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23
Algebraic multigrid methods based on compatible relaxation and energy minimization
 In Proceedings of the 16th International Conference on Domain Decomposition Methods. SpringerVerlag
, 2005
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Algebraic analysis of twogrid methods: the nonsymmetric case
, 2008
"... Twogrid methods constitute the building blocks of multigrid methods, which are among the most efficient solution techniques for solving large sparse systems of linear equations. In this paper, an analysis is developed that does not require any symmetry property. Several equivalent expressions are p ..."
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Cited by 12 (8 self)
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Twogrid methods constitute the building blocks of multigrid methods, which are among the most efficient solution techniques for solving large sparse systems of linear equations. In this paper, an analysis is developed that does not require any symmetry property. Several equivalent expressions are provided that characterize all eigenvalues of the iteration matrix. In the symmetric positive definite case, these expressions reproduce the sharp twogrid convergence estimate obtained by Falgout, Vassilevski and Zikatanov [Numer. Lin. Alg. Appl., 12 (2005), pp. 471–494], and also previous algebraic bounds which can be seen as corollaries of this estimate. These results allow to measure the convergence by checking “approximation properties”. In this work proper extentions of the latter to the nonsymmetric case are presented. Sometimes approximation properties for the symmetric positive definite case are summarized in loose terms; e.g.: Interpolation must be able to approximate an eigenvector with error bound proportional to the size of the eigenvalue [SIAM J. Sci. Comput., 22 (2000), pp. 1570–1592]. It is shown that this can be applied to nonsymmetric problems too, understanding “size” as “modulus”. Eventually, an analysis is developed, for the nonsymmetric case, of the theoretical foundations of “compatible relaxation”, according to which a Fine/Coarse partitioning may be checked and possibly improved.
Algebraic analysis of aggregationbased multigrid
, 2009
"... Convergence analysis of twogrids methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggreg ..."
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Cited by 10 (6 self)
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Convergence analysis of twogrids methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggregate a parameter which in some sense measures its quality. The procedure is purely algebraic and can be used to control a posteriori the quality of automatic coarsening algorithms. Assuming the aggregation pattern sufficiently regular, it is further shown that the resulting bound is asymptotically sharp for a large class of elliptic boundary value problems, including problems with variable and discontinuous coefficients. In particular, the analysis of typical examples shows that the convergence rate is insensitive to discontinuities under some reasonable assumptions on the aggregation scheme.
On algebraic multilevel methods for nonsymmetric systems  convergence results, Electronic Trans
 Numer. Anal
, 2008
"... Abstract. We analyze algebraic multilevel methods applied to nonsymmetric Mmatrices. Two types of multilevel approximate block factorizations are considered. The first one is related to the AMLI method. The second method is the multiplicative counterpart of the AMLI approach which we call the mult ..."
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Abstract. We analyze algebraic multilevel methods applied to nonsymmetric Mmatrices. Two types of multilevel approximate block factorizations are considered. The first one is related to the AMLI method. The second method is the multiplicative counterpart of the AMLI approach which we call the multiplicative algebraic multilevel (MAMLI) method. The MAMLI method is closely related to certain geometric and algebraic multigrid methods, such as the AMGr method. Although these multilevel methods work very well in practice for many problems, not much is known about theoretical convergence properties for nonsymmetric problems. Here, we establish convergence results and comparison results between AMLI and MAMLI multilevel methods applied to nonsymmetric Mmatrices.
Convergence analysis of geometrical multigrid methods for solving datasparse boundary element equations
 Computing and Visualization in Science, 2007. available online: DOI
"... The convergence analysis of multigrid methods for boundary element equations arising from negativeorder pseudodifferential operators is quite different from the usual finite element multigrid analysis for elliptic partial differential equations. In this paper, we study the convergence of geometric ..."
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The convergence analysis of multigrid methods for boundary element equations arising from negativeorder pseudodifferential operators is quite different from the usual finite element multigrid analysis for elliptic partial differential equations. In this paper, we study the convergence of geometric multigrid methods for solving largescale, datasparse boundary element equations arising from the adaptive cross approximation to the single layer potential equations. Keywords integral equations of first kind, single layer potential operator, boundary element method, adaptive cross approximation, geometric multigrid, preconditioners, iterative solvers. 1
CBS constants for graphLaplacians and application to multilevel methods for discontinuous Galerkin systems
 J. Complexity
, 2006
"... Abstract. The goal of this work is to derive and justify a multilevel preconditioner for symmetric discontinuous approximations of second order elliptic problems. Our approach is based on the following simple idea. The finite element space V of piecewise polynomials of certain degree that are disco ..."
