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Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method
We characterize the class $CG(s)$ of matrices A for which the linear system $A{\bf x} = {\bf b}$ can be solved by an s-term conjugate gradient method. We show that, except for a few anomalies, the…
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A taxonomy for conjugate gradient methods
TLDR
It is shown that any CG method for $Ax = b$ is characterized by an hpd inner product matrix B and a left preconditioning matrix C and how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well-known connection between CG and Lanczos.
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First-order system least squares for second-order partial differential equations: part I
TLDR
The least-squares approach developed here applies directly to convection--diffusion--reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-Squares methodology.
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Algebraic Multigrid Based on Element Interpolation (AMGe)
TLDR
This work develops a theoretical foundation for AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type finite element methods for partial differential equations and presents numerical results that demonstrate the efficacy of the method.
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The Tchebychev iteration for nonsymmetric linear systems
SummaryIn this paper an iterative method for solving nonsymmetric linear systems based on the Tchebychev polynomials in the complex plane is discussed. The iteration is shown to converge whenever the…
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An incomplete factorization technique for positive definite linear systems
TLDR
The technique combines an incomplete factorization method called the shifted incomplete Cholesky factorization with the method of generalized conjugate gradients and is shown to be more efficient on a set of test problems than either direct methods or explicit iteration schemes.
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A Technique for Accelerating the Convergence of Restarted GMRES
TLDR
A new technique is presented for accelerating the convergence of restarted GMRES by disrupting this alternating pattern of residual vectors, which resembles a full conjugate gradient method with polynomial preconditioning.
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Smoothed Aggregation Multigrid for Markov Chains
TLDR
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.
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First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity
Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a…
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Adaptive Smoothed Aggregation ( α SA ) Multigrid ∗
TLDR
An extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the near-kernel components is unavailable is introduced in an adaptive process that uses the method itself to determine near- kernel components and adjusts the coarsening processes accordingly.
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