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59
Adaptive Smoothed Aggregation (αSA) Multigrid
, 2005
"... Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the part ..."
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Cited by 31 (7 self)
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Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) multigrid methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the nearkernel or nearnullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the nearkernel components is unavailable. This extension is accomplished in an adaptive process that uses the method itself to determine nearkernel components and adjusts the coarsening processes accordingly.
MULTILEVEL ADAPTIVE AGGREGATION FOR MARKOV CHAINS, WITH APPLICATION TO WEB RANKING
"... Abstract. A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smooth aggregation and adaptive algebraic multigrid methods for sparse linear systems, and is also close ..."
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Cited by 17 (8 self)
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Abstract. A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smooth aggregation and adaptive algebraic multigrid methods for sparse linear systems, and is also closely related to certain extensively studied iterative aggregation/disaggregation methods for Markov chains. In contrast to most existing approaches, our aggregation process does not employ any explicit advance knowledge of the topology of the Markov chain. Instead, adaptive agglomeration is proposed that is based on strength of connection in a scaled problem matrix, in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level. Strength of connection is determined as in the algebraic multigrid method, and the aggregation process is fully adaptive, with optimized aggregates chosen in each step of the iteration and at all recursive levels. The multilevel method is applied to a set of stochastic matrices that provide models for web page ranking. Numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods. The tests also provide more insight into why Google’s PageRank model is a successful model for determining a ranking of web pages.
SMOOTHED AGGREGATION MULTIGRID FOR MARKOV CHAINS
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2009
"... A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid meth ..."
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Cited by 16 (8 self)
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A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures wellposedness of the coarselevel problems: the coarselevel operators are singular Mmatrices on all levels, resulting in strictly positive coarselevel corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.
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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Dendro: Parallel algorithms for multigrid and AMR methods on 2:1 balanced octrees
"... Abstract—In this article, we present Dendro, a suite of parallel algorithms for the discretization and solution of partial differential equations involving secondorder elliptic operators. Dendro uses trilinear finite element discretizations constructed using octrees. Dendro, which is built on top o ..."
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Cited by 14 (7 self)
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Abstract—In this article, we present Dendro, a suite of parallel algorithms for the discretization and solution of partial differential equations involving secondorder elliptic operators. Dendro uses trilinear finite element discretizations constructed using octrees. Dendro, which is built on top of PETSc (Argonne National Laboratories), comprises of four main modules: a bottomup octree generation and 2:1 balancing module, a meshing module, a geometric multiplicative multigrid module, and a module for adaptive mesh refinement (AMR). The first two components constitute prior work that we have published elsewere. Here, we focus on the multigrid and AMR modules. The key features of Dendro are coarsening/refinement, interoctree transfers of scalar and vector fields, and parallel partition of multilevel octree forests. We describe an algorithm for constructing the coarser multigrid levels starting with an arbitrary 2:1 balanced fine grid octree discretization. Also, we describe matrixfree implementations for the discretized finite element operators and the intergrid transfer operations. The current implementation of Dendro is most appropriate for problems with smooth variable coefficients. We present scalability results for a Poisson problem, a linear elastostatics problem, and for a timedependent heat equation. We use the first two equations to illustrate the effectiveness of the multigrid solver. We use the third equation to illustrate the performance of the AMR components. We present results on up
An algebraic multigrid preconditioner for a class of singular Mmatrices
 SIAMJ. Sci. Comput
, 1982
"... Abstract. We apply algebraic multigrid (AMG) as a preconditioner for solving large singular linear systems of the type (I − T T)x = 0 with GMRES. Here, T is assumed to be the transition matrix of a Markov process. Although AMG and GMRES were originally designed for the solution of regular systems, w ..."
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Cited by 12 (0 self)
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Abstract. We apply algebraic multigrid (AMG) as a preconditioner for solving large singular linear systems of the type (I − T T)x = 0 with GMRES. Here, T is assumed to be the transition matrix of a Markov process. Although AMG and GMRES were originally designed for the solution of regular systems, with adequate adaptation their applicability can be extended to problems as described above.
Efficient multilevel eigensolvers with applications to data analysis tasks
 IEEE Trans. on Pattern Anal. and Machine Intell
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Regularizationrobust preconditioners for timedependent PDE constrained optimization problems
, 2011
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A greedy strategy for coarsegrid selection
, 2006
"... Efficient solution of the very large linear systems that arise in numerical modelling of realworld applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considere ..."
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Cited by 10 (4 self)
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Efficient solution of the very large linear systems that arise in numerical modelling of realworld applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered, much recent interest has been in the development of purely algebraic approaches that may be applied in many situations, without problemspecific tuning. Here, we consider an algebraic approach to finding the fine/coarse partitions needed in multilevel approaches. The algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms, multilevel block factorization and algebraic multigrid. While no guarantee on the rate of coarsening is given, the splitting is shown to always yield an effective preconditioner in the twolevel sense. Numerical performance of twolevel and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations, and the advantages of each of these two algorithmic settings are explored. 1
A parallel geometric multigrid method for finite elements on octree meshes
, 2008
"... Abstract. In this article, we present a parallel geometric multigrid algorithm for solving elliptic partial differential equations (PDEs) on octree based conforming finite element discretizations. We describe an algorithm for constructing the coarser multigrid levels starting with an arbitrary 2:1 b ..."
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Abstract. In this article, we present a parallel geometric multigrid algorithm for solving elliptic partial differential equations (PDEs) on octree based conforming finite element discretizations. We describe an algorithm for constructing the coarser multigrid levels starting with an arbitrary 2:1 balanced finegrid octree discretization. We also describe matrixfree implementations for the discretized finite element operators and the intergrid transfer operations. The key component of our scheme is an octree meshing algorithm, which handles “hanging ” vertices in a manner that naturally supports conforming trilinear shape functions. Our MPIbased implementation has scaled to billions of elements on thousands of processors on the Cray XT3 MPP system “Bigben ” at the Pittsburgh Supercomputing Center (PSC) and the Intel 64 Linux Cluster “Abe ” at the National Center for Supercomputing Applications (NCSA). Although we do not discuss adaptive mesh refinement here, the proposed method can be used efficiently in such problems since it has a low setup cost.