Residual dynamic mode decomposition: robust and verified Koopmanism

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dc.contributor.authorColbrook, Matthew J. J.en
dc.contributor.authorAyton, Lorna J. J.en
dc.contributor.authorSzoke, Mateen
dc.date.accessioned2023-03-29T13:25:24Zen
dc.date.available2023-03-29T13:25:24Zen
dc.date.issued2023-01-17en
dc.description.abstractDynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.en
dc.description.notesThis work was supported by a FSMP Fellowship at Ecole Normale Superieure (M.J.C.); EPSRC Early-Career Fellowship EP/P015980/1 (L.J.A.); and NSF Grant CBET-1802915 (M.S.).M.J.C. would like to thank both the Cecil King Foundation and the London Mathematical Society for a CecilKing Travel Scholarship, part of which funded a visit to Virginia Tech. M.S. would like to thank the National Science Foundation, in particular Dr R. Joslin, for their support (grant CBET-1802915) that enabled the capture of the PIV dataset.en
dc.description.sponsorshipFSMP Fellowship at Ecole Normale Superieure; EPSRC Early-Career Fellowship [EP/P015980/1]; NSF [CBET-1802915]; Cecil King Foundation; London Mathematical Society; National Science Foundation [CBET-1802915]en
dc.description.versionPublished versionen
dc.format.mimetypeapplication/pdfen
dc.identifier.doihttps://doi.org/10.1017/jfm.2022.1052en
dc.identifier.eissn1469-7645en
dc.identifier.otherA21en
dc.identifier.urihttp://hdl.handle.net/10919/114219en
dc.identifier.volume955en
dc.language.isoenen
dc.publisherCambridge University Pressen
dc.rightsCreative Commons Attribution 4.0 Internationalen
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/en
dc.subjectcomputational methodsen
dc.subjectturbulent boundary layersen
dc.subjectbig dataen
dc.titleResidual dynamic mode decomposition: robust and verified Koopmanismen
dc.title.serialJournal of Fluid Mechanicsen
dc.typeArticle - Refereeden
dc.type.dcmitypeTexten

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