POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem

The ROMES method for statistical modeling of reduced-order-model error

This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive `error indicators' to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic) uncertainty introduced by the reduced-order model. To model normed errors, the method employs existing rigorous error bounds and residual norms as indicators; numerical experiments show that the method leads to a near-optimal expected effectivity in contrast to typical error bounds. To model errors in general outputs, the method uses dual-weighted residuals---which are amenable to uncertainty control---as indicators. Experiments illustrate that correcting the reduced-order-model output with this surrogate can improve prediction accuracy by an order of magnitude; this contrasts with existing `multifidelity correction' approaches, which often fail for reduced-order models and suffer from the curse of dimensionality. The proposed error surrogates also lead to a notion of `probabilistic rigor', i.e., the surrogate bounds the error with specified probability