Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Numerical Linear Algebra with Applications, 6:257--280, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....E is a zero vector, then s ij = c ij . Definition 4. 2 A node v i of the vertex set V is said to be independent from a subset V I of V if and only if a ij = 0 and a ji = 0 for all v j 2 V I : An immediate consequence of the independence is the following corollary that is first proved in [42]. Corollary 4.3 If a node v i in V 2 is independent from all the nodes in V 1 , then s ii = c ii ; i.e. the values of the ith row of C will not be modified in the partial LU factorization. We now modify our threshold tolerance reordering strategy slightly to a diagonal threshold strategy, ....

....v i in V 2 is independent from all the nodes in V 1 , then s ii = c ii ; i.e. the values of the ith row of C will not be modified in the partial LU factorization. We now modify our threshold tolerance reordering strategy slightly to a diagonal threshold strategy, similar to that discussed in [42]. We assume that the node v i is in V 1 if ja ii j ffl and D is still a diagonal matrix. With such a modification, we have jd i j ffl for 1 i m. Denote by M = max 1i;jn fja ij jg the size of the largest elements in absolute value of A. Proposition 4.4 The size of the elements of the Schur ....

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Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Numer. Linear Algebra Appl., 6(4):257--280, 1999. 23

....convergence rate. The FIDAP and Flat matrices tested in Sections 5.2 and 5.3 have small or zero main diagonal entries. The poor convergence performance of both PBILU2 and SLU is mainly due to the instability associated with ILU factorizations of these matrices. Diagonal thresholding strategies [32, 38] can be employed in PBILU2 to exclude the rows with small diagonals from the submatrix B, so that its ILU factorization will be stable. The parallel implementation of diagonally thresholded PBILU2 will be investigated in our future study. We plan to extend our parallel two level block ILU ....

Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Numer. Linear Algebra Appl., 6(4):257-- 280, 1999.

....L and U factors [24] In Gaussian elimination such a problem may be avoided by employing a partial or full pivoting strategy. In multilevel preconditioning technique a diagonal threshold strategy may be used to force the nodes (rows) with small size diagonal elements into the coarse level system [41, 43]. Thus, all rows in the vertex cover should have large absolute diagonal values. Moreover, there is a concept of strong coupling (connection) among a group of nodes. A node j is said to be strongly connected to a node i if ja i;j j is large. In multilevel block preconditioning technique (BILUM ....

Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Technical Report UMSI 98/7, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1998.

....we assume that the partial LU factorization described above is exact; i.e. no dropping strategy is enforced. We also assume that, in the reordered matrix, the D submatrix is diagonal. Such a reordering can be achieved by an independent set search as in a multielimination strategy of Saad [33, 39]. Thus, the factorization (4) is reduced to D F E C = I 0 ED Gamma1 I D F 0 A 1 : 5) We now assume that all indices are local to individual submatrices. In other words, when we say the ith row of the matrix F , we mean the ith row of the submatrix F , not the ith row ....

....E is a zero vector, then s ij = c ij . Definition 4. 2 A node v i of the vertex set V is said to be independent to a subset V I of V if and only if a ij = 0 and a ji = 0 for all v j 2 V I : An immediate consequence of the independentness is the following corollary that is first proved in [39]. Corollary 4.3 If a node v i in V 2 is independent to all the nodes in V 1 , then s ii = c ii ; i.e. the values of the ith row of C will not be modified in the partial LU factorization. We now modify our threshold tolerance reordering strategy slightly to a diagonal threshold strategy, ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Numer. Linear Algebra Appl. to appear.

....having the smallest degree. Thus, we can define a new algorithm ( Minimal degree algorithm ) MULTI LEVEL BLOCK ILU PRECONDITIONERS 5 by replacing Line 6 of Algorithm 3.2 by the following line 1 : 6a. Choose s 2 adj(j) such that deg(s) minfdeg(i) i 2 adj(j)g It was recently suggested in [40], that stability may be further improved by insisting that the magnitude of the diagonal elements a ss resulting from the strategy in Line 6, be greater than some threshold tolerance . Larger enhances the stability of the factorization, but may reduce the size of the independent set. The best ....

....Line 6, be greater than some threshold tolerance . Larger enhances the stability of the factorization, but may reduce the size of the independent set. The best strategy may be to use a threshold parameter to ensure that the diagonals are not zero or too small (this may be applied to ILUM too [40]) The above algorithms for finding block independent sets may be modified appropriately to include the threshold strategy. There are other heuristics that may be used to form blocks, such as choosing the node whose row has the most (least) diagonal dominance. We have experimented with this and ....