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Abstract. The goal of this work is to derive and justify a multilevel preconditioner for symmetric discontinuous approximations of second order elliptic problems. Our approach is based on the following simple idea. The finite element space V of piecewise polynomials of certain degree that are discontinuous on the partition T0 is projected onto the space of piecewise constants on the same partition. This will constitute the finest space in the multilevel method. The projection of the discontinuous Galerkin system on this space is associated to the socalled “graphLaplacian”. In 2D this is a very simple Mmatrix with −1 as off diagonal entries and current diagonal entries equal to the number of the neighbours through the interfaces of the current finite element. Then after consecutive aggregation of the finite elements we produce a sequence of spaces of piecewise constant functions. We develop the concept of hierarchical splitting of the unknowns and using local analysis we derive uniform estimates for the constant in the strengthen CauchyBunyakowskiSchwarz (CBS) inequality. As a measure of the angle between the spaces of the splitting, this further is used to justify a multilevel preconditioner of the discontinuous Galerkin system in spirit of the work [4] of Axelsson and Vassilevski. key words: discontinuous Galerkin, second order elliptic equation, graphLaplacian, multilevel preconditioning, CBS constant (1.1) 1.
ALGEBRAIC MULTILEVEL PRECONDITIONERS FOR THE GRAPH LAPLACIAN BASED ON MATCHING IN GRAPHS
"... Abstract. This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vecto ..."
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Abstract. This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the twolevel convergence rate where the coarse level graph is obtained by matching. The twolevel convergence of the method is then used to establish the convergence of an Algebraic Multilevel Iteration that uses the twolevel scheme recursively. On structured grids, the method is proven to have convergence rate ≈ (1 − 1 / log n) and O(nlog n) complexity for each cycle, where n denotes the number of unknowns in the given problem. Numerical results of the algorithm applied to various graph Laplacians are reported. It is also shown that all the theoretical estimates derived for matching can be generalized to the case of aggregates containing more than two vertices. 1.
COLLOCATION COARSE APPROXIMATION (CCA) IN MULTIGRID
"... Abstract. The two common approaches to defining coarse operators in multigrid numerical algorithms are discretization coarse approximation (DCA) and (Petrov)Galerkin coarse approximation (GCA). Here, a new approach called collocation coarse approximation (CCA) is introduced, which—like GCA—is algeb ..."
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Abstract. The two common approaches to defining coarse operators in multigrid numerical algorithms are discretization coarse approximation (DCA) and (Petrov)Galerkin coarse approximation (GCA). Here, a new approach called collocation coarse approximation (CCA) is introduced, which—like GCA—is algebraically defined and able to cater to difficult features such as discontinuous coefficients, but, unlike GCA, allows explicit control over the coarsegrid sparsity pattern (stencil) and therefore control over the computational complexity of the solver. CCA relies on certain basis functions for which the coarse approximation to the finegrid problem is exact. Numerical experiments for twodimensional diffusion problems including jumping coefficients demonstrate the potential of the resulting multigrid algorithms.
Algebraic analysis of V–cycle multigrid
, 2007
"... We consider multigrid methods for symmetric positive definite linear systems. We develop an algebraic analysis of V–cycle schemes with Galerkin coarse grid matrices. This analysis is based on the Successive Subspace Correction convergence theory which we revisit. We reformulate it in a purely algebr ..."
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We consider multigrid methods for symmetric positive definite linear systems. We develop an algebraic analysis of V–cycle schemes with Galerkin coarse grid matrices. This analysis is based on the Successive Subspace Correction convergence theory which we revisit. We reformulate it in a purely algebraic way, and extend its scope of application to, e.g., algebraic multigrid methods. This reformulation also yields more accurate bounds. Considering a model problem, we show that our results can give a satisfactorily sharp prediction of actual multigrid convergence. Key words. multigrid, V–cycle, successive subspace correction, convergence analysis
Algebraic theory of twogrid methods
"... The algebraic theory of twogrid methods has been initiated by Achi Brandt in 1986 [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, it has been used in many works to analyze algebraic multigrid methods and guide their developments. The theory has also been improved and extended in a number ..."
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The algebraic theory of twogrid methods has been initiated by Achi Brandt in 1986 [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, it has been used in many works to analyze algebraic multigrid methods and guide their developments. The theory has also been improved and extended in a number of ways. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflationbased preconditioners and twolevel domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is selfcontained thanks to a collection of new proofs, often shorter than the original ones.