M. Benzi, J. K. Cullum, and M. Tuma. Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput., 22(4):1318--1332 (electronic), 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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 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 ....

....(7) and (8) will all lead to the same S. In practice, 5) is the most common scheme for defining Incomplete LU factorizations, see, e.g. 20] or [22] Typically, 5) produces the smallest amount of fill in compared with the other formulas. The update (8) has also been used in a number of papers [24, 2, 6, 8]. In the symmetric positive definite case, it is guaranteed to produce a stable ILU factorization, see [24] 2.1 Update variants In order to simplify the description of the algorithms to be considered we make the following observation which allows us to express all four types of updates just ....

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M. Benzi, J. K. Cullum, and M. Tuma. Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput., 22:1318--1332, 2000.

.... 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 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 ....

....that are less than a certain drop tolerance. Moreover it has been pointed out in [4] that dropping entries of p,q produces poor results. We will still consider this variant for generality and because it has an interesting direct connection with ILU. In [4, 5] p, q were defined via (10) while in [14, 3] for the case of symmetric positive definite matrices (11) is used (In this situation, W Z) Clearly, the strict biorthogonality property of the exact factorized inverse does not hold anymore if a drop tolerance is introduced. Interestingly, however, stability can be proved 9 for H matrices, ....

M. Benzi, J. K. Cullum, and M. Tma. Robust approximate inverse preconditioning for the conjugate gradient method. Technical report LA-UR-99-2899, Los Alamos National Laboratory, Scientific Computing Group, 1999.

.... 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 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 ....

....that are less than a certain drop tolerance. Moreover it has been pointed out in [4] that dropping entries of p# q produces poor results. We will still consider this variant for generality and because it has an interesting direct connection with ILU. In [4, 5] p# q were defined via (10) while in [14, 3] for the case of symmetric positive definite matrices (11) is used (In this situation, W j Z) Clearly, the strict biorthogonality property of the exact factorized inverse does not hold anymore if a drop tolerance is introduced. Interestingly,however, stability can be proved 9 for H matrices, ....

M. Benzi, J. K. Cullum, and M. T ffi uma. Robust approximate inverse preconditioning for the conjugate gradient method. Technical report LA--UR--99--2899, Los Alamos National Laboratory, Scientific Computing Group, 1999.

....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 ....

....without knowledge of the true inverse A 1 . Note that to avoid forming (M T # AM 1 # ) explicitly, which may incur significant fill in, an approximate inverse algorithm that works on a linear operator (not necessarily a matrix) is required. One example, used in this research, is SAINV[7]. To summarize, an overview of the multiresolution approximate inverse method is given in figure 2.1. 3. The General Algorithm. 4 R. BRIDSON, W. P. TANG Fig. 3.1. The transform algorithms of the Lifting Scheme The forward transform (standard basis # multiresolution basis) Start with the ....

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M. Benzi, J. K. Cullum, and M. Tuma, Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci. Comput., 22 (2000), pp. 1318--1332.

....alternative. The intention of this paper is to take a closer look at incomplete LU decompositions and especially on how entries are dropped. The main key used here for analyzing dropping in the incomplete LU decomposition is its strong relation [7, 8] to factored sparse approximate inverse methods [3, 4, 2, 16, 21]. In an earlier paper [8] comparisons between an incomplete LU decompositions with pivoting and a factored approximate inverse with pivoting have shown several examples where the approximate inverse was superior to the ILU . So apparently ILUs may gain more stability from approximate inverses by ....

....pivoting. For simplicity let us consider the algorithms without pivoting at this stage. Recently it has been shown in [7] that Algorithm 1 has a strong relation to sparse approximate inverse preconditioners. Without going into the details, we will roughly describe the idea of AINV type algorithms [3, 4, 2, 8]. The idea is to directly compute upper triangular matrices W;Z such that W AZ = D, with a diagonal matrix D. The version which we will focus on is the so called right looking AINV, where W and Z are updated by a rank 1 update. Essentially a biorthogonalization process for W and Z is ....

