Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing, 20:2103--2121, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....reason that A j may be singular e.g. the diagonal of matrix A = T 0 in (9) may have zero entries. Some extensions based on the idea of [13] may be applied as discussed in Section 6. 10 24 Remark 2 We remark that for a class of general sparse linear systems, Saad, Zhang, Botta, Wubs et al. [28, 12, 32, 33] have proposed a recursive multi level preconditioner (named as ILUM) similar to this Algorithm 1. The rst di erence is that we need to apply one level of wavelets to achieve a nearly sparse matrix while these works start from a sparse matrix and permute it to obtain a desirable pattern suitable ....

Y. Saad and J. Zhang (1999), BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 20 (6), pp.2103-2121.

....structured, the interpolation (see later) is simple linear, and the approximate inverse is trivial. There are also classes of algebraic multi resolution methods, where the actual mesh and PDE are forgotten and only the matrix A is available. Examples of this include algebraic multigrid[31] BILUM[33], and repeated red black ILU[8] 1.9 Roadmap Chapter two begins with a brief review of classical wavelets. The main work is an exposition of a popular construction of second generation wavelets, going into the details of the transform algorithms and presenting new ideas about construction on ....

....and M # is upper triangular, this can now be interpreted as an incomplete LDU factorization: A # M M # The transformation to the multi resolution basis is now seen as an incomplete factorization preconditioner, using triangular solves with approximate factors. This is analogous to BILUM[33] or repeated red black ILU[8] where the triangular factors are found with a multi level algebraic algorithm rather than the interpolation approach here. Inspired from this analogy, an interesting extension to this thesis would be an algebraic version of the multi resolution approximate inverse ....

Y. Saad and J. Zhang, BILUM: block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems, to appear in SIAM J. Sci. Comput.

....equations (in some cases extended with a k turbulence model) to Rayleigh B enard ow, and to convection di usion problems in two and three dimensions. For a comparison of a variety of solvers including MRILU on Laplace like equations see [2] Similar methods like MRILU are described in [1, 8, 10]. Furthermore, there is also a link to algebraic multigrid, e.g. 7] As was recognized also by others, e.g. 8] multilevel ILU methods can be parallelized. The basic steps in the factorization phase form the search of an independent set, multiplication of two sparse matrices, transposition, and ....

Y. Saad and J. Zhang. BILUM: Block versions of multi-elimination and multilevel ILU preconditioner for general sparse linear systems. Technical Report UMSI 97-126, University of Minnesota, Minneapolis, 1997.

....grid correction) is applied. These algebraic multigrid methods can be used in situations where a grid (hierarchy) is not available. Also these methods can be used for developing black box solvers. Recently there have been developed ILU type of preconditioners with a multilevel structure, cf. [5,6,16,21,22]. The multilevel structure is induced by a level wise numbering of the unknowns. In [2,3,17,18] new hybrid methods have been presented, which use ideas both from ILU (incomplete Gaussian elimination) and from multigrid. In the present paper we reconsider the approximate cyclic reduction ....

Saad, Y., Zhang, J.: BILUM: block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. Report UMSI 97/126, Department of Computer Science, University of Minnesota (1997)

....structured, the interpolation (see later) is simple linear, and the approximate inverse is trivial. There are also classes of algebraic multi resolution methods, where the actual mesh and PDE are forgotten and only the matrix A is available. Examples of this include algebraic multigrid[31] BILUM[33], and repeated red black ILU[8] 10 1.9 Roadmap Chapter two begins with a brief review of classical wavelets. The main work is an exposition of a popular construction of second generation wavelets, going into the details of the transform algorithms and presenting new ideas about construction on ....

....# is upper triangular, this can now be interpreted as an incomplete LDU factorization: A # M T # e # 1 M # The transformation to the multi resolution basis is now seen as an incomplete factorization preconditioner, using triangular solves with approximate factors. This is analogous to BILUM[33] or repeated red black ILU[8] where the triangular factors are found with a multi level algebraic algorithm rather than the interpolation approach here. Inspired from this analogy, an interesting extension to this thesis would be an algebraic version of the multi resolution approximate inverse ....

Y. Saad and J. Zhang, BILUM: block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems, to appear in SIAM J. Sci. Comput.

.... preconditioner isn t guaranteed to solve the scalability problem; for example, standard ILU does not scale any better than no preconditioner, and the question of how best to use matrix orderings, dropping strategies, and numerical modifications to improve the scalability of ILU is still open [3, 4, 9, 14, 32, 33, 34]. 1 2 R. BRIDSON, W. P. TANG function G(x, y) on# # satisfying LG(x, y) #(x y) u(x) ## G(x, y)f(y) dy (assuming this exists, as one would expect for an elliptic problem well posed enough to permit numerical solution) The discrete solution is similarly found with the matrix ....

