A robust shifted proper orthogonal decomposition: Proximal methods for decomposing flows with multiple transports

TL;DR

This paper introduces a robust extension of the shifted proper orthogonal decomposition (sPOD) that employs proximal algorithms to better decompose flow data with multiple transports, handling noise and interpolation errors effectively.

Contribution

It develops a new methodology extending sPOD by optimizing co-moving fields with nuclear norm penalization and robustness terms, solved via convex optimization algorithms.

Findings

Effective separation of flow components demonstrated on synthetic benchmarks.

Applicable to 1D and 2D incompressible and reactive flows.

Maintains interpretability similar to POD for individual fields.

Abstract

We present a new methodology for decomposing flows with multiple transports that further extends the shifted proper orthogonal decomposition (sPOD). The sPOD tries to approximate transport-dominated flows by a sum of co-moving data fields. The proposed methods stem from sPOD but optimize the co-moving fields directly and penalize their nuclear norm to promote low rank of the individual data in the decomposition. Furthermore, we add a robustness term to the decomposition that can deal with interpolation error and data noises. Leveraging tools from convex optimization, we derive three proximal algorithms to solve the decomposition problem. We report a numerical comparison with existing methods against synthetic data benchmarks and then show the separation ability of our methods on 1D and 2D incompressible and reactive flows. The resulting methodology is the basis of a new analysis paradigm that results in the same interpretability as the POD for the individual co-moving fields.