2026 Volume 21 Issue 2 Pages JFST0012
In this study, the overdetermined discrete empirical interpolation method (ODEIM), an extension of the discrete empirical interpolation method (DEIM), is reformulated, and parametric studies by varying the numbers of modes and sensors are conducted. Conventional DEIM employs the QR method to determine sensor locations, which places an upper limit on the number of sensors. Therefore, the proposed ODEIM, using the determinant-based greedy (DG) method, allows for an arbitrary number of sensor points, regardless of the number of modes. The stability of reduced-order systems is systematically examined by applying these methods to two highly distinct benchmarks: a numerically unstable, time-dependent free-surface relaxation problem, and an inherently stable, steady-state Laplace equation problem. Through extensive parametric evaluations, this study clarifies the previously unestablished relationships among the number of state modes rx, the number of nonlinear modes rf , and the number of sensors p, thereby establishing practical guidelines to minimize reconstruction errors. For the steady-state Laplace equation problem, accurate estimation results were obtained over a wide range of mode numbers, showing that DEIM is sufficiently effective. On the other hand, for the free-surface relaxation problem including time integration, DEIM often results in large errors or divergence. However, stable and accurate approximation results are found to be obtained by considering the differences in energy decay characteristics between the state and nonlinear fields, and by increasing the sensor count in ODEIM even for the free-surface relaxation problem. The proposed parameter selection criteria provide a practical guideline for ensuring computational stability in overdetermined reduced-order systems. These findings contribute to the development of robust dimensionality reduction techniques for complex fluid simulations.
