Simulation of complex flow field is vital for understanding physical phenomena and towards developing efficient designs and quantify uncertainties. However, high-fidelity simulation on large dynamical systems has still been proven to be too computationally demanding on resources for it to be directly incorporated in design work-flows where many iterations across large parameter spaces are needed. In order to alleviate this problem, model reduction techniques can be employed to facilitate and provide approximate solutions at a significantly smaller computational expense.

In particular, projection-based reduced order models (ROMs) use a Galerkin projection to obtain a surrogate system where the high-dimensional full order model (FOM), i.e. Navier-Stokes equations, is projected onto a low-dimensional manifold of the FOM solution space. The low-dimensional ROM can typically have significantly fewer degrees of freedom compared the the FOM, which can be millions of points. The ROM orthgonal basis is obtained via proper orthogonal decomposition (POD) of a dense matrix of instantaneous snapshots of the FOM flow field. A particular application is the air-wake over a ship which has several million grid cells comprising the computational domain. The first two modes contain 87% and 8% of the total energy of the flow field. The decomposition of the flow field is truncated and can be evolved only a subset of the total number of modes efficiently.

However, if non-linearities are present in the dynamical system (which there are if we consider the Navier-Stokes equations), the reduction in computational costs are inhibited by non-linear terms, which still scale with the dimension of the FOM. This is commonly referred to as the lifting bottleneck but can be overcome with hyper-reduction techniques such as discrete empirical interpolation method (DEIM). DEIM seeks to approximate the non-linear term through sparse sampling by computing the non-linear terms at certain sampling points and interpolating the remaining points in a low-dimensional subspace. The computational cost of the FOM can be mitigated by DEIM with nearly a linear speed up.