Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

Abstract

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time while space-only POD modes in general do not. We also 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. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet.

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

“…Several methods have also been proposed in literature aimed at improving the POD modes [49][50][51][52] . There are also different variants of POD that have been introduced, like spatio-temporal biorthogonal decomposition 53 , spectral POD (SPOD) 54 , frequency based POD that is also called as SPOD 55 , multiscale POD (MPOD) 56 which splits the correlation matrix into the contribution of different scales.…”

Methods for data-driven multiscale model discovery for materials

“…Thus the DMD algorithm can be thought of as a modification of the SVD architecture which attempts to account for both spatial and temporal dynamic activity of the data. Recently, DMD has also been rigorously connected to the spectral proper orthogonal decomposition method [66].…”

Multiscale proper orthogonal decomposition (mPOD) of TR-PIV data—a case study on stationary and transient cylinder wake flows

“…Examples of the first variants are proposed by Lumley and Poje (1997), Maurel et al (2001), Rowley et al (2004), where multiple quantities are involved in the inner product. Examples of the second variants are proposed by Citriniti and George (2000) and Towne et al (2018), where the correlation matrix is computed in the frequency domain using time averaging over short windows, following the popular Welch's periodogram method Welch (1967). It is worth noticing that this decomposition is well known in other fields as Empirical Orthogonal Functions (EOF) decomposition or Principal Component Analysis (PCA).…”