....problems, it is instructive to compare the new algorithm with some other multiresolution methods. As mentioned before, the basis transforms can be expressed as triangular matrices with unit block diagonals, so the algorithm could be viewed as a highly parallel variant of multilevel ILU (e.g. [4, 3, 9, 32, 33, 34]) with an approximate inverse replacing D 1 for the approximate LDU factorization. Another viewpoint comes from noting that the operators # P # I # and (P T # I) within the transforms for # and # correspond to node nested multigrid s prolongation and restriction respectively. The ....

Y. Saad and J. Zhang, BILUM: block versions of multi-elimination and multi-level ILU preconditioners for general sparse linear systems, SIAM J. Sci. Comput., 20, no. 6, pp. 2103--2121.

....even in areas where these are already widely used, such as PDE problems. We conclude by observing that preprocessings of the type considered in this paper could be used to improve the stability and effectiveness of multilevel ILU preconditioners, like Saad s ILUM [48] and its block variants [50], as well as similar methods based on sparse approximate inverses. This is an interesting possibility for further research. Acknowledgements. We would like to thank Iain Duff and Jacko Koster for providing us with the MC64 subroutines. This work was performed while the second author was a ....

Y. Saad and J. Zhang, BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 20 (1999), pp. 2103--2121.

....is applied. These algebraic multigrid methods can be used in situations where a grid (hierarchy) is not available. Also these methods can be used for developing black box solvers. Recently there have been developed ILU type of preconditioners with a multilevel structure, cf. 5] 6] 23] 28] [29]. The multilevel structure is induced by a level wise numbering of the unknowns. Recently, in [3] 4] 25] a few new hybrid methods have been presented, which use ideas both from ILU (incomplete Gaussian elimination) and from multigrid. In [3] a multigraph variant of the well known HBMG (cf. ....

Y. Saad and J. Zhang, BILUM: Block Versions of Multi-Elimination and Multi-Level ILU Preconditioner for General Sparse Linear Systems , report UMSI 97/126, Department of Computer Science, University of Minnesota, 1997.

....(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. BILUM: Block versions of multi-elimination and multilevel ILU preconditioner for general sparse linear systems. SIAM Journal on Scienti c Computing, 20:2103-2121, 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 [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 ....

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Y. Saad and J. Zhang. BILUM: Block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing, 20:2103--2121, 1999.

.... a graph has been an interesting topic in graph theory and has many potential applications in parallel computing and graph partitioning [2, 3, 4] One important application is to extract parallelism from a large sparse matrix during the construction of multilevel incomplete LU (ILU) preconditioners [7, 9]. Technical Report No. 365 03, Department of Computer Science, University of Kentucky, Lexington, KY, 2003. The research work of C. Shen was supported in part by the U.S. National Science Foundation under grant CCR 0092532. E mail: cshen cs.uky.edu. The research work of J. Zhang was ....

Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103--2121, 1999.

....to try to fill this gap. ILUM [16] and a few related methods [7, 5, 1] showed that 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 ....

....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 ....

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Y. Saad and J. Zhang. BILUM: Block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing, 20:2103--2121, 1999.

....way to reduce the iteration counts for small h is to increase the amount of fill ins. There is always a trade off between computational efficiency and storage space. Near gridindependent convergence rate may be achievable by employing multi level structure in constructing the ILU preconditioners [17]. Multigrid method. For completeness, we also solved the linear systems using a standard multigrid method [1] In particular, we used the red black Gauss Seidel as the smoother and performed one relaxation on each level before projection and after the interpolation, i.e. this is a V(1,1) cycling ....

.... are other iterative methods, especially the algebraic multigrid method and the multilevel ILU preconditioned Krylov subspace methods that can efficiently solve the convection diffusion equations with high Reynolds number and discretized by the upwind scheme and the fourth order compact scheme [17]. In these methods, only the fine grid linear system is from the discretization of a continuous problem, the coarse level matrices are generated algebraically by the Galerkin method or other techniques and thus no effective cell Reynolds number problem is presented on the coarse grids. 14 Table ....

Y. Saad and J. Zhang. BILUM: block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput. to appear.

....is the standard LU factorization of the D submatrix. We point out that A 1 is constructed by updating C row by row with drop tolerance applied even on updates. Hence, this method constructs the Schur complement indirectly, in contrast to some alternative methods, e.g. the BILUM preconditioner in [40], in which the Schur complement is constructed explicitly by matrix matrix multiplications. Sparsity and computation costs are kept low by adapting the dual dropping strategy of ILUT with respect to the computation of each row of A 1 . In particular, small size fill in with respect to is dropped ....