Y. Saad and J. Zhang, Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems, Technical Report UMSI 98/7, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1998.

....to pair node j with a nearest neighbor having the smallest degree. Thus, we can define a new algorithm ( Minimal degree algorithm ) by replacing Line 6 of Algorithm 3.2 by the following line 1 : 6a. Choose s 2 adj(j) such that deg(s) minfdeg(i) i 2 adj(j)g It was recently suggested in [40], that stability may be further improved by insisting that the magnitude of the diagonal elements a ss resulting from the strategy in Line 6, be greater than some threshold tolerance . Larger enhances the stability of the factorization, but may reduce the size of the independent set. The best ....

....Line 6, be greater than some threshold tolerance . Larger enhances the stability of the factorization, but may reduce the size of the independent set. The best strategy may be to use a threshold parameter to ensure that the diagonals are not zero or too small (this may be applied to ILUM too [40]) The above algorithms for finding block independent sets may be modified appropriately to include the threshold strategy. There are other heuristics that may be used to form blocks, such as choosing the node whose row has the most (least) diagonal dominance. We have experimented with this and ....

Y. Saad and J. Zhang, Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems, Technical Report UMSI 98/7, Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN, 1998.

....1. The diagonal threshold strategy is enforced by excluding a row i from the block independent set if t i ffl. Such an implementation only uses the matrix values once to compute the measure ft i g. The graph of the matrix is used to build a block independent set, along with the array ft i g. In [22] it has been shown that in the point version of BILUM, a row that is excluded from the independent set because of its small diagonal value and that has no link to any other nodes Multi Level Preconditioner for Coating Problems 7 in the independent set will not be modified during the computation ....

....value with (0:0001 )r i , where r i is computed as the average nonzero values of the current row. Such a safeguard procedure prevents the ILUT factorization from breaking down. But too many replacements yield an inaccurate preconditioner. Multi Level Preconditioner for Coating Problems 9 in [22] for a different multi level preconditioning technique with blocks of size 1 (the blocks are actually points) It is shown in [22] that a row with a zero diagonal element is to remain such a status if it is not excluded as a node with links to the nodes of the independent set. However, the ....

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Saad, Y., and Zhang, J., Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems, Technical Report UMSI 98/7, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1998.

....by increasing the number of coarse level iterations so that a new preconditioner does not have to be recomputed. 2. 2 Relation to algebraic multigrid method Although the coarse level system in multilevel preconditioning techniques is used as fine level system in algebraic multigrid method [40]. It has been shown in [50] that such a difference is not of vital importance as long as the approximate solution with respect to the fine and coarse level systems can be obtained in an efficient way. In multilevel preconditioning techniques, the solution with D ff is usually obtained without ....

....of D Gamma1 ff to be very large. These large elements will cause difficulties in the subsequent recursive block ILU factorizations and may result in an unstable preconditioner [17] One way to alleviate this problem is to guarantee a certain degree of diagonal dominance of the D ff submatrix [40]. We can incorporate a diagonal threshold strategy in the search of the independent set so that only the nodes with a certain degree of diagonal dominance are allowed to be included in the independent set. Similar diagonal threshold strategies have been investigated by Saad and Zhang [40, 42] It ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Numer. Linear Algebra Appl. to appear.

....between A and LU. The ILUT implementation gives an easy representation of the residual matrix. Proposition 3.2 The elements of the residual matrix R as in Equation (7) are those elements dropped in Algorithm 3.4. Proof. The proof can be formulated from the arguments in [50, p. 274] and [54]. Clearly, Algorithm 3.4 will fail when any individual ILUT fails on at least one of the blocks due to zero pivots. There are at least three strategies to deal with this situation. First, one can use pivoting as in ILUTP [50] a variant of ILUT which incorporates column pivoting. Second, ....

....ILUT fails on at least one of the blocks due to zero pivots. There are at least three strategies to deal with this situation. First, one can use pivoting as in ILUTP [50] a variant of ILUT which incorporates column pivoting. Second, we may use a diagonal threshold strategy as was done in ILUM [54]. In this technique nodes with small absolute diagonal values are put in the vertex cover. This strategy may reduce the size of the independent set. Third, we may replace a small (absolute) diagonal value by a larger one and proceed with the normal ILUT. The third strategy is suitable and almost ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Technical Report UMSI 98/7, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1998.

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Y. Saad and J. Zhang. Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems. Numerical Linear Algebra with Applications, 6:257--280, 1999.

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