An efficient data assimilation algorithm using the Allen-Cahn equation

Data assimilation combines observation data and the physical model to provide an estimate of the state which is more accurate than what could be obtained by using just the data or the model alone. We will establish a robust algorithm for the Allen-Cahn (AC) equation by considering the data assimilation term. The proposed approach includes a feedback control scheme for the AC equation, which forces the numerical solutions to preserve the observed data. We utilize the Crank-Nicolson formula and a stable numerical method to solve the data assimilation method using the AC equation. The proposed numerical algorithm has second-order temporal and spatial accuracy. Our computational scheme has a low computational burden because of the linearity of the elliptic equation. The discrete system of equations is unconditionally energy stable. The efficiency of our proposed method has been verified through a number of computational tests. The proposed approach is the first simple algorithm to consider the data assimilation method using the AC equation. The computational results indicate that our method performs well with low spatial resolution observational measurements and can be used for studying the image inpainting and shape transformation processing.