Kalman filters for PDEs is an old topic which is theoretically seducing but leads to a so-called curse of dimensionality in its numerical implementation. In order to circumvent this curse of dimensionality, it is now classical to base the numerical strategies on reduced order modeling. This raises the question of the accuracy of the resulting estimator.
Alternatives approaches based on reduced-basis decomposition of the covariance were also developed in the past years, but here with an additional question of stability of the resulting estimator.
Using H-matrix based discretization strategies, a recent numerical tool developed for integral equations discretization, we here show how a full Kalman estimator can now be envisioned PDE models and their associated large discretization, hence freshening up the interest for this old theory.