45-58.

....explicitly, each iteration step requires only one additional M v matrix vector multiply; therefore, it is 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 ....

L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditionings: I. Theory, SIAM J. Matrix Anal. Appl. 14, 1993, pp. 45-58.

....system (1) is given by x = Zu . Note that the coefficient matrix in (2) is symmetric positive definite. In this paper, we consider exclusively approximate inverses in factorized form. Several techniques have been proposed to compute factorized sparse approximate inverse preconditioners; see, e. g, [32], 10] 42] 25] 31] and the references therein. There are two major issues in computing a sparse approximate inverse: the first is the choice of an appropriate sparsity pattern for the approximate inverse factor Z , and the second is the actual computation of the entries of Z . Determining a ....

....can be difficult, especially for unstructured problems. Techniques for guessing a sparsity pattern a priori have been investigated, for instance, in [19] Once a sparsity pattern has been found, least squares techniques can be used to compute an approximate inverse with the given sparsity pattern [32]. Alternatively, the sparsity pattern can be computed dynamically, together with the nonzero entries, by applying a drop tolerance in the course of an inverse factorization algorithm. This is the approach taken in the AINV algorithm [10] and in this paper we will focus on variants of this ....

[Article contains additional citation context not shown here]

L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditioning I: Theory, SIAM J. Matrix Anal. Applic., 14 (1993), pp. 45--58.

....= Zu . Note that the coefficient matrix in (2) is symmetric positive definite. In this paper, we consider exclusively approximate inverses in factorized form. Several techniques have been proposed to compute factorized sparse approximate inverse preconditioners; see, e. g, 32] 10] 42] 25] [31] and the references therein. There are two major issues in computing a sparse approximate inverse: the first is the choice of an appropriate sparsity pattern for the approximate inverse factor Z , and the second is the actual computation of the entries of Z . Determining a good sparsity pattern ....

....the cost of applying the preconditioner without too much affecting the rate of convergence of the PCG iteration. A simple criterion is to drop entries below a prescribed drop tolerance. A posteriori filtration has been shown to improve the efficiency of the FSAI preconditioned iteration in [31]. Here we consider post filtration for the SAINV preconditioner. Because SAINV, unlike FSAI, is already computed using a drop tolerance, the usefulness of applying post filtration is not immediately obvious. Effects of post filtration on point and block SAINV preconditioning are shown in Fig. 10 ....

L. Yu. Kolotilina, A. A. Nikishin and A. Yu. Yeremin, Factorized sparse approximate inverse preconditioning IV: Simple approaches to rising efficiency, Numer. Linear Algebra Applic., 6 (1999), pp. 515--531.

....sequentially computes a number of eigenpairs by CG minimizations of the Rayleigh quotient over subspaces of decreasing size. When effectively preconditioned, we found [3] that the efficiency of DACG well compares with that of established packages, like ARPACK. We exploit two preconditioners, FSAI [7] and AINV [2] falling into the class of approximate inverse preconditioners. These preconditioners have been extensively studied by many authors in order to accelerate, in a parallel environment, iterative methods for solving linear systems Ax = b. These preconditioners explicitly compute an ....

....for example when A is an H matrix (see [2] a unit upper triangular matrix Z and a diagonal matrix with positive diagonal entries D, are obtained. Thus, a sparse approximate inverse of A, M = Z D Gamma1 Z T = ZZ T is produced, being Z = ZD Gamma1=2 . The FSAI preconditioner [7] consists of computing a lower triangular matrix G L by solving the matrix equation ( G L A) ij = ffi ij ; i; j) 2 S L ; S L being a prescribed lower triangular sparsity pattern, including the main diagonal. We set S L = f(i; j) A ij 6= 0; i jg, i.e. the same pattern as A. The final ....

L. Y. Kolotilina and A. Y. Yeremin. Factorized sparse approximate inverse preconditioning I. Theory. SIAM J. Matrix Anal., 14:45--58, 1993.

