Henrik Rosenberger

This work introduces a novel method to generate snapshot data for operator inference that guarantees the exact reconstruction of intrusive projection-based reduced-order models (ROMs). To ensure exact reconstruction, the operator inference least squares matrix must have full rank, without regularization. Existing works have achieved this full rank using heuristic strategies to generate snapshot data and a-posteriori checks on full rank but without a guarantee of success. Our novel snapshot data generation method provides this guarantee thanks to two key ingredients: first, we identify ROM states that induce full rank, then we generate snapshots corresponding to exactly these states by simulating multiple trajectories for only a single time step. This way, the number of required snapshots is minimal and orders of magnitude lower than typically reported with existing methods. The method avoids non-Markovian terms and does not require reprojection. Since the number of snapshots is minimal, the least squares problem simplifies to a linear system that is numerically more stable. In addition, because the inferred operators are exact, properties of the intrusive ROM operators such as symmetry or skew-symmetry are preserved. Numerical results for differential equations involving 2nd, 3rd, and 8th order polynomials demonstrate that the novel snapshot data generation method leads to exact reconstruction of the intrusive reduced order models. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/h3rror/exactOpInf and in the supplementary materials ( exactOpInf-main.zip [6.26MB]). [Formula: see text]

The application of operator inference to parametric problems whose operators exhibit affine parameter dependency results in a minimization problem with two issues. First, the solution to the minimization problem is generally not unique, so we might obtain a different solution than the one we are interested in. Second, the large dimensions of the minimization problem can cause prohibitively high computational costs. In this work, we resolve these issues by extending previous work on exact operator inference to the parametric case. Concretely, we propose a snapshot data generation method that guarantees the uniqueness of the solution and thereby the exact reconstruction of the corresponding intrusive operators of projection-based reduced order models. Furthermore, we show that the monolithic problem that comprises the snapshot data of all sampled parameters can be decoupled into separate problems for only one parameter sample each. This decoupling results in smaller subproblems that are embarrassingly parallel, numerically more stable and can be solved more efficiently. In numerical experiments, we demonstrate this decoupling for the heat equation with two different temperature-, parameter- and space-dependent thermal conductivities.

The present work presents a stable proper orthogonal decomposition (POD)-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh–Bénard convection in a square geometry for three Rayleigh numbers: 104 (steady state), 3×105 (periodic), and 6×106 (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM’s stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.