....for the matrix A. An alternative is based on the observation that A 1 is another sparse matrix and we can apply the same procedures to A 1 that have been applied to A to yield an even smaller Schur complement A 2 . This is the philosophy of multilevel ILU preconditioning techniques developed in [36, 40, 41]. However, for this moment, we only discuss the possible construction of a two level preconditioner. A two level preconditioner. The easiest way to construct a two level preconditioner is to apply the ILUT factorization technique to the matrix A 1 . One question will be naturally asked: is the ....

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Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103--2121, 1999.

....the level set strategy is inadequent. Furthermore, ILU(0) preconditioner may not be accurate enough and the subsequent preconditioned iterations may converge slowly or may not converge at all. Thus, higher 2 accuracy preconditioners have been advocated by a few authors for increased robustness [8, 21, 37, 45, 24, 30]. However, higher accuracy preconditioners usually means that more fill in entries are kept in the preconditioners and the couplings among the nodes are increased as well [24] The increased couplings reduce inherent parallelism and new ordering techniques must be employed to extract parallelism ....

.... 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 ....

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Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103-- 2121, 1999.

....robustness of ILU type preconditioning techniques with those of scalability of multi level preconditioners. Two techniques in this class are the Schur complement technique (Schur ILU) developed in [19] and the point and block multi elimination ILU preconditioners (ILUM, BILUM) discussed in [16, 21]. The framework of the Schur ILU preconditioner is that of a distributed sparse linear system, in which equations are assigned to different processors according to a mapping determined by a graph partitioner. The matrix of the related Schur complement system is also regarded as a distributed ....

....have also been developed in finite element analysis or for unstructured meshes [1, 5] The ILUM preconditioner has been extended to a block version (BILUM) in which the B block in (1) is block diagonal. This method utilizes independent sets of small clusters (or blocks) instead of single nodes [4, 21]. In some difficult cases, the performance of this block version is substantially superior to that of the scalar version. The major difference between our approach and the approaches of Botta and Wubs [4] and Reusken [13] is in the choice of variables for the reduced system. In [4] and [13] the ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUM: Block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. Technical Report UMSI 97/126, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1997.

....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. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103--2121, 1999.

.... used for the first step is described in [91] This has been extended to use domain decomposition reordering at every step [73] A block version of ILUM using blocks of size up to 2 by 2 by finding independent sets of edges has been reported to be more robust on convection diffusion problems [134]. An algorithm to pair a node with a neighbor that has minimum degree gave the best results. Algorithms that chose edges in the independent set based on their edge weight were not as effective. A block version of ILUM that can use large sparse blocks has also recently been developed [133] ....

Y. Saad and J. Zhang. BILUM: Block versions of multi-elimination and multi-level ILU preconditioners for general sparse linear systems. Technical Report UMSI 97/126, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1997.

....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. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103-- 2121, 1999.

....of the coefficients of the first derivatives is not too large. However, for robust solution of the sparse linear systems from various discretizations, a preconditioned Krylov subspace method can be used. The preconditioned Krylov subspace method, using a multilevel block ILU preconditioner [11], is generally more expensive than the multigrid method, when the latter works. But the former is more robust with respect to the 8 variation of coefficients and of the discretization schemes. Some interesting comparisons of the advantages and compromise of different discretization schemes and ....

Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103--2121, 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. BILUM: Block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. Technical Report UMSI 97/126, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1997. To appear.

....to nonuniform grids [9, 17] We used it with a multigrid method in [6] and obtained good results for some test problems with large Re. The three difference schemes, CDS, UPS, and FCS, are easy to implement. The resulting linear systems are frequently used as test examples for iterative methods [4, 14]. In a recent study, the relative advantages and disadvantages of these three schemes on uniform grids are compared, in terms of computational efficiency, solution accuracy, and the algebraic properties of the discretized linear systems [19] A preconditioned Krylov subspace method was used in ....

....level operator [12] One approach in this category is to construct a multilevel preconditioner to work with a Krylov subspace method. Algebraic multilevel preconditioning techniques aim at solving general sparse matrices. A multilevel block incomplete LU preconditioning (BILUM) was introduced in [14], and has been used to solve the convection diffusion equation discretized by UPS and FCS [14] Suppose the coefficient matrix of a linear system is A, BILUM first reorder the matrix into a two by two form A D F E C ; where D is a block diagonal matrix. This block partitioning may ....

[Article contains additional citation context not shown here]

Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 20(6):2103--2121, 1999.

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Y. Saad and J. Zhang. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing, 20:2103--2121, 1999.

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Y. Saad and J. Zhang. BILUM: block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. SIAM J. Sci. Cornput., 1999. to appear.

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Y. Saad and J. Zhang. BILUM: block versions of multi-elimination and multi-level ILU preconditioner for general sparse linear systems. SIAM J. Sci. Comput., 1999. to appear.

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