Nonlinear Model Order Reduction on Quadratic Manifolds via Greedy Algorithms with Dimension-Dependent Regularization

Abstract

Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can be an efficient and accurate emulator. However, for parametric partial differential equations with slowly decaying Kolmogorov $n$-width, the dimension of the linear subspace required for a reasonable accuracy becomes very large, which undermines computational efficiency. To address this limitation, quadratic manifold methods have recently been proposed. These data-driven methods first identify a quadratic mapping by minimizing the linear projection error over a large set of snapshots, often with the aid of regularization techniques to solve the associated minimization problem, and then use this mapping to construct ROMs. In this paper, we propose and test a novel enhancement to this quadratic manifold approach by introducing a first-of-its-kind double-greedy algorithm on the regularization parameters coupled with a standard greedy algorithm on the physical parameter. Our approach balances the trade-off between the accuracy of the quadratic mapping and the stability of the resulting nonlinear ROM, leading to a highly efficient and data-sparse algorithm. Numerical experiments conducted on equations such as linear transport, acoustic wave, advection-diffusion, and Burgers' demonstrate the accuracy, efficiency, and stability of the proposed algorithm.

Figures (9)

• Figure 1: The relative linear-/QM-ROM and reconstruction errors as a function of reduced basis dimensions $r$, for three regularization parameters $\lambda=1e-6, 1e4, 1e6$ (from Left to Right). For $\lambda=1e-6$, the QM-ROM is unstable for $r\geq 5$, producing infinite relative errors that are not shown in the left panel. For $\lambda=1e6$ in the right panel, the QM-ROM barely outperforms the linear-ROM. Moderate $\lambda=1e4$ provides the best balance between stability and accuracy among these three cases.
• Figure 2: Average relative errors, error estimators, regularization parameters, and indices of selected parameters as functions of the reduced basis dimension $r$ for $N_{\text{incre}}=2, r_0=1$, and $l_{\text{sam}} =2$. Case 1 to Case 3 (Top to Bottom row) and $n_{\lambda}=2, 3, 4$ for Case 1 to Case 3.
• Figure 3: Average relative errors, error estimators, regularization parameters, indices of selected parameters as functions of the reduced basis dimension $r$ for different $N_{\text{incre}}$. Case 1 to Case 3 (Top to Bottom row).
• Figure 4: Average relative errors, error estimators, regularization parameters, indices of selected parameters as functions of reduced basis dimension $r$ for $l_{\text{sam}}=5$, $\Xi_\lambda=10^{-3:1:3}$, and $n_{\lambda}=2$.
• Figure 5: Supremacy in accuracy of Greedy-Quadratic ROM: FDM solution, Greedy-Quadratic ROM and Greedy-Linear ROM approximations at $t=6$ with $r=29$ for $\sigma =0.47915$ (from Left to Right) for the linear acoustic wave equation.