Adaptive $h$-refinement for reduced-order models
For engineers and scientists using reduced-order models, this method allows online error control and adaptation to new phenomena without expensive full-order solves.
This work introduces an adaptive h-refinement method for reduced-order models that enriches the reduced-basis space online without full-order-model solves, enabling the model to meet any prescribed error tolerance. Experiments on the inviscid Burgers equation show it captures moving shocks not in the original basis.
This work presents a method to adaptively refine reduced-order models \emph{a posteriori} without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive $h$-refinement: it enriches the reduced-basis space online by `splitting' a given basis vector into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed offline via recursive $k$-means clustering of the state variables using snapshot data. The method identifies the vectors to split online using a dual-weighted-residual approach that aims to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or full-order-model solves. Further, it enables the reduced-order model to satisfy \emph{any prescribed error tolerance} regardless of its original fidelity, as a completely refined reduced-order model is mathematically equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.