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34
Multigrid techniques for unstructured meshes
 in VKI Lecture Series VKILS
, 1995
"... An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practica ..."
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Cited by 24 (3 self)
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An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practical applications. Advanced multigrid topics, such as the use of algebraic multigrid methods, and the combination of multigrid techniques with adaptive meshing strategies are dealt with in subsequent sections. These represent current areas of research, and the unresolved issues are discussed. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to
Stochastic Process Algebras as a Tool for Performance and Dependability Modelling
 in IEEE International Computer Performance and Dependability Symposium
, 1995
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State Space Construction and SteadyState Solution of GSPNs on a SharedMemory Multiprocessor
, 1997
"... A common approach for the quantitative analysis of a generalized stochastic Petri net (GSPN) is to generate its entire state space and then solve the corresponding continuoustime Markov chain (CTMC) numerically. This analysis often suffers from two major problems: the state space explosion and the ..."
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Cited by 23 (4 self)
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A common approach for the quantitative analysis of a generalized stochastic Petri net (GSPN) is to generate its entire state space and then solve the corresponding continuoustime Markov chain (CTMC) numerically. This analysis often suffers from two major problems: the state space explosion and the stiffness of the CTMC. In this paper we present parallel algorithms for sharedmemory machines that attempt to alleviate both of these difficulties: the large main memory capacity of a multiprocessor can be utilized and long computation times are reduced by efficient parallelization. The algorithms comprise both CTMC construction and numerical steadystate solution. We give experimental results obtained with a Convex SPP16 sharedmemory multiprocessor that show the behavior of the algorithms and the parallel speedups obtained.
PANDA  Petri Net Analysis and Design Assistant
, 1997
"... Introduction Modeling is an important technique for developing and evaluating IT systems today. If answers for meaningful questions are required, constructing useful models of hard and software architectures results in complex structures combining logical dependencies and probabilistic attributes. ..."
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Cited by 20 (2 self)
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Introduction Modeling is an important technique for developing and evaluating IT systems today. If answers for meaningful questions are required, constructing useful models of hard and software architectures results in complex structures combining logical dependencies and probabilistic attributes. A wellestablished mathematical theory for the design and analysis of such models using stochastic processes exists, but for its application expert knowledge of the special mathematical background is necessary. Therefore, several modeling methodologies have been developed to facilitate the application of stochastic theories. Stochastic Petri nets have been shown to be a successful approach for modeling systems whose behavior is probabilistic in some sense and whose investigation involves timing aspects. The advantages of stochastic Petri nets are a solid formal theory including algorithms for numerical analysis, combined with a notation that permits easy visualizatio
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.
On the Utility of the MultiLevel Algorithm for the Solution of Nearly Completely Decomposable Markov Chains
, 1994
"... Recently the MultiLevel algorithm was introduced as a general purpose solver for the solution of steady state Markov chains. In this paper we consider the performance of the MultiLevel algorithm for solving Nearly Completely Decomposable (NCD) Markov chains, for which specialpurpose iterative agg ..."
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Cited by 13 (1 self)
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Recently the MultiLevel algorithm was introduced as a general purpose solver for the solution of steady state Markov chains. In this paper we consider the performance of the MultiLevel algorithm for solving Nearly Completely Decomposable (NCD) Markov chains, for which specialpurpose iterative aggregation/disaggregation algorithms such as the KouryMcAllisterStewart (KMS) method have been developed that can exploit the decomposability of the the Markov chain. We present experimental results indicating that the generalpurpose MultiLevel algorithm is competitive, and can be significantly faster than the specialpurpose KMS algorithm when GaussSeidel and Gaussian Elimination are used for solving the individual blocks.
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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ALGEBRAIC MULTIGRID FOR MARKOV CHAINS
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2009
"... An algebraic multigrid (AMG) method is presented for the calculation of the stationary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpolation formula is proposed t ..."
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Cited by 9 (6 self)
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An algebraic multigrid (AMG) method is presented for the calculation of the stationary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums. It is shown how the adoption of a previously described lumping technique maintains the irreducible singular Mmatrix character of the coarselevel operators on all levels. Together, these properties are sufficient to guarantee the wellposedness of the algorithm. Numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems.