Modal Decomposition of Flow Data via Gradient-Based Transport Optimization

In the context of model reduction, we study an optimization problem related to the approximation of given data by a linear combination of transformed modes, called transformed proper orthogonal decomposition (tPOD). In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three numerical examples using a periodic shift operator as transformation.