Low-Rank Dynamic Mode Decomposition using Riemannian Manifold Optimization

We present a method for non-intrusive data-driven reduced order modeling of high-dimensional dynamical systems using a new low-rank extension of Dynamic Mode Decomposition (DMD). A matrix optimization problem with a rank-constraint on the solution is formulated and results in a non-convex optimization problem. We propose two methods to solve the optimization problem. The first is an iterative subspace projection method that is computationally efficient but can only give the optimal solution under certain conditions. In the second method we perform Riemannian optimization on Grassmanian manifolds. Using a model equation for fluid flows, we evaluate the performance of the proposed methods on complex linearized Ginzburg-Landau equations in the supercritical globally unstable regime.