Computational solutions to the high-dimensional Kalman filtering problem are described in the setting of the MEG inverse problem. The overall objective of the described work is to localize and estimate dynamic brain activity from observed extraneous magnetic fields recorded at an array of sensor positions on the scalp and to do so in a manner that takes advantage of the true underlying statistical continuity in the current sources. To this end, we outline inverse mapping procedures that combine models of current dipoles with dynamic state-space estimation algorithms. While these algorithms are eminently well-suited to this class of dynamic inverse problems, they possess computational limitations that need to be addressed either by approximation or through the use of high performance computational resources. In this work we describe such a high performance computing (HPC) solution to the Kalman filter and demonstrate its applicability to the magnetoencephalography (MEG) inverse problem.