.... categorized first by their result: some algorithms produce a single sparse matrix approximating A 1 e.g. SPAI[22] Chow and Saad s MR method[15, 16] Tang and Wan s local inverse[35] and others produce factored approximations (approximate inverses of the triangular factors) e.g. FSAI[25], AINV[4] The factored form has the advantages of guaranteed non singularity, extra sparsity from good orderings, and apparently more effect per nonzero thanks to its more implicit nature. However, the non factored form has the advantages of robustness with respect to orderings bad pivots are ....

....that works when the matrix is known only as a linear operator. Actually, a little more is known: the adjoint of the operator (M T # AM 1 # ) T = M T # A T M 1 # may be used in the algorithm as well. This rules out the Frobenius norm minimization 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 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 ....

L. Kolotilina and A. Yeremin, Factorized sparse approximate inverse preconditionings I. theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....such preconditioning techniques is parallelism. Another motivation is that the ILU preconditioners, whichhave been developed for M matrices [19] often fail for indefinite matrices. A few of these approximate inverse techniques are based on minimizing the norm kI ; AMk 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 diagonal D, such that Z AW = D, see 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 ....

Y. Kolotilina and Y. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

....on high performance hardware; they are also a valuable general purpose alternative to ILU for tough problems where ILU breaks down from instabilities. Several algorithms for computing sparse approximations to A 1 , or to its inverse triangular factors L 1 and U 1 , have been proposed: e.g. [5, 6, 7, 19, 27, 30, 36]. Unfortunately, for linear systems arising from elliptic PDE s, there appears to be an inherent problem in the explicit nature of these preconditioners, a fundamental conflict between accuracy and sparsity. As problem sizes increase, their performance (either in terms of convergence rate at a ....

L. Kolotilina and A. Yeremin, Factorized sparse approximate inverse preconditionings I. theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....since their application only requires (easily parallelized) matrix vector multiplication. These methods can be divided into two classes: those that form an approximation to the inverse matrix (e.g. 4] 11] 14] and those that approximate the inverses of the matrix s LU factors (e.g. 1] 2] [12]) This second class has the benefit of guaranteeing that that the preconditioner is nonsingular, and more importantly it seems that the factored form is more e#ective per nonzero[3] However, the inverse factors are critically dependent on the ordering of the matrix indeed, in general they will ....

....A 1 # ZD 1 W T where Z and W are unit upper triangular, and D 1 is diagonal. In [2] Benzi and Tuma began an investigation of ordering schemes for AINV, which we elaborate here. Notice that many of the results found here should apply to other factored approximate inverse schemes (e.g. see [12]) 3. Goals. Intuitively, for a more e#ective preconditioner we need a more accurate approximation to the true inverse factors. However, sparsity constraints do not allow # The work was supported by the Natural Sciences and Engineering Council of Canada, the Information Technology Research Centre ....

L. Kolotilina and A. Yeremin, Factorized sparse approximate inverse preconditionings I. theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....explicitly, each iteration step requires only one additional M v matrix vector multiply; therefore, it is 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 ....

L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditionings: I. Theory, SIAM J. Matrix Anal. Appl. 14, 1993, pp. 45-58.

....first proposed in the late 1970s [15] were in this category. It is only with the advent of massive parallel processing that such methods were considered as serious contenders of the now standard ILU methods [86] A flurry of publications followed this work and the work by Kolotilina and Yeremin [105, 106]. To cite just a few, 46, 85] define strategies for determining the best pattern for the inverse, 41, 40, 17, 16] define alternative schemes. While at the beginning, these preconditioning methods were received with much skepticism, it is fair to say that substantial progress has been made and a ....

....reported that approximate inverse schemes can often be competitive with ILU factorization methods even in a sequential environment. One idea for constructing an approximate inverse is to find a sparse matrix M such that kAM Gamma Ik is small for some convenient norm. Kolotilina and Yeremin [106] presented an algorithm in which the inverse was delivered in factored form, which has the advantage that singularity of M is avoided. In [46] an algorithm is presented which uses the 1 norm for the minimization. We also mention Chow and Saad [41] who use GMRES for the minimization of kAM Gamma ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I.Theory. SIAM J. Matrix Anal. Applic., 14:45--58, 1993.

