Approximate Kalman filtering for large-scale systems with an application to hyperthermia cancer treatments

TL;DR

A real-time feasible state-estimation scheme for a class of large-scale systems that approximates the steady state Kalman filter, focusing on systems where the state-vector is the result of discretizing the spatial domain, as typically seen in Partial Differential Equations.

Abstract

Accurate state estimates are required for increasingly complex systems, to enable, for example, feedback control. However, available state estimation schemes are not necessarily real-time feasible for certain large-scale systems. Therefore, we develop in this paper, a real-time feasible state-estimation scheme for a class of large-scale systems that approximates the steady state Kalman filter. In particular, we focus on systems where the state-vector is the result of discretizing the spatial domain, as typically seen in Partial Differential Equations. In such cases, the correlation between states in the state-vector often have an intuitive interpretation on the spatial domain, which can be exploited to obtain a significant reduction in computational complexity, while still providing accurate state estimates. We illustrate these strengths of our method through a hyperthermia cancer treatment case study. The results of the case study show significant improvements in the computation time, while simultaneously obtaining good state estimates, when compared to Ensemble Kalman filters and Kalman filters using reduced-order models.

Figures (6)

• Figure 1: Transparent front view of the phantom model (left) and a phantom model placed in the applicator (right).
• Figure 2: Model-based heat loads $b_1(\bm{r})$ and $b_2(\bm{r})$ on the central transversal slice.
• Figure 3: Temperature probe locations on the central transversal slice of the phantom and model. The white and gray regions in Figure \ref{['fig:henk_model']} denote the pelvis and soft-tissue, respectively.
• Figure 4: Conditional expected value conditioned on a state in the soft-tissue (a) and a state in the pelvis (b).
• Figure 5: Probe temperature and estimated temperature at the probe locations for different observers.