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

2007/08/03 by Thorsten Rohwedder, Rohwedder, Thorsten, Reinhold Schneider +3
Computer Science · Mathematics · #65J10 #65N25 #65N55 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.0708.0517

openalex publication_date 2007/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we discuss an abstract iteration scheme for the calculation of\nthe smallest eigenvalue of an elliptic operator eigenvalue problem. A short and\ngeometric proof based on the preconditioned inverse iteration (PINVIT) for\nmatrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. We\nshow that convergence is retained up to any tolerance if one only uses\napproximate applications of operators which leads to the perturbed\npreconditioned inverse iteration (PPINVIT). We then analyze the Besov\nregularity of the eigenfunctions of the Poisson eigenvalue problem on a\npolygonal domain, showing the advantage of an adaptive solver to uniform\nrefinement when using a stable wavelet base. A numerical example for PPINVIT,\napplied to the model problem on the L-shaped domain, is shown to reproduce the\npredicted behaviour.\n

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