Abstract. For the iterative solution of large sparse linear systems we develop a theory for a family of augmented and deflated Krylov space solvers that are coordinate based in the sense that the given problem is transformed into one that is formulated in terms of the coordinates with respect

solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this two-part article we propose a new method for steady-state PDE-constrained optimization, based on the idea of full space SQP with reduced

generate a Krylov space for a singular operator that is associated with the projected problem. The deflated biconjugate gradient requires two such Krylov spaces, but it also allows us to solve two dual linear systems at once. Deflated Lanczos-type product methods fit in our new framework too. The question

We investigate the application of Krylov space methods to the solution of shifted linear systems of the form (A + σ)x − b = 0 for several values of σ simultaneously, using only as many matrix-vector operations as the solution of a single system requires. We find a suitable description

Abstract. We consider the linear system which arises from discretiza-tion of the pressure Poisson equation with Neumann boundary condi-tions, coming from bubbly flow problems. In literature, preconditioned Krylov iterative solvers are proposed, but these show slow convergence for relatively large

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 this paper I describe a new optimal Krylov subspace solver for shifted unitary matrices called the Shifted Unitary Orthogonal Method (SUOM). This algorithm is used as a benchmark against any improvement like the two-grid algorithm. I use the latter to show that the overlap operator can

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varies from one step to the next so that a exible Krylov solver is required. In this paper, we present a new numeri al te hnique for non-symmetri problems that ombines these two features. We illustrate the numeri al behavior of the new solver both on a set of small a ademi test examples as well

. It has been widely observed that Krylov space solvers based on two three-term recurrences can give signicantly less accurate residuals than mathematically equivalent solvers implemented with three two-term recurrences. In this paper we attempt to clarify and justify this dierence theoretically