The conformal ‘bratwurst’ mapsand associated Faber polynomials

Koch, Tino; Liesen, Jörg

doi:10.1007/pl00005401

Cited by 18 publications

(29 citation statements)

References 32 publications

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“…Hence the bound (28) Fig. 3 Tight inclusion of the eigenvalues of the GRCAR matrix by two elements of the class of sets introduced in [45,49]. studied below in the context of nonnormal matrices can in principle be used in the normal case as well.…”

Section: Normal Matrices -Minres and Gmresmentioning

confidence: 98%

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Convergence analysis of Krylov subspace methods

Liesen

Tichý

2004

GAMM-Mitteilungen

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Key words Krylov subspace methods, convergence analysis. MSC (2000) 15A06, 65F10, 41A10One of the most powerful tools for solving large and sparse systems of linear algebraic equations is a class of iterative methods called Krylov subspace methods. Their significant advantages like low memory requirements and good approximation properties make them very popular, and they are widely used in applications throughout science and engineering. The use of the Krylov subspaces in iterative methods for linear systems is even counted among the "Top 10" algorithmic ideas of the 20th century. Convergence analysis of these methods is not only of a great theoretical importance but it can also help to answer practically relevant questions about improving the performance of these methods. As we show, the question about the convergence behavior leads to complicated nonlinear problems. Despite intense research efforts, these problems are not well understood in some cases. The goal of this survey is to summarize known convergence results for three well-known Krylov subspace methods (CG, MINRES and GMRES) and to formulate open questions in this area.

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Section: Normal Matrices -Minres and Gmresmentioning

confidence: 98%

“…To overcome such difficulties, a parameterized class of non-convex sets with analytic boundaries is constructed in [49] (also see [45]), for which the Faber polynomials are explicitly known. These polynomials give rise to analytic and easily computable bounds for the min-max approximation problem; see [49] for details.…”

Section: Normal Matrices -Minres and Gmresmentioning

confidence: 99%

Convergence analysis of Krylov subspace methods

Liesen

Tichý

2004

GAMM-Mitteilungen

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“…For Krylov methods, δ * should typically be chosen much smaller [20]; however, an analytical treatment in that case is highly nontrivial, see e.g. [16].…”

Section: Optimal Equivalent Operator Preconditioningmentioning

confidence: 99%

An Analysis of Equivalent Operator Preconditioning for Equation-Free Newton–Krylov Methods

Samaey¹,

Vanroose²

2010

SIAM J. Numer. Anal.

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We consider the computation of a fixed point of a time-stepper using NewtonKrylov methods, and propose and analyze equivalent operator preconditioning for the resulting linear systems. For a linear, scalar advection-reaction-diffusion equation, we investigate in detail how the convergence rate depends on the choice of preconditioner parameters and on the time discretization. The results are especially valuable when computing fixed points of a coarse time-stepper in the equation-free multiscale framework, in which one simulates an unavailable coarse-scale model by wrapping a set of computational routines around appropriately initialized fine-scale simulations. Both analytical results and numerical experiments are presented, showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner.

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“…[14], while curves of type 2 are regarded in [8]. The marked points are hyperbolic Menke points of order 8.…”

Section: Distribution Of Hyperbolic Menke Pointsmentioning

confidence: 99%