1,094 research outputs found
A Combined Preconditioning Strategy for Nonsymmetric Systems
We present and analyze a class of nonsymmetric preconditioners within a
normal (weighted least-squares) matrix form for use in GMRES to solve
nonsymmetric matrix problems that typically arise in finite element
discretizations. An example of the additive Schwarz method applied to
nonsymmetric but definite matrices is presented for which the abstract
assumptions are verified. A variable preconditioner, combining the original
nonsymmetric one and a weighted least-squares version of it, is shown to be
convergent and provides a viable strategy for using nonsymmetric
preconditioners in practice. Numerical results are included to assess the
theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure
On large-scale diagonalization techniques for the Anderson model of localization
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model
of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the JacobiāDavidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete
LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization
by several orders of magnitude
A biconjugate gradient type algorithm on massively parallel architectures
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine
QMR: A Quasi-Minimal Residual method for non-Hermitian linear systems
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. A novel BCG like approach is presented called the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported
Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections
This work focuses on the iterative solution of sequences of KKT linear
systems arising in interior point methods applied to large convex quadratic
programming problems. This task is the computational core of the interior point
procedure and an efficient preconditioning strategy is crucial for the
efficiency of the overall method. Constraint preconditioners are very effective
in this context; nevertheless, their computation may be very expensive for
large-scale problems, and resorting to approximations of them may be
convenient. Here we propose a procedure for building inexact constraint
preconditioners by updating a "seed" constraint preconditioner computed for a
KKT matrix at a previous interior point iteration. These updates are obtained
through low-rank corrections of the Schur complement of the (1,1) block of the
seed preconditioner. The updated preconditioners are analyzed both
theoretically and computationally. The results obtained show that our updating
procedure, coupled with an adaptive strategy for determining whether to
reinitialize or update the preconditioner, can enhance the performance of
interior point methods on large problems.Comment: 22 page
Absolute value preconditioning for symmetric indefinite linear systems
We introduce a novel strategy for constructing symmetric positive definite
(SPD) preconditioners for linear systems with symmetric indefinite matrices.
The strategy, called absolute value preconditioning, is motivated by the
observation that the preconditioned minimal residual method with the inverse of
the absolute value of the matrix as a preconditioner converges to the exact
solution of the system in at most two steps. Neither the exact absolute value
of the matrix nor its exact inverse are computationally feasible to construct
in general. However, we provide a practical example of an SPD preconditioner
that is based on the suggested approach. In this example we consider a model
problem with a shifted discrete negative Laplacian, and suggest a geometric
multigrid (MG) preconditioner, where the inverse of the matrix absolute value
appears only on the coarse grid, while operations on finer grids are based on
the Laplacian. Our numerical tests demonstrate practical effectiveness of the
new MG preconditioner, which leads to a robust iterative scheme with minimalist
memory requirements
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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