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26
GMRES on (Nearly) Singular Systems
 SIAM J. Matrix Anal. Appl
, 1994
"... . We consider the behavior of the gmres method for solving a linear system Ax = b when A is singular or nearly so, i.e., illconditioned. The (near) singularity of A may or may not affect the performance of gmres, depending on the nature of the system and the initial approximate solution. For singu ..."
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Cited by 39 (2 self)
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. We consider the behavior of the gmres method for solving a linear system Ax = b when A is singular or nearly so, i.e., illconditioned. The (near) singularity of A may or may not affect the performance of gmres, depending on the nature of the system and the initial approximate solution. For singular A, we give conditions under which the gmres iterates converge safely to a leastsquares solution or to the pseudoinverse solution. These results also apply to any residual minimizing Krylov subspace method that is mathematically equivalent to gmres. A practical procedure is outlined for efficiently and reliably detecting singularity or illconditioning when it becomes a threat to the performance of gmres. Key words. gmres method, residual minimizing methods, Krylov subspace methods, iterative linear algebra methods, singular or illconditioned linear systems AMS(MOS) subject classifications. 65F10 1. Introduction. The generalized minimal residual (gmres) method of Saad and Schultz [1...
Numerical Experiments with Iteration and Aggregation for Markov Chains
 ORSA Journal on Computing
, 1996
"... This paper describes an iterative aggregation/disaggregation method for computing the stationary probability vector of a nearly completely decomposable Markov chain. The emphasis is on the implementation of the algorithm and on the results that are obtained when it is applied to three modelling exam ..."
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Cited by 27 (8 self)
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This paper describes an iterative aggregation/disaggregation method for computing the stationary probability vector of a nearly completely decomposable Markov chain. The emphasis is on the implementation of the algorithm and on the results that are obtained when it is applied to three modelling examples that have been used in the analysis of computer/communication systems. Where applicable, a comparison with standard iterative and direct methods for solving the same problems, is made. Key words: Large Markov Chain Models; NearCompleteDecomposability; Iteration and Aggregation; Numerical Experiments. Research supported in part by NSF (DDM8906248) Introduction Let Q be the infinitesimal generator of an irreducible continuoustime Markov chain and let ß be its stationary probability vector. Thus q ij denotes the rate of transition from state i to state j; q ii = \Gamma P j 6=i q ij and ß i is the probability that the system is in state i at statistical equilibrium. It may be sho...
On The Use Of Two QMR Algorithms For Solving Singular Systems And Applications In Markov Chain Modeling
 in Markov chain modeling, Journal of Numerical Linear Algebra with Applications 2
, 1994
"... Recently, Freund and Nachtigal proposed the quasiminimal residual algorithm (QMR) for solving general nonsingular nonHermitian linear systems. The method is based on the Lanczos process, and thus it involves matrixvector products with both the coefficient matrix of the linear system and its tr ..."
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Cited by 21 (5 self)
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Recently, Freund and Nachtigal proposed the quasiminimal residual algorithm (QMR) for solving general nonsingular nonHermitian linear systems. The method is based on the Lanczos process, and thus it involves matrixvector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transposefree QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported. Keywords: Krylov subspace methods, nonHermitian matrices, singular systems, quasimini...
Structured Analysis Approaches for Large Markov Chains  A Tutorial
 Applied Numerical Mathematics
, 1996
"... The tutorial introduces structured analysis approaches for continuous time Markov chains (CTMCs) which are a means to extend the size of analyzable state spaces significantly compared with conventional techniques. It is shown how generator matrices of large CTMCs can be represented in a very compact ..."
