Y. Saad and J. Zhang. BILUTM: A domain-based multi-level block ILUT preconditioner for general sparse matrices. SIAM Journal on Matrix Analysis and Applications, 21, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(1) by a (sparse) direct solver, 2) by using a standard preconditioned Krylov solver, or (3) by performing a backward forward solution associated with an accurate ILU (e.g. ILUT) preconditioner. In particular, a multi level ILU type procedure could be used to solve A i i = r i approximately [16,3,4,24,23]. 2.2 Schur Complement Techniques Schur complement techniques refer to methods which iterate on the interdomain interface unknowns only, implicitly using interior unknowns as intermediate variables. These techniques are at the basis of what will be described in the next sections. Schur ....

Y. Saad and J. Zhang. BILUTM: A domain-based multi-level block ILUT preconditioner for general sparse matrices. SIAM Journal on Matrix Analysis and Applications, 21, 2000.

....the LU factors of the Schur complement matrix S. It would be interesting to modify the Gaussian elimination process in order to obtain the matrix S directly instead of its factored form. Such a modification, called the restricted version of Gaussian elimination was intro duced and exploited in [20]. We reproduce it here for the sake of completeness. ALGORITHM 2.4 Restricted Gaussian Elimination algorithm 1. For i = 1, n G 2. For k = 1,min (i 1,hA) 3. mi, k = mi,k mk, k 4. For j = k 1, n6 Do 5. mi,j : mi,j mi,k mk,j 6. EndFor(j) 7. EndFor(k) 8. EndFor(i) In this ....

....5. mi,j : mi,j mi,k mk,j 6. EndFor(j) 7. EndFor(k) 8. EndFor(i) In this algorithm, the standard elimination process is carried out for the first n rows of Mg. For the remaining rows, the elimination is carried out only to column n . It is easy to see from what was stated above (see also [20]) that the (2,2) block in the matrix M resulting from this factorization is actually the Schur complement matrix associated with the C block. Thus, sparse Gaussian elimination techniques compute LU factorizations of these Schur complement matrices. Assume now that a dropping strategy is used to ....

Y. Saad, J. Zhang, BILUTM: A domain-based multi-level block ILUT preconditioner for general sparse matrices, Technical Report UMSI-97-118, Minnesota Supercomputing Institute, Minneapolis, MN, 1998.

....is their excellent scalability with respect to mesh size. Their scope however is limited. A number of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite [3, 5, 9, 7, 19, 22, 23, 26, 27, 28]. Multigrid methods are difficult to surpass when they work. However, their implementation requires multilevel grids and specialized tuning is often needed. The Algebraic Multigrid (AMG) methods were introduced in the seventies initially by Ruge and Stuben [18] to remedy these limitations. ....

....Recently, a collection of ILU factorizations was introduced in the literature which drew much attention. These methods possess features of multilevel methods as well as some features of ILU factorizations. ILUM [19] is one such approach and recent work by Botta and co workers [8, 9] and [22, 23], indicates that this type of approach can be fairly robust and scale well with problem size, unlike standard ILU preconditioners. The idea was extended to a block version (BILUM) using dense blocks [22] and then this was further extended into BILUTM which treats the diagonal blocks as sparse ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: A domain-based multi-level block ILUT preconditioner for general sparse matrices. SIAM Journal on Matrix Analysis and Applications, 21, 2000. to appear in SIMAX.

....corresponding to the BIS ordering and a block ILU factorization of the form P l A l P T l = E l C l I 0 E l (LB l UB l ) I 0 A l 1 ; 1) where LB l UB l is an ILU factorization of the block diagonal matrix B l . This process is similar to the sequential BILUTM factorization [10]. The parallel implementation of this generic process is described in the following algorithm: Algorithm 2.1 The PBILUM Factorization. 1. If l is not the last level, Then 2. Find a global block independent set. 3. Permute the local matrix in each processor. 4. Generate the local reduced ....

