Kalman filtering for state estimation of advection diffusion PDE from sparse observation

Kalman filter is an important algorithm used in control theory. It takes an initial state as input and a series of observations over time and output the hidden state. The advection-diffusion equation is a PDE that characterizes the combination effect of advection and diffusion of a given object in the solvent. Such a problem is within the domain that the Kalman filter can solve. In this report, I will first derive the Kalman filter algorithm, then examine its application to an advection-diffusion equation. I will use different metrics to quantify the numerical performance of the algorithm. The contribution of this report lies in the combination of the Kalman filter algorithm with the advection equation. Also, an ample amount of graphs that can visually tell us the evolving trend of the state