....M and I respectively. Thus solving (2) leads to solving n independent least squares problems, min m j kAm j Gamma e j k 2 ; j = 1; n; 3) which can be done in parallel. Another possibility is to use a weighted Frobenius norm which had been investigated intensively by Kolotilina et al. [26] [25] 27] A complete survey can be found in [1] Other constructions of approximate inverse are discussed in [11] 10] 30] 4] 5] 19] 8] A comparison of approximate inverse preconditioners and ILU(0) on Harwell Boeing matrices can be found in [20] In this paper, however, we shall focus ....

....[11] For the former approach, they solve the least squares problems (3) by QR factorization, which may sound costly. But since m j is sparse, the cost of QRF can be greatly reduced. Moreover, they derive algorithms to determine the positions of fill in adaptively. Similar methods can be found in [26], 25] 27] 22] 23] in which case, the sparsity pattern of M is typically fixed as banded or the nonzeros of A. For Chow and Saad s approach, they use standard iterative method (e.g. GMRES) to find an approximate solution to Am j = e j ; and apply some dropping strategy to m j to control ....

L.Yu. Kolotilina and A.Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

....[15, 16, 21] Parallel implementations it is not a trivial task to write a high performance parallel implementation of ILU preconditioners. It is even more di#cult to construct an e#ective portable (between di#erent architectures) implementation. Recently, interest in a SAIP has emerged [8, 11, 14, 31, 32, 33, 35, 38, 39, 40]. The motivation of this renewed interest is largely from parallel processing. Instead of using ILU to approximate the matrix A, we seek a sparse approximation of A 1 . The forward and backward solution process in the ILU preconditioning step is replaced by a simple (sparse) matrix vector ....

....The search for an optimal sparsity pattern would be a much more expensive proposition than the solution of Ax = b itself. E#ective heuristics are required. The success of several new methods relies on their elegant schemes in determining the sparsity pattern and the solution of the approximation [5, 14, 11, 33, 39]. Their new insight to this old idea has o#ered new promise. Some di#cult problems can be e#ectively solved using these new techniques. There are two kinds of approaches to constructing sparse approximate inverses. The first and usually more e#ective one is the factorized sparse approximate ....

[Article contains additional citation context not shown here]

L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....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 ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory. SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.

....thus A is preconditioned to become GLAG T L . A weighted Frobenius norm is used in this case, and results in a small linear system for each row for a given sparsity pattern. The structure of GL is chosen to be the same as the structure of the lower triangular part of A. In their more recent work [19], fill in elements may be added, but are chosen in a way so that constructing GL is not much more expensive, with the simple hope that the additional degrees of freedom will improve the preconditioner quality. Preconditioners for general systems may be constructed by approximating the left and ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings II. Solution of 3D FE systems on massively parallel computers. Research Report EM-RR 3/92, Elegant Mathematics, Inc. Bothell, Washington, 1992.

....the number of new rows in the least squares subproblem. In similar and recent work by Huckle and Grote [11] the reduction in the residual norm is tested for each candidate fill in element, but fill in may be introduced more than one at a time. In other related work, Kolotilina and Yeremin [18] consider symmetric, positive definite systems and construct factorized sparse approximate inverse preconditioners which are also symmetric, positive definite. The preconditioner has the form M = G T L GL where GL is lower triangular, and thus A is preconditioned to become GLAG T L . A ....

....possible to construct factorized approximate inverses of the form min M 2i 1 kI Gamma M 2i Delta Delta Delta M 4 M 2 AM 1 M 3 Delta Delta Delta M 2i 1 k 2 F (13) which alternate from left to right factors. This latter form is reminiscent of the symmetric form of Kolotilina and Yeremin [18]. However, we only did experiments with the first factorized form (12) which we consider from here on. Since the product M 1 M 2 Delta Delta Delta M i is never formed explicitly, the factorized approach effectively uses less memory for the preconditioner at the cost of multiplying with each ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory. SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.

