A Matrix-Free Kalman Filtering Approach for Real-Time Estimation of Thermal Fields in Diffusion-Dominated Systems

Real-time estimation of temperature fields is essential in the monitoring and control of battery modules, power electronics, and other diffusion-dominated thermal systems. However, classical Kalman filtering becomes infeasible at high-state dimension because full-covariance updates incur quadratic memory and cubic computational cost in state dimension. This work develops a matrix-free information-form Kalman filter (KF) that avoids dense covariance operations by reformulating the correction step as a randomized Kaczmarz (RK) iteration. The resulting scheme relies only on sparse matrix–vector products and a diagonal precision approximation, achieving memory complexity that scales linearly with state dimension while retaining the statistical structure of the Kalman framework. The RK iteration also acts as an implicit regularizer, improving robustness in moderately ill-conditioned thermal inverse problems. Numerical experiments on representative two-dimensional thermal diffusion models demonstrate significant computational speedup and memory reduction compared with the classical KF while maintaining competitive estimation accuracy. Under moderate process noise, the method exhibits robustness benefits attributed to implicit regularization. These results indicate that matrix-free information-form filtering offers a practical and computationally efficient approach for large-scale thermal state estimation in real-time applications.