....algorithms for constructing the block independent set (BIS) efficiently from a distributed sparse matrix. Two approaches to accomplishing this task have been discussed in [12] One is a sequential BIS search algorithm, which is analogous to the sequential BIS algorithms introduced for the BILUM in [10]. Another one is a parallel BIS algorithm. The simple parallel BIS algorithm implemented in [12] suffers from the double node removal problem, which effectively reduces the size of the BISes constructed at each level. This simple parallel BIS algorithm generates the last level reduced system which ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 21(1):279--299, 1999.

....this approach is fairly robust and that it scales well with problem size [7, 5] unlike standard ILU preconditioners. The idea was extended to a block version (BILUM) using a sort of domain decomposition strategy [23] A number of follow up articles demonstrated the effectiveness of this approach [22, 20, 21]. Our tests indicate that the block approach is generally more efficient and more robust than a standard ILUT preconditioned GMRES [17] as well as its scalar sibling, ILUM. For hard problems, these attributes often come with the added benefit of reduced memory usage. Although these ....

....from techniques which approximately solve the Schur complement system associated with interface variables. In this paper we extend this idea by using the Algebraic Recursive Multilevel Solvers (ARMS) framework. 2 Sequential ARMS basic notions The multi level ILU preconditioners developed in [16, 5, 6, 22, 23] exploit the property that a set of unknowns that are not coupled to each other can be eliminated simultaneously in Gaussian elimination. Such sets are termed independent sets , see e.g. 14] In [23] the ILUM factorization described in [16] was generalized by resorting to group independent ....

Y. Saad and J. Zhang. BILUTM: A domain-based multi-level block ILUT preconditioner for general sparse matrices. Technical Report umsi-98-118, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN,

....of the Schur complement matrix GammaS. It would be interesting to modify the Gaussian elimination process in order to obtain the matrix GammaS directly instead of its factored form. Such a modification, called the restricted version of Gaussian elimination was introduced and exploited in [21]. We reproduce it here for the sake of completeness. Algorithm 2.4 Restricted Gaussian Elimination algorithm 1. For i = 1; nG 2. For k = 1; min (i Gamma 1; nA ) 3. m i;k = m i;k =m k;k 4. For j = k 1; nG Do 5. m i;j : m i;j Gamma m i;k m k;j 6. EndFor(j) 7. EndFor(k) 8. EndFor(i) ....

....m i;j Gamma m i;k m k;j 6. EndFor(j) 7. EndFor(k) 8. EndFor(i) In this algorithm, the standard elimination process is carried out for the first nA rows of M g : For the remaining rows, the elimination is carried out only to column nA : It is easy to see from what was stated above (see also [21]) that the (2,2) block in the matrix M resulting from this factorization is actually the Schur complement matrix associated with the C block. Thus, sparse Gaussian elimination techniques compute LU factorizations of these Schur complement matrices. Assume now that a dropping strategy is used to ....

Y. Saad, J. Zhang, BILUTM: A domain--based multi--level block ILUT preconditioner for general sparse matrices, Technical Report UMSI--97--118, Minnesota Supercomputing Institute, Minneapolis, MN, 1998.

....and the forward backward solutions in ARMS are recursive. In addition ARMS allows inter level iterations (referred to as W cycles in the multigrid literature) though these tend to be fairly expensive if the number of levels is high. For more details on this multilevel preconditioner see [9] and [8] A particular instance of the ARMS preconditioner as well as the ARMS performance for a given iterative algorithm are controlled by several parameters, such as the block size and number of levels specifying the block and level preconditioner structures, respectively. To use deflated ....

Y. Saad and S. Zhang, BILUTM: A Domain-Based Multi-Level Block ILUT Preconditioner for General Sparse Matrices, Tech. Rep. UMSI 98/118, Supercomputer Institute, Univ. Minnesota, 1998.

