On-the-fly reduced order modeling of finite-time nonlinear sensitivities

We present an on-the-fly reduced order modeling framework for computing the evolution of finite-time nonlinear sensitivities in dynamical systems. Unlike solving a linearized system that assumes infinitesimal perturbations around a base trajectory, this framework places no limitation on the size of the perturbation as nonlinear interactions are considered. We propose a model-driven low-rank approximation that leverages a time-dependent basis by extracting correlations between sensitivities on-the-fly. To this end, we derive forward low-rank evolution equations for an orthonormal state basis, correlation matrix, and orthonormal parametric basis. The resulting equations are Jacobian-free and leverage the same nonlinear solver that is used to compute the evolution of the base state. For nonlinear sensitivities with arbitrarily time dependent base state, we demonstrate that low-rank structure often exists, and can be extracted in real time by solving forward evolution equations. This is in direct contrast to traditional reduced order modeling techniques that leverage a static basis extracted from high-fidelity data during an offline stage. In this work, we demonstrate the efficacy of the method for a number of applications including transition for compressible flow.