558 research outputs found
Convergence of restarted Krylov subspaces to invariant subspaces
The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of non-normality. Associated with a desired set of eigenvalues is a maximum ``reachable invariant subspace'' that can be developed from the given starting vector. Convergence for this distinguished subspace is bounded in terms involving a polynomial approximation problem. Elementary results from potential theory lead to convergence rate estimates and suggest restarting strategies based on optimal approximation points (e.g., Leja or Chebyshev points); exact shifts are evaluated within this framework. Computational examples illustrate the utility of these results. Origins of superlinear effects are also described
Deflated GMRES for Systems with Multiple Shifts and Multiple Right-Hand Sides
We consider solution of multiply shifted systems of nonsymmetric linear
equations, possibly also with multiple right-hand sides. First, for a single
right-hand side, the matrix is shifted by several multiples of the identity.
Such problems arise in a number of applications, including lattice quantum
chromodynamics where the matrices are complex and non-Hermitian. Some Krylov
iterative methods such as GMRES and BiCGStab have been used to solve multiply
shifted systems for about the cost of solving just one system. Restarted GMRES
can be improved by deflating eigenvalues for matrices that have a few small
eigenvalues. We show that a particular deflated method, GMRES-DR, can be
applied to multiply shifted systems. In quantum chromodynamics, it is common to
have multiple right-hand sides with multiple shifts for each right-hand side.
We develop a method that efficiently solves the multiple right-hand sides by
using a deflated version of GMRES and yet keeps costs for all of the multiply
shifted systems close to those for one shift. An example is given showing this
can be extremely effective with a quantum chromodynamics matrix.Comment: 19 pages, 9 figure
A framework for deflated and augmented Krylov subspace methods
We consider deflation and augmentation techniques for accelerating the
convergence of Krylov subspace methods for the solution of nonsingular linear
algebraic systems. Despite some formal similarity, the two techniques are
conceptually different from preconditioning. Deflation (in the sense the term
is used here) "removes" certain parts from the operator making it singular,
while augmentation adds a subspace to the Krylov subspace (often the one that
is generated by the singular operator); in contrast, preconditioning changes
the spectrum of the operator without making it singular. Deflation and
augmentation have been used in a variety of methods and settings. Typically,
deflation is combined with augmentation to compensate for the singularity of
the operator, but both techniques can be applied separately.
We introduce a framework of Krylov subspace methods that satisfy a Galerkin
condition. It includes the families of orthogonal residual (OR) and minimal
residual (MR) methods. We show that in this framework augmentation can be
achieved either explicitly or, equivalently, implicitly by projecting the
residuals appropriately and correcting the approximate solutions in a final
step. We study conditions for a breakdown of the deflated methods, and we show
several possibilities to avoid such breakdowns for the deflated MINRES method.
Numerical experiments illustrate properties of different variants of deflated
MINRES analyzed in this paper.Comment: 24 pages, 3 figure
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