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Cited by 19 (8 self)
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The tutorial introduces structured analysis approaches for continuous time Markov chains (CTMCs) which are a means to extend the size of analyzable state spaces significantly compared with conventional techniques. It is shown how generator matrices of large CTMCs can be represented in a very compact form, how this representation can be exploited in numerical solution techniques and how numerical analysis profits from this exploitation. Additionally, recent results covering implementation issues, tool support, and advanced analysis techniques are surveyed. 1 Introduction Analysis of continuous time Markov chains (CTMCs) is a well established approach to analyze the performance, dependability and performability of computer and communication systems. Systems are modeled using specification techniques like queueing networks (QNs), stochastic Petri nets (SPNs), formal specification techniques to mention only a few. Unfortunately, the size of CTMCs underlying most realistic examples can be ...
Numerical Methods for Computing Stationary Distributions of Finite Irreducible Markov Chains
 of Advances in Computational Probability
, 1997
"... Introduction In this chapter our attention will be devoted to computational methods for computing stationary distributions of finite irreducible Markov chains. We let q ij denote the rate at which an nstate Markov chain moves from state i to state j. The n \Theta n matrix Q whose offdiagonal ele ..."
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Cited by 16 (0 self)
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Introduction In this chapter our attention will be devoted to computational methods for computing stationary distributions of finite irreducible Markov chains. We let q ij denote the rate at which an nstate Markov chain moves from state i to state j. The n \Theta n matrix Q whose offdiagonal elements are q ij and whose i th diagonal element is given by \Gamma P n j=1;j 6=i q ij is called the infinitesimal generator of the Markov chain. It may be shown that the stationary probability vector ß, a row vector whose kth element denotes the stationary probability of being in state k, can be obtained by solving the homogeneous system of equations ßQ<F34
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 12 (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.
Transient Solutions of Markov Processes by Krylov Subspaces
 2ND INTERNATIONAL WORKSHOP ON THE NUMERICAL SOLUTION OF MARKOV CHAINS
, 1989
"... In this note we exploit the knowledge embodied in infinitesimal generators of Markov processes to compute efficiently and economically the transient solution of continuous time Markov processes. We consider the Krylov subspace approximation method which has been analysed by Y. Saad for solving linea ..."
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Cited by 9 (1 self)
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In this note we exploit the knowledge embodied in infinitesimal generators of Markov processes to compute efficiently and economically the transient solution of continuous time Markov processes. We consider the Krylov subspace approximation method which has been analysed by Y. Saad for solving linear differential equations. We place special emphasis on error bounds and stepsize control. We discuss the computation of the exponential of the Hessenberg matrix involved in the approximation and an economic evaluation of the Padé method is presented. We illustrate the usefulness of the approach by providing some application examples.
Experimental Study Of Parallel Iterative Solutions Of Markov Chains With Block Partitions
, 1999
"... Experiments are performed which demonstrate that parallel implementations of block stationary iterative methods can solve singular systems of linear equations in substantially less time that the sequential counterparts. Furthermore, these experiments illustrate the behavior of different partitions o ..."
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Cited by 8 (0 self)
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Experiments are performed which demonstrate that parallel implementations of block stationary iterative methods can solve singular systems of linear equations in substantially less time that the sequential counterparts. Furthermore, these experiments illustrate the behavior of different partitions of matrices representing Markov chains, when parallel iterative methods are used for their solution. Several versions of block iterative methods are tested.
Additive Schwarz iterations for Markov chains
 SIAM J. MATRIX ANAL. APPL
, 2005
"... A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular Mmatrices with onedimensional null space and is applicable in particular to systems representing ergodic Markov chains, a ..."
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Cited by 7 (3 self)
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A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular Mmatrices with onedimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial differential equations. Additive Schwarz can be seen as a generalization of block Jacobi, where the set of indices defining the diagonal blocks have nonempty intersection; this is called the overlap. The presence of overlap is known to accelerate the convergence of the methods in the nonsingular case. By providing convergence results, as well as some characteristics of the induced splitting, we hope to encourage the use of this additional computational tool for the solution of Markov chains and other singular systems. We present several numerical examples showing that additive Schwarz performs better than block Jacobi. For completeness, a few numerical experiments with block Gauss–Seidel and multiplicative Schwarz are also included.
Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model
, 2009
"... One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like ..."
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Cited by 7 (3 self)
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One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like