Efficient Hyperreduction for Large‐Scale Problems: Exploiting Reducible Constraint Manifolds in Empirical Quadrature Procedure

ABSTRACT We develop efficient hyperreduction methods for projection‐based model reduction of nonlinear partial differential equations (PDEs) with a large number of parameters and/or large parametric extents. Our formulation is based on the empirical quadrature procedure (EQP), which solves an optimization problem that involves “residual‐matching constraints” over a training parameter set to find a sparse quadrature rule that yields rapid yet accurate approximations of the PDE residual, and solves the constrained optimization via non‐negative least squares (NNLS). Specifically, we extend the EQP and NNLS to provide more efficient offline training for problems that (i) demand tight hyperreduction tolerances, (ii) involve a large number of residual‐matching constraints, and/or (iii) involve a high‐dimensional parameter space. To address (i), we develop second‐order accurate constraints for EQP and a rounding‐error stable NNLS formulation that efficiently provides a solution to the optimization problem with a tight tolerance. To address (ii), we develop NNLS with constraint reduction (NNLS‐CR), which exploits the fact that many constraints are often redundant and systematically constructs a reduced orthogonal set of constraints that still represents all the original constraints. To address (iii), we introduce an EQP method that adaptively constructs the training parameter set and solves the associated constrained optimization problem using a version of NNLS‐CR that admits incremental constraint update. We demonstrate the offline efficiency of the methods, as well as the parametric robustness of the resulting ROMs, using parameterized Navier–Stokes and Reynolds‐averaged Navier–Stokes equations in four different contexts: Shape parameter sweep; flight parameter sweep; ensemble‐based data assimilation; and forward uncertainty quantification.