Goal-oriented model reduction for parametrized time-dependent nonlinear partial differential equations

Abstract

We present a projection-based model reduction formulation for parametrized time-dependent nonlinear partial differential equations (PDEs). Our approach builds on the following ingredients: reduced bases (RB), which provide rapidly convergent approximations of the parameter-temporal solution manifold; reduced quadrature (RQ) rules, which provide hyperreduction of the nonlinear residual; and the dual-weighted residual (DWR) method, which provides an error representation formula for the quantity of interest. To find the RQ rules, we develop an empirical quadrature procedure (EQP) for time-dependent problems; we analyze the output error due to hyperreduction using a space–time DWR framework and identify appropriate constraints so that the output error due to hyperreduction is controlled. We in addition equip our reduced model with an online-efficient DWR a posteriori error estimate for the output; we again analyze the error in the hyperreduced dual problem and DWR expression to find appropriate constraints for the EQP so that the error in the error estimate is controlled. In the offline stage, the RBs and RQs, as well as the finite element mesh, are simultaneously constructed using a POD-greedy algorithm that leverages the online-efficient output error estimate. We demonstrate the framework for parametrized unsteady flows in a lid-driven cavity and over a NACA0012 airfoil. Reduced models achieve over two orders of magnitude reduction in the number of degrees of freedom, number of quadrature points, and wall-clock computational time, while achieving less than 0.5% output error and providing efficient error estimates in predictive settings.