Dynamical Low-Rank Approximations for Kalman Filtering

We propose a dynamical low rank approximation of the Kalman-Bucy process (DLR-KBP), which evolves the filtering distribution of a partially continuously observed linear SDE on a small time-varying subspace at reduced computational cost.

This reduction is valid in presence of small noise and when the filtering distribution concentrates around a low dimensional subspace.

We further extend this approach to a DLR-ENKF process, where particles are evolved in a low dimensional time-varying subspace at reduced cost.

This allows for a significantly larger ensemble size compared to standard EnKF at equivalent cost, thereby lowering the Monte Carlo error and improving filter accuracy.

Theoretical properties of the DLR-KBP and DLR-ENKF are investigated, including a propagation of chaos property.

Numerical experiments demonstrate the effectiveness of the technique.