Compress‐and‐restart block Krylov subspace methods for Sylvester matrix equations

@article{Kressner2020CompressandrestartBK,
  title={Compress‐and‐restart block Krylov subspace methods for Sylvester matrix equations},
  author={Daniel Kressner and Kathryn Lund and Stefano Massei and Davide Palitta},
  journal={Numerical Linear Algebra with Applications},
  year={2020},
  volume={28}
}
Block Krylov subspace methods (KSMs) comprise building blocks in many state‐of‐the‐art solvers for large‐scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise… 
View on Wiley
arxiv.org
2 Citations
Methods Citations
1

2 Citations

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

This work illustrates how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm and shows how to fully merge the two iterative procedures to obtain a novel, efficient implementation of the low-rank ADI method.

Limited‐memory polynomial methods for large‐scale matrix functions

Various limited‐memory methods for the approximation of the action of a large‐scale matrix function on a vector are reviewed, with emphasis on polynomial methods, whose memory requirements are known or prescribed a priori.

References

SHOWING 1-10 OF 65 REFERENCES

Fast Singular Value Decay for Lyapunov Solutions with Nonnormal Coefficients

It is shown that a larger departure from normality can actually correspond to faster decay of singular values: if the singular values decay slowly, the numerical range cannot extend far into the right-half plane.

On the ADI method for Sylvester equations

Krylov-subspace methods for the Sylvester equation

Minimal residual methods for large scale Lyapunov equations

Boundary Element Methods

In Chap. 3 we transformed strongly elliptic boundary value problems of second order in domains \( \Omega \subset \mathbb{R}^3\) into boundary integral equations. These integral equations were…

Accuracy and stability of numerical algorithms, Second Edition

This second edition gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic and combines algorithmic derivations, perturbation theory, and rounding error analysis.

Restarted GMRES for Shifted Linear Systems

This work develops a variant of the restarted GMRES method exhibiting the same advantage and investigates its convergence for positive real matrices in some detail and applies it to speed up "multiple masses" calculations arising in lattice gauge computations in quantum chromodynamics, one of the most time-consuming supercomputer applications.

A survey of direct methods for sparse linear systems

The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems.

GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

An Error Analysis for Rational Galerkin Projection Applied to the Sylvester Equation

  • B. Beckermann
  • Mathematics, Computer Science
    SIAM J. Numer. Anal.
  • 2011
A new formula for the residual of Galerkin projection onto rational Krylov spaces applied to a Sylvester equation is suggested, and a relation to three different underlying extremal problems for rational functions is established.
...