244 research outputs found
Recycling BiCGSTAB with an Application to Parametric Model Order Reduction
Krylov subspace recycling is a process for accelerating the convergence of
sequences of linear systems. Based on this technique, the recycling BiCG
algorithm has been developed recently. Here, we now generalize and extend this
recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving
sequences of dual linear systems, while the focus here is on efficiently
solving sequences of single linear systems (assuming non-symmetric matrices for
both recycling BiCG and recycling BiCGSTAB).
As compared with other methods for solving sequences of single linear systems
with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based
recycling algorithms, like recycling BiCGSTAB, have the advantage that they
involve a short-term recurrence, and hence, do not suffer from storage issues
and are also cheaper with respect to the orthogonalizations.
We modify the BiCGSTAB algorithm to use a recycle space, which is built from
left and right approximate invariant subspaces. Using our algorithm for a
parametric model order reduction example gives good results. We show about 40%
savings in the number of matrix-vector products and about 35% savings in
runtime.Comment: 18 pages, 5 figures, Extended version of Max Planck Institute report
(MPIMD/13-21
Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems : Part I: Theory
We study block-diagonal preconditioners and an efficient variant of constraint preconditioners for general two-by-two block linear systems with zero (2,2)-block. We derive block-diagonal preconditioners from a splitting of the (1,1)-block of the matrix. From the resulting preconditioned system we derive a smaller, so-called related system that yields the solution of the original problem. Solving the related system corresponds to an efficient implementation of constraint preconditioning. We analyze the properties of both classes of preconditioned matrices, in particular their spectra. Using analytical results, we show that the related system matrix has the more favorable spectrum, which in many applications translates into faster convergence for Krylov subspace methods. We show that fast convergence depends mainly on the quality of the splitting, a topic for which a substantial body of theory exists. Our analysis also provides a number of new relations between block-diagonal preconditioners and constraint preconditioners. For constrained problems, solving the related system produces iterates that satisfy the constraints exactly, just as for systems with a constraint preconditioner. Finally, for the Lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a Newton iteration. Our theoretical results are confirmed by numerical experiments on a constrained optimization problem.
We consider the general, nonsymmetric, nonsingular case. Our only additional requirement is the nonsingularity of the Schur-complement--type matrix derived from the splitting that defines the preconditioners. In particular, the (1,2)-block need not equal the transposed (2,1)-block, and the (1,1)-block might be indefinite or even singular. This is the first paper in a two-part sequence. In the second paper we will study the use of our preconditioners in a variety of applications
Subspace Recycling for Sequences of Shifted Systems with Applications in Image Recovery
For many applications involving a sequence of linear systems with slowly
changing system matrices, subspace recycling, which exploits relationships
among systems and reuses search space information, can achieve huge gains in
iterations across the total number of linear system solves in the sequence.
However, for general (i.e., non-identity) shifted systems with the shift value
varying over a wide range, the properties of the linear systems vary widely as
well, which makes recycling less effective. If such a sequence of systems is
embedded in a nonlinear iteration, the problem is compounded, and special
approaches are needed to use recycling effectively.
In this paper, we develop new, more efficient, Krylov subspace recycling
approaches for large-scale image reconstruction and restoration techniques that
employ a nonlinear iteration to compute a suitable regularization matrix. For
each new regularization matrix, we need to solve regularized linear systems,
, for a sequence of regularization parameters,
, to find the optimally regularized solution that, in turn, will
be used to update the regularization matrix.
In this paper, we analyze system and solution characteristics to choose
appropriate techniques to solve each system rapidly. Specifically, we use an
inner-outer recycling approach with a larger, principal recycle space for each
nonlinear step and smaller recycle spaces for each shift. We propose an
efficient way to obtain good initial guesses from the principle recycle space
and smaller shift-specific recycle spaces that lead to fast convergence. Our
method is substantially reduces the total number of matrix-vector products that
would arise in a naive approach. Our approach is more generally applicable to
sequences of shifted systems where the matrices in the sum are positive
semi-definite
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