Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study

Abstract

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 equations, 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 an inverse problem in the case of a uniform static grid.

Data assimilation for the Navier-Stokes equations using local observables

“…Conversely, when an algorithm works in practice, it suggests there might be some analysis to support it. Computational work has demonstrated that nudging over the entire computational domain works much better than required in the rigorous estimates [2,56,28,40,47,48] In our pseudospectral implementation, we have h = 2π/N, so strictly speaking Theorem 2.1 would require N ∼ G exp( √ N /2), which is far from obtainable. Yet our computational results are promising.…”