Effects of spatial resolution of the data on dynamic mode decomposition and its quantitative analysis

Dynamic mode decomposition (DMD), based on Koopman analysis, is a tool capable of spatiotemporal analysis for various spatial resolutions, from one-dimensional signals to three-dimensional computational fluid dynamics (CFD) and experimental data. Outputs of the DMD consist of amplitudes, frequencies, decaying rates, and spatial modes. However, the effects of spatial resolution (time-series data in one-dimensional signal and spatial grid in two-dimensional data) and quantitative analysis of DMD are limited to one-dimensional signal data. In this study, the effects of spatial resolution with a fixed time scale of data and correction using scaling factors root root 2/ A on DMD amplitudes and A on DMD spatial mode strengths are investigated, where A is the number of the time-series data in one-dimensional signal or the number of the spatial grid in two-dimensional data. First, proofs of the scaling factors for one-dimensional(line layout) and two-dimensional(grid layout) data are presented. Second, the effect of spatial resolution on the amplitudes and spatial mode strengths and their scaled results are confirmed using one-dimensional artificial signal data, two-dimensional artificial signal field data, two-dimensional vortex shedding simulation data, and two-dimensional pulsating flow experimental data with various data resolutions. The results show that the amplitude increases proportionally with the spatial resolution, and the spatial mode strength is inversely proportional to the time series or spatial resolution of the data in all cases. As a result of applying the scaling factors to one-dimensional artificial signal and two-dimensional artificial signal field data, the amplitudes and spatial modes contain the same values regardless of the change in spatial resolutions. The scaled amplitudes and spatial mode strengths on vortex shedding simulation and two-dimensional laminar pulsating jet show good agreements with slight differences, regardless of the spatial resolution change. The proposed scaling factor can be applied to compare data quantitatively obtained with different spatial resolutions.

Automated summary

This study investigates the effects of spatial resolution on DMD amplitudes and spatial mode strengths, and proposes scaling factors to correct for the resolution differences. The results show that the proposed scaling factors successfully normalize the amplitudes and spatial modes, allowing for quantitative comparison of data obtained with different spatial resolutions. This study is significant for analyzing spatiotemporal data in various fields.