Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases

The energy-conserving sampling and weighting (ECSW) method is a\nhyperreduction method originally developed for accelerating the performance of\nGalerkin projection-based reduced-order models (PROMs) associated with\nlarge-scale finite element models, when the underlying projected operators need\nto be frequently recomputed as in parametric and/or nonlinear problems. In this\npaper, this hyperreduction method is extended to Petrov-Galerkin PROMs where\nthe underlying high-dimensional models can be associated with arbitrary finite\nelement, finite volume, and finite difference semi-discretization methods. Its\nscope is also extended to cover local PROMs based on piecewise-affine\napproximation subspaces, such as those designed for mitigating the Kolmogorov\n$n$-width barrier issue associated with convection-dominated flow problems. The\nresulting ECSW method is shown in this paper to be robust and accurate. In\nparticular, its offline phase is shown to be fast and parallelizable, and the\npotential of its online phase for large-scale applications of industrial\nrelevance is demonstrated for turbulent flow problems with $O(10^7)$ and\n$O(10^8)$ degrees of freedom. For such problems, the online part of the ECSW\nmethod proposed in this paper for Petrov-Galerkin PROMs is shown to enable\nwall-clock time and CPU time speedup factors of several orders of magnitude\nwhile delivering exceptional accuracy.\n