A. Reusken. On the approximate cyclic reduction preconditioner. SIAM Journal on Scientific Computing, 21:565--590, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....each level [28] Hierarchical smoothers compute corrections only for parts of the unknowns. Recently there have been several attempts to improve on the robustness of the corresponding algorithm. One approach introduces generalized hierarchical bases which are better suited to the problem at hand [7, 17, 41]. Here, one bene ts from an intimate connection between hierarchical bases and an approximate Gaussian elimination [8, 24] which implicitly leads to problem adapted and physically meaningful coarse grid operators. However, these methods still su er from the bad stability properties (now w.r.t. ....

....located in the ne without coarse grid points. However, for the non perturbed Laplacian AMG approximately leads back to geometric coarsening and bilinear interpolations. Hence, we do not expect an AMG based hierarchical multiscale solver to be optimal. For the approximate cyclic reduction method [41], which is similar in spirit to AMG based HBMG, this drawback is also conceded in [43] Of course, a corresponding hierarchical W cycle would stabilize the method similarly to an AMLI approach [2, 3] but for complicated problems it can be much too expensive. This is due to the fact that the ....

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A. Reusken. On the approximate cyclic reduction preconditioner. Technical Report 144, Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, 1997.

....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 preconditioner of [17,18] This method is based on the recursive application of a two level method, as ....

....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 preconditioner of [17,18]. This method is based on the recursive application of a two level method, as in cyclic reduction or in a multigrid V cycle method. For the de nition of a two level structure we use two important concepts: a reduced graph and a maximal independent set. The partitioning of the set of unknowns, ....

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Reusken, A.: On the approximate cyclic reduction preconditioner. SIAM J. Sci. Comput., to appear

.... solver can be obtained by using matrix dependent prolongation and restriction operators (see [12, 19, 25, 27] An interesting topic for future research is the combination of this new hierarchy of local and inherently parallel smoothers with algebraic multigrid techniques (see for instance [18, 20, 21, 24]) In Section 2 we brie y review the SPAI Algorithm and show how sparse approximate inverses are used as smoothers in multigrid. In Section 3 we prove that for SPAI 0 the smoothing property ( 15] holds under reasonable assumptions on the matrix A. More precisely, for A symmetric and positive de ....

A. Reusken, On the approximate cyclic reduction preconditioner, SIAM J. Scientic Comp. 21 , 2000, pp. 565-590.

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A. Reusken. On the approximate cyclic reduction preconditioner. SIAM Journal on Scientific Computing, 21:565--590, 1999.

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