(untitled) (Communications in Computational Physics, 2021)
Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equation, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to the estimation of a physical length scale in the case of a uniform static grid.