3 research outputs found
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
On fixed-point, Krylov, and block preconditioners for nonsymmetric problems
The solution of matrices with block structure arises in numerous
areas of computational mathematics, such as PDE discretizations based on
mixed-finite element methods, constrained optimization problems, or the
implicit or steady state treatment of any system of PDEs with multiple
dependent variables. Often, these systems are solved iteratively using Krylov
methods and some form of block preconditioner. Under the assumption that one
diagonal block is inverted exactly, this paper proves a direct equivalence
between convergence of block preconditioned Krylov or fixed-point
iterations to a given tolerance, with convergence of the underlying
preconditioned Schur-complement problem. In particular, results indicate that
an effective Schur-complement preconditioner is a necessary and sufficient
condition for rapid convergence of block-preconditioned GMRES, for
arbitrary relative-residual stopping tolerances. A number of corollaries and
related results give new insight into block preconditioning, such as the fact
that approximate block-LDU or symmetric block-triangular preconditioners offer
minimal reduction in iteration over block-triangular preconditioners, despite
the additional computational cost. Theoretical results are verified numerically
on a nonsymmetric steady linearized Navier-Stokes discretization, which also
demonstrate that theory based on the assumption of an exact inverse of one
diagonal block extends well to the more practical setting of inexact inverses.Comment: Accepted to SIMA