....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, 6, 8, 18, 15] or its factors [1, 19], 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 be sparse. The ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory. SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.

....years, a number of techniques have been developed which directly approximate the inverse of A. The popularity of these methods is due mostly to their suitability for parallel computing environments. A few of these approaches are based on minimizing the norm kI Gamma AMk in some appropriate norm [17, 15, 13, 7] while others directly solve the equation Z AW = D, where the unknown matrices Z; W are unit upper triangular and D is a diagonal matrix, see e.g. 22, 4, 5, 1, 16] The latter class of methods in particular has a similar algebraic behavior which is already well known for incomplete ....

Y. Kolotilina and Y. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

....of preconditioning to speed convergence, is a popular choice. Approximate inverse preconditioners, whose application requires only (easily parallelized) matrix vector multiplication, are of particular interest today. Several methods of constructing approximate inverses have been proposed (e.g. [2, 3, 9, 20, 22, 24]) falling into two categories: those that directly form an approximation to A 1 and those that form approximations to the inverses of the matrix s LU factors. This second category currently shows more promise than the first for three reasons. First, it is easy to ensure that the factored ....

....is diagonal. However, the purely structural results presented in section 2 apply equally to other factored approximate inverse schemes. Whether the numerical results carry over is still to be determined. For example, conflicting evidence has been presented in [5] and [16] about the e#ect on FSAI [22], which perhaps will be resolved only when the issue of sparsity pattern selection for FSAI has been settled. Some preliminary work in studying the e#ect of ordering on the performance of AINV has shown promising results [3] A more recent work by the same authors is [5] We carry this research ....

L. Kolotilina and A. Yeremin, Factorized sparse approximate inverse preconditionings I. theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....(Mx) b; M Gamma1 L AM Gamma1 R (MRx) M Gamma1 L b: With appropriate choices of preconditioners, M or M L MR , the linear system solvers converge faster on the above equations than on the original equation. Research on preconditioners for linear systems continues to be very active [1, 3, 22, 28, 37, 55]. One motivation of this paper is to take advantage of this research and apply it for solving eigenvalue problems. Work supported by NSF DMR 95 25885 and by the Minnesota Supercomputer Institute. 1 Most of the successful iterative linear system solvers are preconditioned projection methods, ....

....an approximate solution z of this preconditioning equation. Some of the common strategies include applying iterative methods on the preconditioning equation [2, 19, 35] constructing an approximation to A that is easy to invert [1, 3, 37, 55] and constructing an approximation to the inverse of A [9, 20, 22]. This paper will attempt to exploit results of this research and apply them to precondition eigenvalue problems. One of our first goals for this purpose is to identify an effective preconditioning equation for eigenvalue problems. Davidson s method is one of the best known preconditioned ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14(1):45--58, 1993. 19

....in section 6, we show the e#ectiveness of the SAI as a smoother for multigrid on a variety of problems, including anisotropic problems, discontinuous coe#cient problems, and unstructured grid problems. 2. SAI smoothers. Various techniques have been proposed for an e#ective SAI preconditioner [2, 4, 9, 11, 18, 20, 29, 30, 37]. However, the goal of constructing an e#ective smoother is very di#erent from finding a good preconditioner. For a powerful preconditioner, the capability for removing both the high and low frequency errors is essential. In contrast, a good smoother may just dampen the high frequency errors ....

L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

....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 kI Gamma AMk 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 diagonal D, such that Z AW = D, see e.g. 21, 4, 5, 3, 14] The latter class of preconditioners turns out to have an algebraic behavior that is similar to the ....

Y. Kolotilina and Y. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

....is 1, and the additive Schwarz uses full restriction and ignores off processor values during interpolation. All these options can be modified by the user either on command line or inside the program. Approximate inverse is a relatively new type of preconditioning strategy for parallel machines [9, 13, 17, 30]. Explicit forms of the approximate inverse preconditioners can be easily applied as parallel sparse matrix vector multiplication which can usually be performed faster than triangular solutions required by all incomplete LU preconditioners. Approximate inverse preconditioners can be easily ....

