Circulant preconditioners with unbounded inverses

AbstractThe eigenvalue and singular-value distributions for matrices S−1nAn and C−1nAn are examined, where An, Sn, and Cn are Toeplitz matrices, simple circulants, and optimal circulants generated by the Fourier expansion of some function f. Recently it has been proven that a cluster at 1 exists whenever f is from the Wiener class and strictly positive. Both restrictions are now weakened. A proof is given for the case when f may take the zero value, and hence the circulants are to have unbounded inverses. The main requirements on f are that it belong to L2 and be in some sense, sparsely vanishing. Specifically, if f is nonnegative and circulants Sn (or Cn) are positive definite, then the eigenvalues of S−1nAn (or C−1nAn) are clustered at 1. If f is complex-valued and Sn (or Cn) are nonsingular, then the singular values of S−1nAn (or C−1nAn) are clustered at 1 as well. Also proposed and studied are the improved circulants. It is shown that (improved) simple circulants can be much more advantageous than optimal circulants. This depends crucially on the smoothness properties of f. Further, clustering-on theorems are given that pertain to multilevel Toeplitz matrices preconditioned by multilevel simple and optimal circulants

Time stepping free numerical solution of linear differential equations: Krylov subspace versus waveform relaxation

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation method based on block Krylov subspaces. Second, we compare this new implementation against Krylov subspace methods combined with the shift and invert technique

Preconditioning for nonsymmetry and time-dependence

In this short paper, we decribe at least one simple and frequently arising situation |that of nonsymmetric real Toeplitz (constant diagonal) matrices| where we can guarantee rapid convergence of the appropriate iterative method by manipulating the problem into a symmetric form without recourse to the normal equations. This trick can be applied regardless of the nonnormality of the Toeplitz matrix. We also propose a symmetric and positive definite preconditioner for this situation which is proved to cluster eigenvalues and is by consequence guaranteed to ensure convergence in a number of iterations independent of the matrix dimension

Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation (WR) method based on block Krylov subspaces. Second, we compare this new WR-Krylov implementation against Krylov subspace methods combined with the shift and invert (SAI) technique. Some analysis and numerical experiments are presented. Since the WR-Krylov and SAI-Krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved, both methods can be seen as iterative across-time methods. The key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size