....eliminated. This can be done by modifying the ILUT algorithm of Saad [37] and restricting the elimination process to the columns corresponding to V 1 , when the row index is greater than the size of V 1 . Such a process is called a partial Gaussian elimination or a partial LU factorization in [41]. Note that, due to the partial Gaussian elimination, all rows in the (E C) submatrix can be processed independently (in parallel) This is because all nodes in the E submatrix that are to be eliminated use only the computed (I)LU factorization of the (D F ) part. Note also that the diagonal ....

....submatrix can be processed independently (in parallel) This is because all nodes in the E submatrix that are to be eliminated use only the computed (I)LU factorization of the (D F ) part. Note also that the diagonal values of the rows of the C submatrix are never used as pivots. It can be shown [41] that such a partial Gaussian elimination process modifies C into the (incomplete) Schur complement of A. In exact arithmetic, C would be changed into A 1 = C Gamma ED Gamma1 F = C Gamma EU Gamma1 L Gamma1 F; 3) where LU is the standard LU factorization of the D submatrix. We point ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 21(1):279--299, 1999.

.... can be run respectfully on distributed parallel computers are scarce [9] Recently, a class of high accuracy preconditioners that combine the inherent parallelism of domain decomposition, the robustness of ILU factorization, and the scalability potential of multigrid method have been developed [30, 31]. The multilevel block ILU preconditioners (BILUM and BILUTM) have been tested to show promising convergence rate and scalability for solving certain problems. The construction of these preconditioners are based on block independent set ordering and recursive block ILU factorization with Schur ....

....within each level, their parallel implementations have not yet been reported. In this study, we mainly address the issue of implementing the multilevel block ILU preconditioners in a distributed environment using distributed sparse matrix template [26] The BILUTM preconditioner of Saad and Zhang [31] is modified to be implemented as a two level block ILU preconditioner on distributed memory parallel architectures (PBILU2) We used Saad s PSPARSLIB library 1 with MPI as basic communication routines. Our PBILU2 preconditioner is compared with one of the most favorable Schur complement based ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 21(1):279--299, 1999.

....of the Schur complement matrix GammaS. It would be interesting to modify the Gaussian elimination process in order to obtain the matrix GammaS directly instead of its factored form. Such a modification, called the restricted version of Gaussian elimination was introduced and exploited in [20]. We reproduce it here for the sake of completeness. Algorithm 2.4 Restricted Gaussian Elimination algorithm 1. For i = 1; nG 2. For k = 1; min (i Gamma 1; nA ) 3. m i;k = m i;k =m k;k 4. For j = k 1; nG Do 5. m i;j : m i;j Gamma m i;k m k;j 6. EndFor(j) 7. EndFor(k) 8. EndFor(i) ....

....m i;j Gamma m i;k m k;j 6. EndFor(j) 7. EndFor(k) 8. EndFor(i) In this algorithm, the standard elimination process is carried out for the first nA rows of M g : For the remaining rows, the elimination is carried out only to column nA : It is easy to see from what was stated above (see also [20]) that the (2,2) block in the matrix M resulting from this factorization is actually the Schur complement matrix associated with the C block. Thus, sparse Gaussian elimination techniques compute LU factorizations of these Schur complement matrices. Assume now that a dropping strategy is used to ....

Y. Saad, J. Zhang, BILUTM: A domain--based multi--level block ILUT preconditioner for general sparse matrices, Technical Report UMSI--97--118, Minnesota Supercomputing Institute, Minneapolis, MN, 1998.

....Such adjustment inevitably increases the number of iterations needed to reach convergence, 18 as we showed in Section 4.3, so a balance is needed here. More sophisticated preconditioning techniques that exploit multigrid or multilevel concepts should be considered for very large scale problems [32, 30]. We have taken some very preliminary steps into these adaption questions, but they remain a focal point for our continuing work. 6 Acknowledgements This research has been supported in part by NSF grants # 791AT 51067A and CCR 9902022, by the State of Texas Advanced Technology Program and by a ....

