Appell Integral Transfrom ( the easy way to make new martingales), and some of its applications
A class of algorithms known as extended dynamic mode decomposition
(EDMD) has been shown to be empirically effective at identifying
intrinsic modes of a dynamical system from time-series data. The
algorithm amounts to constructing an NxN matrix by observing a
dynamical system through N observables at a sequence of M phase
space points. While empirically successful, there are few
rigorous results on the convergence this algorithm. Moreover, the
relationship between M and N remains obscure.
In this talk I will focus on analytic expanding circle maps and show that
spectral data of the EDMD matrices can be linked to spectral data (for
example eigenvalues, also known as Ruelle resonances in this context)
of the Perron-Frobenius operator or its dual, the Koopman operator,
associated to the underlying map, provided both operators are
considered on suitable function spaces. In particular, I will show that for
equidistantly chosen phase space points, spectra of the EDMD
matrices converge to the Ruelle resonances at exponential speed in N,
provided that the number of data points M is chosen to be a constant
multiple of N, where the constant depends on complex expansion
properties of the underlying analytic circle map.
This is joint work with Wolfram Just and Julia Slipantschuk.
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