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- AMLI method (2)
- algebraic multigrid methods (2)
- Algebraic multi-level methods (1)
- Algebraic multilevel methods (1)
- CG (1)
- GMRES (1)
- Krylov subspace methods (1)
- MINRES (1)
- Perron-Frobenius theorem (1)
- augmentation (1)
We present a new extension of the well-known
Perron-Frobenius theorem to regular matrix pairs $(E,A)$.
The new extension is based on projector chains and is motivated from
the solution of positive differential-algebraic systems or descriptor
systems. We present several examples where the new condition holds, whereas conditions
in previous literature are not satisfied.
We establish theoretical comparison results for algebraic multi-level
methods applied
to nonsingular non-symmetric M-matrices.
We consider two types of multi-level approximate block factorizations or AMG methods, the
AMLI and the MAMLI method.
We compare
the spectral radii
of the iteration matrices of these methods. This comparison
shows, that the spectral radius of the MAMLI method is less than or equal to
the spectral radius of the AMLI method.
Moreover, we establish how the quality of the approximations
in the block factorization effects the spectral radii of the
iteration
matrices. We prove comparisons results
for different approximation of the fine grid block as well as for the
used Schur
complement. We also establish a theoretical comparison between the
AMG methods and the classical block Jacobi and block Gauss-Seidel methods.
Here we analyze algebraic multilevel methods applied to non-symmetric M-matrices.
We consider two types of multilevel approximate block factorizations.
The first one is related to the AMLI method.
The second method
is the multiplicative counterpart of the AMLI approach which we call
multiplicative algebraic multilevel method, the MAMLI method. The MAMLI method is closely related to certain geometric
and algebraic multigrid methods like the AMGr method. Although these
multilevel methods work very well
in practice for many problems, there is not that much known about
theoretical convergence
properties for non-symmetric problems.
Here, we establish
convergence results and comparison results between AMLI and MAMLI
multilevel methods
applied to non-symmetric M-matrices.
Deflated and augmented Krylov subspace methods: Basic Facts and a Breakdown-free deflated MINRES
(2011)
In this paper we consider deflation and augmentation techniques for accelerating
the convergence of Krylov subspace methods for the solution of nonsingular linear
algebraic systems. The two techniques are conceptually different from
preconditioning. Deflation "removes" certain parts from the operator, while
augmentation adds a subspace to the Krylov subspace. Both approaches have been
used in a variety of methods and settings. For Krylov subspace methods that
satisfy a (Petrov-) Galerkin condition we show that augmentation can in general
be achieved implicitly by projecting the residuals appropriately and correcting
the approximate solutions in a final step. In this context, we analyze known
methods to deflate CG, GMRes and MinRes. Our analysis reveals that the recently
proposed RMinRes method can break down. We show how such breakdowns can be
avoided by choosing a special initial guess, and we derive a breakdown-free
deflated MinRes method. In numerical experiments we study the properties of
different variants of MinRes analyzed in this paper.