Adaptive $h$-refinement for reduced-order models

Comparison of solutions computed by proper orthogonal decomposition–Galerkin with and without $h$-adaptivity.

Abstract

This work presents a method to adaptively refine reduced‐order models 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 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.

Publication
International Journal for Numerical Methods in Engineering, Vol. 102, No. 5, p.1192–1210 (2015)