L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14(1):45--58, 1993.

....generating excessive fill in. The extensive set of di#cult test problems we consider at the end shows that our algorithm produces a sparse and e#ective preconditioner. The computation of approximate inverses, based on minimizing (4) has been proposed by several authors. Kolotilina and Yeremin [11]; Kolotilina, Nikishin, and Yeremin [12] and Lifshitz, Nikishin, and Yeremin [13] compute a factorized sparse approximate inverse but only consider fixed sparsity patterns. Grote and Simon solve 840 M. J. GROTE AND T. HUCKLE (4) explicitly but only allow for a banded sparsity pattern in M [9] ....

....inverse. It is easy, however, to reformulate the algorithm to compute only the lower triangular part of M . This yields a symmetric preconditioner, but the algorithm then loses its inherent parallelism. An interesting alternative would be to compute a factorized sparse approximate inverse, as in [11], but to leave the sparsity open as in the SPAI algorithm. The convergence of most iterative methods heavily depends on the distribution of the eigenvalues or the singular values of the preconditioned matrix [6] Indeed, if most eigenvalues are clustered about 1 and only a few outliers are ....

L. YU.K OLOTILINA AND A. YU.YEREMIN, Factorized sparse approximate inverse preconditionings, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

.... is needed (in the real life computations the required accuracy is about 1 5 ) The problem of approximate evaluation of the inverse matrices is very useful when one is interested in finding special preconditioning matrices used to accelerate the convergence of basic iterative methods (see, [KY93]) There are several basic advantages of these algorithms. It is well known that Monte Carlo algorithms are parallel algorithms. They have high parallel efficiency when parallel computers are used [MU49] DT93] DT93b] Monte Carlo algorithms are also very efficient when the problem under ....

....the classical methods. The presented estimates show that Monte Carlo algorithms are preferable when one need to have a coarse estimation of the inverse matrix. The problem is very important when one is interested in finding factorized sparse approximate inverse preconditioners (see, for example [KY93]) If a system of p processors is available, then the Monte Carlo techniques take time TMC (m; k; n; p) O m 2 p kn : Clearly, the rate of convergence (respectively, the average length k of the Markov chain) depends on the spectral radius of the matrix. As long as the spectral radius ....

L.Yu. Kolotilina, A. Yu. Yeremin, Factorized Sparse Approximate Inverse Preconditionings I. Theory, SIAM J. Matrix Anal. Appl., vol. 14 (1993), No 1, pp. 45--58.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings i. theory. SIAM J. Matrix Anal. Appl., 14(1):45--58, 1993.

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L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditioning I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45--58.

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Kolotilina LY, Yeremin AY. Factorized sparse approximate inverse preconditioning I: theory. SIAM Journal of Matrix Analysis and its Applications 1993; 14:45-58.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

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L. Y. Kolotilina and A. Y. Yeremin, Factorized Sparse Approximate Inverse Preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14, 1993, pp. 45-58.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory. SIAM Journal on Matrix Analysis and Applications, 14:45--58, 1993.

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L. Y. Kolotilina and A. Y. Yeremin. Factorized sparse approximate inverse preconditionings. I. Theory. SIAM J. Matrix Anal. Appl., 14(1):45--58, 1993.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings. SIAM J. Matrix Anal. Applic., 14:45--58, 1993.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings i. theory. SIAM J. Matrix Anal. Appl., 14:45--58, 1993.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings ii. solution of 3d fe systems on massively parallel computers. Technical Report EM-RR 3/92, Elegant Mathematics, Inc., Bothell, Washington, 1992.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings II. solution of 3D FE systems on massively parallel computers. International Journal of High Speed Computing, 7(2):191--215, 1995.

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L. Yu. Kolotilina and A. Yu. Yeremin. Factorized sparse approximate inverse preconditionings I. Theory. SIAM Journal on Matrix Analysis and Applications, 14(1), 45--58, 1993.

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