A Fast Spectral Formulation of the Multiscale Proper Orthogonal Decomposition

Abstract: Multiscale Proper Orthogonal Decomposition (mPOD) decomposes fluid flows into energy-optimal modes within prescribed frequency bands by combining Proper Orthogonal Decomposition with a multiresolution analysis (MRA). In its classical formulation, mPOD relies on a filter bank of finite impulse response (FIR) filters, enabling lossless reconstruction while mitigating Gibbs oscillations and temporal ringing. However, the smooth transition bands required for this purpose introduce partial spectral overlap between adjacent scales and require, for each band, the solution of an eigenvalue problem spanning the full temporal dimension. This work introduces a fast spectral formulation of the mPOD that substantially reduces the computational cost. The proposed approach replaces time-domain FIR filters with compact spectral masks enforcing strictly disjoint frequency supports, thereby exactly decoupling the problem across scales. This leads to a block-diagonal correlation operator in spectral space, so that each band can be treated independently. The resulting eigenvalue problems reduce to small systems whose size depends on the number of active frequencies per band rather than the full time dimension. The approach is validated on a synthetic dataset highlighting spectral windowing effects and on experimental particle image velocimetry (PIV) data of a cylinder wake at Reynolds number (\mbox{Re} \approx 5000). In both cases, the proposed formulation accurately recovers the modal structures and singular values of the classical mPOD while reducing the computational cost by orders of magnitude.