M. Benzi, J. K. Cullum, and M. Tuma. Robust approximate inverse preconditioning for the conjugate gradient method. Technical report LA{UR{99-2899, Los Alamos National Laboratory, Scientic Computing Group, 1999.

....part. This paper explores how to get the best performance out of an approximate inverse preconditioner, particularly on modern superscalar workstations. The algorithm we turn our attention to is Benzi and Tuma s AINV[1, 2] or more specifically, a slight variation on the stabilized version SAINV[4]. We previously explored the issue of ordering in [7] noting that for good orderings the set up time for the preconditioner can be reduced dramatically. Here we go into details on that and other techniques for boosting the performance of the method. We note that in [1, 2] Benzi and Tuma had ....

....mode. For this article, compressed column storage format is assumed: each matrix is a collection of n sparse column vectors. However, the inner products rZ j and W T j c are more efficiently computed if one of the vectors is stored in full mode; while a sparse sparse operation could 1 SAINV in [4] is actually a generalization of modified Gram Schmidt; this variation is a slightly faster but typically equal quality algorithm. REFINING AN APPROXIMATE INVERSE 3 theoretically be faster, a typical implementation s more complicated branching and memory accesses make it slower on today s ....

M. Benzi, J. Cullum, and M. Tuma, Robust approximate inverse preconditioning for the conjugate gradient method, submitted to SIAM J. Sci. Comput. REFINING AN APPROXIMATE INVERSE 16

....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 LU decompositions, e.g. they are stable for M and H matrices. In [6] relations between factorized approximate inverses and incomplete LU decompositions have been ....

....and modifications are performed only once for each row, thus making it possible to use very simple sparse row storage schemes such as the Compressed Sparse Row (CSR) format. Algorithms for directly computing W;Z such that W AZ = D, with a diagonal matrix D have recently been suggested in [4, 5, 1, 16]. Here we choose to outline a version that has been used for the symmetric positive definite case. Algorithm 2 (Factorized Approximate Inverse) Let A = A ij ) ij 2 R n;n , drop tolerance . Compute A Gamma1 ZD Gamma1 W . p = q = 0; 0) 2 R n ; Z = W = I n . for i = 1 ....

M. Benzi, J. K. Cullum, and M. Tuma. Robust approximate inverse preconditioning for the conjugate gradient method. Technical report LA--UR--99--2899, Los Alamos National Laboratory, Scientific Computing Group, 1999.

.... 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 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 ....

....are less than a certain drop tolerance. Moreover it has been pointed out in [4] that dropping entries of p; q produces poor results. We will still consider this variant for generality and because it has an interesting direct connection with ILU . In [4, 5] p; q were defined via (10) while in [14, 3] for the case of symmetric positive definite matrices (11) is used (In this situation, W j Z) Clearly, the strict biorthogonality property of the exact factorized inverse does not hold anymore if a drop tolerance is introduced. Interestingly, however, stability can be proved 9 for H matrices, ....

M. Benzi, J. K. Cullum, and M. T ffi uma. Robust approximate inverse preconditioning for the conjugate gradient method. Technical report LA--UR--99--2899, Los Alamos National Laboratory, Scientific Computing Group, 1999.

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Benzi M, Cullum JK, Tuma M. Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput, 22 (2000), 1318-1332.

....that is, we have an incomplete inverse factorization of C of the form C Z D Z where D is diagonal with entries d j = z j C z j 0. This is a factored sparse approximate inverse that can be used as a preconditioner for the conjugate gradient algorithm applied to Cx = f ; see [2, 14]. The preconditioner is guaranteed to be positive de nite (since d j 0 for all j) and is easily applied in parallel, since its application only requires matrix vector products. It is generally known as the SAINV (for Stabilized Approximate Inverse) preconditioner. Note that the construction ....

