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The Spectral Structure of a Nonlinear Operator and Its Approximation

In: Sustained Simulation Performance 2015

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  • Uwe Küster

    (High Performance Computing Center Stuttgart)

Abstract

Whereas linear operators enable a deep structural analysis by their spectra and the associated eigenspace decomposition, similar seems to be impossible for relevant nonlinear operators. It turns out that there is a very general functional analytical loophole for the spectral approach by the Koopman operator. The involved theory is complicated and not yet applied to the numerically important nonlinear operators. Some approaches are using the dynamical mode decomposition of Peter Schmid for the calculation of generalized eigenmodes of nonlinear equations. We show some deficiencies of this approach with respect to the spectrum of the Koopman operator and remedies by using a Krylov space based approximation procedure for eigenvalues and eigenvectors.

Suggested Citation

  • Uwe Küster, 2015. "The Spectral Structure of a Nonlinear Operator and Its Approximation," Springer Books, in: Michael M. Resch & Wolfgang Bez & Erich Focht & Hiroaki Kobayashi & Jiaxing Qi & Sabine Roller (ed.), Sustained Simulation Performance 2015, pages 109-123, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-20340-9_9
    DOI: 10.1007/978-3-319-20340-9_9
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