GalKF

GalKF, short for Galerkin Kalman Filter, is a family of reduced-order state-estimation methods that combine Galerkin projection with Kalman filtering to estimate the state of high-dimensional dynamical systems from noisy measurements. The approach seeks to improve computational tractability by projecting the full state onto a low-dimensional subspace spanned by a chosen basis, such as proper orthogonal decomposition (POD) or Fourier modes.

In the typical linear setting, the full model is xdot = A x + w and y = C x + v. Projection onto the r-dimensional subspace with a basis Phi yields the reduced state a with x ≈ Phi a, the reduced system adot = A_r a + w_r where A_r = Phi^T A Phi and w_r = Phi^T w, and the observed output is y = C Phi a + v, or equivalently y = C_r a + v with C_r = C Phi. A standard Kalman filter is then applied to the reduced model to estimate a, and the full-state estimate is reconstructed as x_hat = Phi a_hat. For nonlinear dynamics, GalKF can incorporate linearization or use nonlinear reduced-order filters (e.g., extended or unscented Kalman filters) in the reduced coordinates.

Applications are common in areas with very large state spaces, including high-dimensional fluid flows, weather or climate modeling, structural health monitoring, and certain robotics problems. Advantages include substantially lower computational cost and the ability to exploit dominant modes of the system; limitations arise from dependence on the chosen basis to capture the essential dynamics and potential loss of accuracy if important behavior lies outside the subspace.

See also Kalman filter, reduced-order modeling, Galerkin projection.