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

## TL;DR

This study explores a modified data assimilation algorithm for reaction-diffusion equations where measurement points move over time, showing improved efficiency with fewer data points in computational experiments.

## Contribution

It introduces a moving cluster data assimilation approach and demonstrates its effectiveness for reaction-diffusion equations, reducing the number of required measurement points.

## Key findings

- Convergence achieved with fewer measurement points using moving data points.
- The algorithm performs well in 1D Allen-Cahn reaction-diffusion equations.
- Application to inverse problems with static grids shown.

## 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.