Y. Saad and J. Zhang. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 21(1):279--299, 1999. 20

....entries corresponding to the nonzero positions of A are computed and stored [12] Hence, M is as sparse as A and contains 5 diagonals. This simple IC preconditioner works so well for the current problem that we consider the implementation of other powerful preconditioning strategies unwarranted [18, 19]. The IC algorithm we used is a generic IC procedure for general sparse matrices taken from [7] For reference convenience, it is reproduced in Algorithm 4.1, where we use the notations M = m i;j ) and A = a i;j ) Algorithm 4.1 Procedure for incomplete Cholesky factorization. 1. m 1;1 = p a ....

Y. Saad and J. Zhang. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 21(1):279--299, 1999.

....is their excellent scalability with respect to mesh size. Their scope however is limited. A number of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite [2, 4, 7, 8, 17, 19, 20]. Multigrid methods can be extremely efficient when they work. However, their implementation requires multi level grids and specialized tuning is often needed. The Algebraic MultiGrid (AMG) This work was supported in part by NSF under grant CCR 9618827, and in part by the Minnesota Supercomputer ....

....which is as general purpose as the ILU Krylov combination. Recently, a class of preconditioners that drew much attention is a collection of ILU factorizations which possess certain features of multigrid techniques. ILUM [17] is one such approach and recent work by Botta and co workers [6, 7] and [19, 20], indicates that this type of approach can be fairly robust and scale well with problem size, unlike standard ILU preconditioners. This method combines the generality of Krylov methods and the scalability of multigrid methods. The idea was recently extended to a block version (BILUTM) with a ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: A domain-based multi-level block ILUT preconditioner for general sparse matrices. Technical Report UMSI 98/118, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1998. to appear.

....the sparsity of the successive matrices, a dropping strategy is introduced to discard small elements. The resulting multilevel incomplete LU factorization can be used as a preconditioner for a Krylov subspace method. For detailed discussions on multilevel preconditioning techniques, we refer to [14, 13]. The major difference between BILUM and MG is that MG uses the same discretization scheme on all the levels, but BILUM only uses the discretization scheme on the finest level. BILUM generates coarse level systems through algebraic factorizations. 4 Numerical Results Two test problems were solved ....

Y. Saad and J. Zhang. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl. to appear.

....without sacrificing overall effectiveness. Similar preconditioners developed in [7, 39] show near grid independent convergence for certain types of problems. Block versions of ILUM have recently been designed using small dense blocks (BILUM) or large domains (BILUTM) as pivots instead of scalars [39, 42, 40]. For some hard to solve problems, BILUM and BILUTM may perform much better than ILUM. Various strategies have been proposed to invert or factor the blocks or domains efficiently. We remark that extracting parallelism from ILU factorizations has been the initial motivation behind the development ....

....and BILUTM may perform much better than ILUM. Various strategies have been proposed to invert or factor the blocks or domains efficiently. We remark that extracting parallelism from ILU factorizations has been the initial motivation behind the development of these multilevel ILU preconditioners [36, 39, 40]. In a recent paper [48] BILUM was tested with several popular Krylov subspace accelerators for solving a few nonsymmetric matrices from applications in computational fluid dynamics. The test results show that the quality of the preconditioner determines the convergence rates of preconditioned ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl. to appear.

....is only performed with respect to the columns in E, and linear combinations for columns in C are performed accordingly. In other words, the elements corresponding to the C submatrix are not eliminated. Such a process is called a partial Gaussian elimination or a partial LU factorization in [38]. Note that, due to the partial Gaussian elimination, all rows in the (E C) submatrix can be processed independently (in parallel) This is because all nodes in the E submatrix that are to be eliminated use only the computed (I)LU factorization of the (D F ) part. Note also that the diagonal ....

