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Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization,10.1007/s10444-009-91

Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization   (Citations: 2)
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In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour.
Journal: Advances in Computational Mathematics - Adv. Comput. Math. , vol. 34, no. 1, pp. 43-66, 2011
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    • ...See also [RSZ] for further references and comments...
    • ...While the first option - although with the same motivation as in the present work - has been adopted in [RSZ], we address here the algorithmic realization of the second option...
    • ...In fact, in [RSZ], from a somewhat dierent perspective, we focus on convergence of preconditioned iteration schemes per se, whereas in the present paper we develop and...
    • ...[KN, RSZ]. Note that the preconditioning is already provided through the formulation as a wellconditioned problem in ‘2(I)...

    Wolfgang Dahmenet al. Adaptive eigenvalue computation: complexity estimates

    • ...ditioner is chosen appropriately; in [40, 11], an adaptive variant was recently proposed...

    Reinhold Schneideret al. Direct minimization for calculating invariant subspaces in density fun...

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