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

TL;DR

The paper addresses decomposition of transport-dominated flows where traditional POD struggles, introducing a robust shifted POD (sPOD) extended with proximal optimization to jointly estimate low-rank co-moving fields and a sparse residual. It formulates the problem as minimizing $\sum_k \lambda_k \|\mathbf{Q}^k\|_* + \lambda_{K+1} \|\mathbf{E}\|_1$ subject to $\mathbf{Q} = \sum_k \mathcal{T}^k(\mathbf{Q}^k) + \mathbf{E}$, and develops three proximal algorithms: joint FB, cyclic BCD-FB, and an ALM-based constrained approach. The first two have descent and critical-point convergence under the Kurdyka-Łojasiewicz framework, while ALM delivers strong empirical performance despite lacking general nonconvex guarantees. Numerical experiments on synthetic data and 1D/2D flows (e.g., wildland-fire and wake flows) demonstrate accurate rank recovery, robust separation of multiple transports, and preserved interpretability akin to POD, enabling transport-aware surrogate modeling of individual phenomena.

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.

Figures (9)

• Figure 1: Illustration of the robust sPOD. The noise is computed by randomly setting 12.5% of the input entries of $\mathbf{Q}$ to $1$. The input data $\mathbf{Q}$ and its decomposition into a low-rank part $\boldsymbol{\tilde{\mathbf{Q}}}=\mathcal{T}^1\mathbf{Q}^1+\mathcal{T}^2\mathbf{Q}^2$, as well as the noise matrix $\mathbf{E}$ are displayed from left to right.
• Figure 2: Impact of the hyperparameters on the relative reconstruction error at each iteration on the multilinear transport test case. The co-moving ranks $R^*_k=\mathrm{rank}(\mathbf{Q}^k)$$k=1,2$ at iteration 500 are stated for each hyperparameter at the end of each line.
• Figure 3: Decay of the relative approximation error in the Frobenius norm.
• Figure 4: Ranks of the estimated co-moving fields at each iteration for the multilinear transport test case.
• Figure 5: Ranks of the estimated co-moving fields at each iteration for the sine waves test case with noise.

Theorems & Definitions (1)

• Remark 1