Low-Rank Ensemble Filters for Elliptic Observations

Mathieu Le Provost, University of California
Shutterstock Fluid Dynamic Graphic

Existing regularization methods of the ensemble Kalman filter (EnKF) assume that the observations are local functions of the state. We present a regularization of the EnKF for non-local observations such as solutions of elliptic partial differential equations.  Elliptic inverse problems are usually highly compressible: low dimensional projections of the observations strongly inform a low-dimensional subspace of the state space. To leverage this structure, we derive a low-rank factorization of the Kalman gain based on the spectrum of the Jacobian of the observation operator.

From the rapid spectral decay, the inference can be performed in the low-dimensional subspace spanned by the dominant eigenvectors. However, the low-rank EnKF (LREnKF) remains biased for nonlinear state-space models. To reduce this bias, we develop a nonlinear generalization of the LREnKF, where the low-dimensional linear transformation is replaced by a sparse and interpretable low-dimensional nonlinear transformation based on measure transport. These low-rank filters are assessed on potential flow problems.