....and the corresponding update in (3.4) can be skipped. It is important to mention that our implementation makes use of structural information on the incomplete inverse factor Z so as to avoid checking which of the inner products are structurally zero, which would be an O(n ) operations; see [2]. Note that the z i vectors are stored (they form the columns of the approximate inverse factor Z) whereas the multipliers h z j ; z i i C =h z j ; z j i C are discarded after they are used to perform an update step. The second preconditioner that can be obtained is, in a sense, the ....

M. Benzi, J. K. Cullum, and M. Tuma, Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci.Comput., 22 (2000), pp. 1318-1332.

....100, Atlanta, GA 30322, USA (e mail: benzi mathcs.emory.edu) Received DRAFT Copyright c fl 2001 John Wiley Sons, Ltd. Revised DRAFT Accepted DRAFT A orthogonalization process and is therefore applicable to general SPD matrices. This A orthogonalization process has been previously used in [1] and [2] to construct robust sparse approximate inverse preconditioners in factored form; here we show that the same algorithm can be used to reliably compute an approximate factorization of A, with only a slight increase in terms of storage. In section 4 we present the results of numerical ....

....preconditioner M = LL is then applied to the original linear system Ax = b. Preconditioners based on diagonally compensated reduction give fairly good results on moderately ill conditioned problems, such as solid elasticity problems [18] and diffusion problems posed on highly distorted meshes [1], but they are ineffective for more difficult problems arising in the finite element analysis of thin shells and other structures; see [15, 19, 1] A more sophisticated approach has been proposed by Tismenetsky [20] with significant improvements and theoretical foundations supplied by Kaporin in ....

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Benzi M, Cullum JK, Tuma M. Robust approximate inverse preconditioning for the conjugate gradient method. SIAM Journal on Scientific Computing 2000; 22: 1318--1332.

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M. Benzi, J. K. Cullum and M. Tuma. Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci. Comput., to appear. M. Benzi, C. D. Meyer and M. Tuma. A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17:1135-1149, 1996.

....together with its stabilized variants. Block algorithms are described in section 3. Section 4 is devoted to experimental results. We present some conclusions in section 5. 2 A stable approximate inverse algorithm In this section we recall the basic AINV technique [10] and its stabilized variants [7], 30] The AINV algorithm builds a factorized sparse approximate inverse of the form M = ZD Gamma1 Z T A Gamma1 (3) 3 where Z is a sparse unit upper triangular matrix and D is diagonal. The algorithm computes Z and D directly from A by means of an incomplete A orthogonalization ....

....resulting preconditioner is often of poor quality. Fortunately, it is possible to prevent breakdowns without the need for any diagonal corrections, simply by formulating the A orthogonalization algorithm in a slightly different manner. This reformulation was recently developed, independently, in [7] and [30] see also [13] for a somewhat different approach. We refer to this algorithm as the SAINV (for stabilized AINV) algorithm. In passing, we also note that there exist other algorithms for constructing factorized approximate inverse preconditioners for positive definite matrices which are ....

[Article contains additional citation context not shown here]

M. Benzi, J. K. Cullum and M. Tuma, Robust approximate inverse preconditioning for the conjugate gradient method, , SIAM J. Sci. Comput., 22 (2000), pp. 1318--1332.

....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. Sufficient conditions are that A be SPD or an H matrix; see [37] 6] 36] [4]. In particular, the AINV preconditioner is well defined if A is diagonally dominant. For general sparse matrices, the existence of the AINV preconditioner is not guaranteed and instabilities, i.e. breakdowns due to very small or zero pivots, can occur during the construction of the ....

M. Benzi, J. K. Cullum and M. Tuma, Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method, Los Alamos National Laboratory Technical Report LA-UR99 -2899, Los Alamos, NM, June 1999.

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M. Benzi, J. K. Cullum, and M. Tuma. Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput., 22(4):1318--1332 (electronic), 2000.

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