Krylov Subspace Methods on Supercomputers

Saad, Yousef

doi:10.1137/0910073

Cited by 205 publications

(89 citation statements)

References 96 publications

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“…All the matrix-vector products with the coefficient matrix A and with each of the (possibly factorized) approximate inverse preconditioners are vectorizable operations. To this end, after the approximate inverse preconditioners have been computed, they are transformed into the JAD, or jagged diagonal, format (see [56,68]). The same is done with the coefficient matrix A.…”

Section: Further Notes On Implementationsmentioning

confidence: 99%

A comparative study of sparse approximate inverse preconditioners

Benzi

Tůma

1999

Applied Numerical Mathematics

224

200

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A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered. A description of the preconditioners is given, and the results of an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications are presented. A comparison with more standard preconditioning techniques, such as incomplete factorizations, is also included. Robustness, convergence rates, and implementation issues are discussed.

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Section: Further Notes On Implementationsmentioning

confidence: 99%

A comparative study of sparse approximate inverse preconditioners

Benzi

Tůma

1999

Applied Numerical Mathematics

224

200

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“…In addition, the forward and back substitution steps, which must be conducted each time the preconditioner is applied, have been fully vectorized with a level-scheduling algorithm. 38 Vectorization is accomplished by keeping a list of all edges that contribute to the nodes in a given level and coloring those edges to allow vectorization. Numerical experiments with the level-scheduling algorithm indicate that the computer time required for the forward and backward substitutions is reduced by a factor of approximately 3.3 in two dimensions and by a factor of approximately 2.8 in three dimensions.…”

Section: Time-advancement Schemementioning

confidence: 99%

Implicit/multigrid algorithms for incompressible turbulent flows on unstructured grids

Anderson¹,

Rausch²,

Bonhaus³

1995

12th Computational Fluid Dynamics Conference

106

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SummaryAn implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the code is a backward Euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned Generalized Minimal Residual (GMRES) technique. For the GMRES calculations, several techniques are investigated for forming the matrix-vector product. Convergence acceleration is achieved through a multigrid scheme that uses non-nested coarse grids that are generated using a technique described in the present paper. Convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow over a NACA 4412 airfoil, a three-element airfoil for which Mach number effects are investigated, and three-dimensional flow over a wing with a partial-span flap.

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“…Thus, not only can the computational cost be dissipated over the lifetime of the learning algorithm, but some columns of the inverse might not be computed at all if the corresponding states are never visited. As a final note on computational matters, efficient techniques for storing the sparse matrix can be found in [19,36] and [3].…”

Section: Lstd-po Actor-critic For Pomdp With Indirectly Observed Costmentioning

confidence: 99%

A least squares temporal difference actor–critic algorithm with applications to warehouse management

Estanjini

Paschalidis

2012

Naval Research Logistics

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This paper develops a new approximate dynamic programming algorithm for Markov decision problems and applies it to a vehicle dispatching problem arising in warehouse management. The algorithm is of the actor-critic type and uses a least squares temporal difference learning method. It operates on a sample-path of the system and optimizes the policy within a prespecified class parameterized by a parsimonious set of parameters. The method is applicable to a partially observable Markov decision process setting where the measurements of state variables are potentially corrupted and the cost is only observed through the imperfect state observations. We show that under reasonable assumptions, the algorithm converges to a locally optimal parameter set. We also show that the imperfect cost observations do not affect the policy and the algorithm minimizes the true expected cost.In the warehouse application, the problem is to dispatch sensor-equipped forklifts in order to minimize operating costs involving product movement delays and forklift maintenance. We consider instances where standard dynamic programming is computationally intractable. Simulation results confirm the theoretical claims of the paper and show that our algorithm converges more smoothly than earlier actor-critic algorithms while substantially outperforming heuristics used in practice.

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Krylov Subspace Methods on Supercomputers

Cited by 205 publications

References 96 publications

A comparative study of sparse approximate inverse preconditioners

A comparative study of sparse approximate inverse preconditioners

Implicit/multigrid algorithms for incompressible turbulent flows on unstructured grids

A least squares temporal difference actor–critic algorithm with applications to warehouse management

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