Projection assisted Dynamic Mode Decomposition of large scale data

Abstract: We have deluge of data in time series format for numerous phenomena. The number of snapshots, resolution and many other factors come into play as we look to identify the dynamics in a given problem. The pre-processing and post-processing steps while working with the data may be related to the resources in terms of the hardware used to collect data and computations that we perform on the data to create a model for the problem. Dynamic Mode Decomposition (DMD) is a data based modeling tool that identifies a matrix to map a quantity at some time instant to the same quantity in future. It is possible to generate a model by projecting the high dimensional spatiotemporal data to a lower dimensional subspace in a probabilistic framework. Sampling and gaussian projection have been used in the past to increase efficiency in the computation. Here, we design an optimized version of DMD that utilizes time delay coordinates and a projection matrix. In our proposal, we discussed about two projection matrices -- one is inspired by the Krylov subspace and the other promotes and leverages sparsity to bring computational benefits in producing a model. Satisfactory results are obtained as they are tested on data related to Double gyre (present in ocean mixing) and on a 2D compressible signal. The motivation behind this scheme of DMD comes from the fact that data from many phenomena are 'big' and 'highly oscillatory.'