The Spectral Structure of a Nonlinear Operator and Its Approximation

Uwe Küster
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Uwe Küster: High Performance Computing Center Stuttgart

A chapter in Sustained Simulation Performance 2015, 2015, pp 109-123 from Springer

Abstract: 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.

Keywords: Nonlinear Operator; Test Vector; Finite Dimensional Space; Hankel Matrix; Dynamic Mode Decomposition (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-20340-9_9

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DOI: 10.1007/978-3-319-20340-9_9

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