Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report UMSI 99/107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....strategy inefficient. Another possible implementation strategy for RILUM is to use a domain based multilevel block ILU factorization (BILUTM) 30] in which the individual blocks are factored by an ILUT strategy. This implementation strategy has been preliminarily reported by Saad and Suchomel [27]. However, as we mentioned in [40] the RILUM implementation based on BILUTM strategy may not yield a grid independent convergence rate, since the first level factorization of BILUTM is not exact. It may also be impossible to implement a preconditioning Schur complement strategy as discussed in ....

Y. SAAD AND B. SUCHOMEL, ARMS: an algebraic recursive multilevel solver for general sparse linear systems, in Abstracts of the Ninth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, 1999, p. 37.

....strategy inefficient. Another possible implementation strategy for RILUM is to use a domain based multilevel block ILU factorization (BILUTM) 29] in which the individual blocks are factored by an ILUT strategy. This implementation strategy has been preliminarily reported by Saad and Suchomel [27]. However, as we mentioned in [40] the RILUM implementation based on BILUTM strategy may not yield a grid independent convergence rate, since the first level factorization of BILUTM is not exact. It may also be impossible to implement a preconditioning Schur complement strategy as discussed in ....

Y. Saad and B. Suchomel, ARMS: an algebraic recursive multilevel solver for general sparse linear systems, in Abstracts of the Ninth Copper Mountain on Multigrid Methods, Copper Mountain, CO, 1999, p. 37.

....system can be solved by a GMRES like accelerator, requiring a solve with S i at each step. 3 Parallel Algebraic Recursive Multilevel Solver Multi level Schur complement techniques available in pARMS [14] are based on techniques which exploit block independent sets, such as those described in [22]. The idea is to create another level of partitioning of each sub domain. An illustration is shown in Figure 3, which distinguishes one more type of interface variables: local interface variables, where we refer to local interface points as interface points between the sub sub domains. Their ....

....points between the sub sub domains. Their couplings are all local to the processor and so these points do not require interprocessor communication. These sub sub domains are not obtained by a standard partitioner but rather by the block independent set reordering strategy utilized by ARMS [22]. Interior points interface Interdomain points points Local interface Fig. 3. A two level partitioning of a domain 3.1 ARMS and pARMS In order to explain the multilevel techniques used in pARMS, it is necessary to discuss the sequential multilevel ARMS technique. In the sequential ARMS, ....

Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107-REVIS, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 2001. Revised version of umsi-99-107.

....When the last level is reached solve with GMRES ILUT. WARMS ) If the current level is not the last, use a few steps of FGMRES to solve the reduced system using WARMS (recursively) as a preconditioner. At the last level ILUT GMRES is again used. WARMS was discussed in the technical report [21]. However, since this algorithm turns out to be fairly expensive for most practical situations we only mention it here for reference. The VARMS preconditioning step is shown in the following algorithm. 6 Algorithm 3.2 VARMS solve(A l ; b l ) Recursive Multi Level Solution 1. Solve L l f l ....

....6w. Solve A l 1 z l = h l using GMRES preconditioned by VARMS(A l 1 ; 3.2 ARMS 2 Inner outer cycles, such as WARMS, require matrix vector operations with the iteration matrix A l at level l. One possible way to do this is simply to store the matrix A l at each level as was discussed in [21]. The permuted form of the Schur complement at level l is the matrix of level l 1, i.e. P l 1 S l P B l 1 F l 1 E l 1 C l 1 : 7) Since the blocks B l 1 ; E l 1 ; F l 1 ; C l 1 are available from the next level, the product S l Theta w may be obtained with two permutations and ....

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Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 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 ....

....the lower level can be any approximate or exact solution technique for solving the system at the next level, i.e. the system A l 1 Theta z l = h 0 l (4) from the forward (restriction) operation. The variations that arise are related to the ways in which this coarser level system is solved. In [21] we have implemented several different 3 Forward Backward Back to original system Original system Reduced system: solve by any means Figure 2: Illustration of a preconditioning operation. options and tested them. The simplest option available is referred to as VARMS [in analogy with the ....

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Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1999.

....to accomplish this are inexpensive when compared for example to the cost of BILU(0) the least expensive factorization. There are other possible applications of the blocking techniques presented here. In preconditioning methods, they can be combined with Algebraic Recursive Multilevel Solvers [12]. In a multilevel ILU context, the successive approximate Schur complements that are generated can become quite dense, and blocking can have a good performance pay off. In fact, this is precisely the strategy utilized in sparse direct solution software to obtain flops rates that are far superior ....

Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1999. 15

....four injected eigenvectors in the case of deflated GMRES(k) We took a random initial guess with the right hand side constructed such that the solution is the vector of all ones. Rational acceleration is applied to the factorization produced by the Algebraic Recursive Multilevel Solver (ARMS) [14]. This choice of the preconditioner is motivated bytheversatility of ARMS and its abilitytosolve efficiently the structural mechanics problems. ARMS is an algebraic multigrid like algorithm that requires no underlying set of grids for defining prolongation and restriction operators. ARMS works by ....

Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1999.

....of problems in tire design is reported in [10] It was found that a variation of the incomplete LU factorization with pivoting serves well as a preconditioner to achieve acceptable iterative convergence for the model M. This paper presents results from applying a multilevel preconditioning scheme [8] and considers larger (model L) problems. This work is supported by Michelin Americas Research and Development Corporation and in part by the Minnesota Supercomputer Institute. y Department of Computer Science, University of Minnesota, Duluth, 320 Heller Hall, 10 University Drive, Duluth, ....

....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 GMRES(k) ....

Y. Saad and B. Suchomel, ARMS: An algebraic recursive multilevel solver for general sparse linear systems, Tech. Rep. umsi-99-107, University of Minnesota Supercomputer Institute, Minneapolis, MN 55415, 1999.

....four injected eigenvectors in the case of deflated GMRES(k) We took a random initial guess with the right hand side constructed such that the solution is the vector of all ones. Rational acceleration is applied to the factorization produced by the Algebraic Recursive Multilevel Solver (ARMS) [14]. This choice of the preconditioner is motivated by the versatility of ARMS and its ability to solve efficiently the structural mechanics problems. ARMS is an algebraic multigrid like algorithm that requires no underlying set of grids for defining prolongation and restriction operators. ARMS works ....

Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1999.

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Y. Saad and B. Suchomel. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report UMSI 99/107, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, 1999.

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