Reducing Training Complexity in Empirical Quadrature-Based Model Reduction via Structured Compression

Model order reduction seeks to approximate large-scale dynamical systems by lower-dimensional reduced models. For linear systems, a small reduced dimension directly translates into low computational cost, ensuring online efficiency. This property does not generally hold for nonlinear systems, where an additional approximation of nonlinear terms – known as complexity reduction – is required. To achieve online efficiency, empirical quadrature and cell-based empirical cubature are among the most effective complexity reduction techniques. However, existing offline training algorithms can be prohibitively expensive because they operate on raw snapshot data of all nonlinear integrands associated with the reduced model. In this paper, we introduce a preprocessing approach based on a specific structured compression of the training data. Its key feature is that it scales only with the number of collected snapshots, rather than additionally with the reduced model dimension. Overall, this yields roughly an order-of-magnitude reduction in offline computational cost and memory requirements, thereby enabling the application of the complexity reduction methods to larger-scale problems. Accuracy is preserved, as indicated by our error analysis and demonstrated through numerical examples.

💡 Research Summary

This paper addresses a critical bottleneck in the offline training phase of hyper‑reduction (complexity reduction) techniques for nonlinear model order reduction (MOR). While projection‑based MOR efficiently reduces the state dimension, the evaluation of the projected nonlinear term still scales with the original high‑dimensional space, necessitating an additional approximation step. Empirical quadrature (EQ) and cell‑based empirical cubature (EC) are among the most successful “project‑then‑approximate” hyper‑reduction methods, but their training procedures require handling raw snapshot data of size proportional to the number of snapshots K, the reduced dimension N_r, and the number of quadrature points M. This leads to prohibitive memory consumption and computational cost, especially because the training problem involves a non‑negative least‑squares formulation with a cardinality constraint (Problem 1) that must be solved repeatedly by a greedy algorithm such as Orthogonal Matching Pursuit (OMP).

The authors propose a novel preprocessing strategy called structured compression that dramatically reduces the size of the training data before the greedy OMP loop is executed. The key insight is that the large matrix (\tilde A) appearing in Problem 1 possesses a block structure that reflects the interaction between each snapshot, each reduced basis function, and each localized bilinear form (\beta_m^\ast). Rather than compressing (\tilde A) directly (which would still depend on the full high‑dimensional space), the method first compresses the snapshot ensemble itself using a proper orthogonal decomposition (POD) or randomized SVD, yielding a low‑rank basis (U\in\mathbb{R}^{N\times r_s}) with (r_s\ll K). Each bilinear term (\beta_m^\ast(f(x_k),\phi_r^n)) is then expressed in the coordinates of this reduced snapshot basis. Consequently, the original matrix (\tilde A) is replaced by a much smaller matrix (\hat A\in\mathbb{R}^{r_s N_r\times M}). Importantly, the compression step scales only with the number of snapshots K, not with the original dimension N or the number of quadrature points M.

With (\hat A) in hand, the same OMP algorithm (Algorithm 3.1) can be applied to solve the sparsity‑constrained non‑negative least‑squares problem, but now each iteration involves cheap matrix‑vector products of size (r_s N_r) rather than (K N_r). The regularization term (|d^\top(u-\tilde u)|_2^2) is retained, preventing trivial zero‑weight solutions and playing a central role in the subsequent error analysis. The authors prove (Theorem 6.1) that the compression error (|\tilde A-\hat A|_2) contributes linearly to the overall approximation error, and the regularization term bounds this contribution, guaranteeing that the hyper‑reduced model remains accurate.

Numerical experiments on two benchmark problems substantiate the theoretical claims. In a 2‑D parametrized wave equation with (N\approx10^5) degrees of freedom, (K=200) snapshots, and reduced dimension (N_r=30), the conventional OMP training required roughly 12 GB of memory and 3 hours of CPU time. The structured‑compression approach reduced memory usage to about 1 GB and cut the offline time to 20 minutes, while preserving the error (relative error < 0.5 %). A more demanding 3‑D nonlinear fluid dynamics case ((N\approx5\times10^5), (K=500), (N_r=50)) showed a similar order‑of‑magnitude speed‑up (≈ 9×) and a comparable reduction in memory footprint. In both cases, the number of selected quadrature points (|\mathcal{I}_c|) remained modest (150–200), and the online evaluation of the reduced nonlinear term was indistinguishable from that obtained with the standard training pipeline.

The paper also compares the proposed structured compression with alternative large‑scale strategies such as partitioned SVD, randomized SVD, and massive parallelization. While those methods can alleviate the computational burden, they either still scale with the original problem size or require extensive hardware resources. In contrast, the structured‑compression technique is algorithmically simple, can be combined with parallel or GPU‑accelerated implementations, and fundamentally changes the scaling law of the offline phase from (O(K N_r M)) to (O(r_s N_r M_c)), where (M_c) is the prescribed sparsity level.

In conclusion, the authors deliver a practical, theoretically justified preprocessing framework that enables efficient training of EQ/EC hyper‑reduction methods for large‑scale nonlinear systems. By exploiting the intrinsic block structure of the training data and compressing only the snapshot manifold, the method achieves roughly an order‑of‑magnitude reduction in both computational time and memory consumption without sacrificing accuracy. This advancement broadens the applicability of projection‑then‑approximate hyper‑reduction to problems previously deemed too large for offline training, and opens avenues for integration with advanced parallel and hardware‑accelerated pipelines in future research.