Ensemble-marginalized Kalman filter for linear time-dependent PDEs with noisy boundary conditions: Application to heat transfer in building walls

TL;DR

This paper introduces the ensemble-marginalized Kalman filter (EnMKF), a novel sequential algorithm for estimating states and parameters of linear PDEs with noisy boundary data, demonstrating improved accuracy and efficiency in heat transfer applications.

Contribution

The paper presents EnMKF, a new marginalized Kalman filter method that outperforms existing approaches in estimating PDE states with noisy boundary conditions, requiring fewer ensemble members.

Findings

EnMKF reduces bias error compared to EnKF.

EnMKF avoids ensemble collapse without inflation.

EnMKF converges with half the ensemble size of EnKF.

Abstract

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.