....submatrix can be processed independently (in parallel) This is because all nodes in the E submatrix that are to be eliminated use only the computed (I)LU factorization of the (D F ) part. Note also that the diagonal values of the rows of the C submatrix are never used as pivots. It can be shown [38] that such a partial Gaussian elimination process modifies C into the (incomplete) Schur complement of A. In exact arithmetic, C would be changed into A 1 = C Gamma ED Gamma1 F = C Gamma EU Gamma1 L Gamma1 F; 3) where LU is the standard LU factorization of the D submatrix. Hence, this ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl. to appear.

....block ILU preconditioner. We introduces some new strategies to increase the robustness of a multi level block ILU preconditioning technique to solve unstructured sparse matrices from the coating problems. The multi level block ILU preconditioner is based on a BILUTM preconditioner introduced in [23] which does not use the matrix values during the construction of the block independent sets. The added new features are shown by numerical experiments to be effective for solving the sparse matrices under our consideration. 2 Multi Level Preconditioning Suppose a large sparse unstructured linear ....

....linear system of n equations Ax = b (1) is to be solved by a preconditioned iterative method. A Krylov subspace method is used as the accelerator [18] and a robust preconditioner is to be constructed. To this end, we construct a multi level block ILU preconditioner similar to that introduced in [23]. The first step in the construction is to employ a heuristic graph algorithm to find a block independent set from the vertex set of the matrix (each row of the matrix A corresponds to a vertex or a node in the vertex set) There have been several block independent set al..gorithms proposed in [21] ....

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Saad, Y., and Zhang, J., BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices, SIAM J. Matrix Anal. Appl., to appear. Multi-Level Preconditioner for Coating Problems 13

....of smaller memory usage. In addition, these preconditioners are highly parallel and their inherent parallelism can be exploited on parallel computers [37] There are other multilevel preconditioning methods that are also derived from multilevel incomplete factorization of the coefficient matrices [2, 5, 12, 16, 18, 30, 39, 50] and that are based on different construction techniques [3, 4, 6, 7, 23, 24] Multilevel and multigrid techniques employ the idea that different error components can be treated efficiently on different level scales. This is the fundamental philosophy behind many multiscale computation techniques. ....

....of the problem sizes [1] In the previous discussion of the preconditioning solution procedure we omitted the permutations and inverse permutations that must be performed before and after the operations on each level. This approach has also been used in the computer programs of BILUM and BILUTM [38, 39]. We may permute the matrices on each level at the construction phase instead. Then only the global permutation is needed before and after the application of the preconditioner [34] The difference between permuting matrices (prepermutation) and permuting vectors (postpermutation) is a tradeoff ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl. to appear.

....to the increased couplings between their rows and columns. The lack of inherent parallelism in traditional ILU type preconditioners promotes the current strong interest in searching for alternative preconditioning techniques. There are parallelizable preconditioners based on multi level techniques [2, 1, 39, 40]. These methods exploit the concept of (block) independent set ordering and construct coarse level system via Schur complement approaches. They are highly efficient and may demonstrate grid independent convergence for certain type of problems [10, 39] In addition, parallelism can be extracted ....

....150 200 250 300 350 400 450 500 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 Number of iterations RAEFSKY4 Matrix dashdot line: tau = 0.1 dashed line: tau = 0.05 solid line: tau = 0.01 dotted line: tau = 0.005 Figure 5: Solving the RAEFSKY4 matrix with different values of . [12, 40, 44]. 5 Conclusions and Remarks We have proposed a sparse approximate inverse technique for computing parallelizable preconditioners of general sparse matrices. This method is extracted from a dense matrix inverse technique that is based on computing two sets of biconjugate vectors. Several ....

Y. Saad and J. Zhang. BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices. Technical Report UMSI 98/118, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1998.

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Y. Saad and J. Zhang. BILUTM: a domain based multi-level block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 1999. to appear.

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Y. Saad and J. Zhang. BILUTM: a domain based multi-level block ILUT preconditioner for general sparse matrices. SIAM J. Matrix Anal. Appl., 1999. to appear.

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