  In this work, we present the ensemble-marginalized Kalman filter (EnMKF), a
sequential algorithm analogous to our previously proposed approach [1,2], for
estimating the state and parameters of linear parabolic partial differential
equations in initial-boundary value problems when the boundary data are noisy.
We apply EnMKF to infer the thermal properties of building walls and to
estimate the corresponding heat flux from real and synthetic data. Compared
with a modified Ensemble Kalman Filter (EnKF) that is not marginalized, EnMKF
reduces the bias error, avoids the collapse of the ensemble without needing to
add inflation, and converges to the mean field posterior using $50\%$ or less
of the ensemble size required by EnKF. According to our results, the
marginalization technique in EnMKF is key to performance improvement with
smaller ensembles at any fixed time.
