Stiefel manifold interpolation for non-intrusive model reduction of parameterized fluid flow problems

Abstract

Many engineering problems are parameterized. In order to minimize the computational cost necessary to evaluate a new operating point, the interpolation of singular matrices representing the data seems natural. Unfortunately, interpolating such data by conventional methods usually leads to unphysical solutions, as the data live on manifolds and not vector spaces. An alternative is to perform the interpolation in the tangent space to the Grassmann manifold to obtain interpolated spatial modes. Temporal modes are afterwards determined via the Galerkin projection of the high-fidelity model onto the interpolated spatial basis. This method, which is known for some fifteen years, is intrusive. Recently, Oulghelou and Allery (JCP, 2021) have proposed a non-intrusive approach (equation-free), but requiring the resolution of two low-dimensional optimization problems after interpolation. In this paper, a non-intrusive alternative based on Interpolation on the Tangent Space of the Stiefel Manifold (ITSSM) is presented. This approach has the advantage of not requiring a calibration phase after interpolation. To assess the method, we compare our results with those obtained using global POD on the one hand, and two methods based on Grassmann interpolation on the other. These comparisons are performed for two classical configurations encountered in fluid dynamics. The first corresponds to the one-dimensional non-linear Burgers' equation. The second example is the two-dimensional cylinder wake flow. We show that the proposed strategy can accurately reconstruct the physical quantities associated with a new operating point. Moreover, the estimation is fast enough to allow real-time computation.