We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). SPOD is derived from a space-time POD problem for stationary flows and leads to modes that each oscillate at a single frequency. We establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients, which are usually treated as deterministic quantities described by an amplitude and phase, should be regarded as statistical quantities, described by their cross-spectral density, in order for the resolvent-mode expansion to properly capture the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical.

Literature:

• Towne, A., O. T. Schmidt, and T. Colonius. “Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis.” Journal of fluid mechanics 847 (2018): 821–867.
  [Bibtex]

  @Article{TowneSchmidtColonius_2018_JFM,
  author = {Towne, A. and Schmidt, O. T. and Colonius, T.},
  title = {Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis},
  journal = {Journal of Fluid Mechanics},
  year = {2018},
  volume = {847},
  pages = {821–867},
  doi = {10.1017/jfm.2018.283},
  file = {:TowneSchmidtColonius_2018_JFM.pdf:PDF},
  publisher = {Cambridge University Press},
  }