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**1 - 4**of**4**### Deflation and augmentation techniques in Krylov subspace methods for the solution of linear systems

, 2013

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### Deflation based preconditioning of linear systems of equations

"... For most real-world problems Krylov space solvers only converge in a reasonable number of iterations if a suitable preconditioning technique is applied. This is particularly true for problems where the linear operator has eigenvalues of small absolute value — a situation that is very common in pract ..."

Abstract
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For most real-world problems Krylov space solvers only converge in a reasonable number of iterations if a suitable preconditioning technique is applied. This is particularly true for problems where the linear operator has eigenvalues of small absolute value — a situation that is very common in practice. One suitable technique for dealing with such problems is to identify an approximately invariant subspace Z that belongs to the set of these small eigenvalues. By using an orthogonal projection along Z the Krylov solver can then be applied only to the orthogonal complement by restricting the operator accordingly. The basis constructed implicitly or explicitly by this restricted operator should then be augmented by a set of basis vectors for Z. There are various ways to handle and implement this approach. They differ not only algorithmically and numerically, but sometimes also mathematically. Some keywords associated with such methods are ‘(spectral) deflation’, ‘augmented basis’, ‘recycling Krylov subspaces’, and ‘singular preconditioning’. While we quickly also review the ‘symmetric case’, where the linear system

### GMRES CONVERGENCE FOR PERTURBED COEFFICIENT MATRICES, WITH APPLICATION TO APPROXIMATE DEFLATION PRECONDITIONING

, 2013

"... How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned sys ..."

Abstract
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How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this approach, we combine our analysis with Stewart’s invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors.

### POLYNOMIAL PRECONDITIONED GMRES AND GMRES-DR

"... We look at solving large nonsymmetric systems of linear equations using polynomial preconditioned Krylov methods. We give a simple way to find the polynomial. It is shown that polynomial preconditioning can significantly improve restarted GMRES for difficult problems, and the reasons for this are ex ..."

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We look at solving large nonsymmetric systems of linear equations using polynomial preconditioned Krylov methods. We give a simple way to find the polynomial. It is shown that polynomial preconditioning can significantly improve restarted GMRES for difficult problems, and the reasons for this are examined. Stability is discussed and algorithms are given for increased stability. Next we apply polynomial preconditioning to GMRES with deflated restarting. It is shown that this is worthwhile for sparse matrices and for problems with many small eigenvalues. Multiple right-hand sides are also considered.