A goal-oriented reduced-order modeling approach for nonlinear systems

In this paper, we develop a novel, goal-oriented reduced-order modeling methodology. The approach uses a low-dimensional basis function set that contains both global and local, goal-oriented basis functions. Compared to reduced-order models using the standard proper orthogonal decomposition (POD) basis, these new goal-oriented POD basis functions lead to better approximations of given quantities of interest (QoI) while maintaining accuracy in the evolution of the state. We demonstrate this approach for two problems involving Burgers equation. In the first problem, the QoI is the spatial average of the solution over various regions. The QoI in the second problem is the feedback control based on a MinMax control design with an extended Kalman filter. In both cases, approximations of the QoI and the state variables are more accurate using the goal-orientated POD than using the standard POD basis with comparable online computational costs.