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SUMMARY:Residual DMD: Rigorous data-driven computation of spectral propert
 ies of Koopman operators for dynamical systems
DTSTART:20220608T161500
DTEND:20220608T171500
DTSTAMP:20260408T034835Z
UID:633452bdf1746234caa86b78ffd7a12330e24ca056d64f94986a5adb
CATEGORIES:Conferences - Seminars
DESCRIPTION:Matthew Colbrook (DAMTP)\nKoopman operators are infinite-dimen
 sional operators that globally linearise non-linear dynamical systems\, ma
 king their spectral information valuable for understanding dynamics. Their
  growing popularity\, dubbed “Koopmania”\, has produced 10\,000s of ar
 ticles over the past decade. However\, Koopman operators can have continuo
 us spectra\, can lack finite-dimensional invariant subspaces\, and approxi
 mations can suffer from spectral pollution (spurious modes). These issues 
 make computing the spectral properties of Koopman operators a considerable
  challenge. In this talk\, we present datadriven algorithms with rigorous 
 convergence guarantees for computing spectral properties of Koopman operat
 ors from trajectory data. We present the first algorithm for computing the
  spectra and pseudospectra of general Koopman operators from trajectory da
 ta without spectral pollution\, namely residual dynamic mode decomposition
  (ResDMD). By combining ResDMD and the resolvent\, we compute smoothed app
 roximations of spectral measures associated with measure-preserving dynami
 cal systems. When computing the continuous and discrete spectrum\, explici
 t convergence theorems provide high-order convergence\, even for chaotic s
 ystems. Kernelized variants of our algorithms allow for dynamical systems 
 with a high-dimensional state-space\, and the error control provided by Re
 sDMD allows a posteriori verification of learnt dictionaries. For example\
 , we compute spectral measures for a protein molecule (20\,046-dimensional
  state-space) and compute nonlinear Koopman modes with error bounds for ch
 aotic turbulent flow past aerofoils (295\,122-dimensional state-space\, Re
 ynolds number > 100\,000).\n\n[1] M.J. Colbrook\, A. Townsend. "Rigorous d
 ata-driven computation of spectral properties of Koopman operators for dyn
 amical systems." arXiv:2111.14889 (2021).
LOCATION:SG 0213 https://plan.epfl.ch/?room==SG%200213
STATUS:CONFIRMED
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