  In this work, we propose a method for the compression of the coupling matrix
in volume\hyp surface integral equation (VSIE) formulations. VSIE methods are
used for electromagnetic analysis in magnetic resonance imaging (MRI)
applications, for which the coupling matrix models the interactions between the
coil and the body. We showed that these effects can be represented as
independent interactions between remote elements in 3D tensor formats, and
subsequently decomposed with the Tucker model. Our method can work in tandem
with the adaptive cross approximation technique to provide fast solutions of
VSIE problems. We demonstrated that our compression approaches can enable the
use of VSIE matrices of prohibitive memory requirements, by allowing the
effective use of modern graphical processing units (GPUs) to accelerate the
arising matrix\hyp vector products. This is critical to enable numerical MRI
simulations at clinical voxel resolutions in a feasible computation time. In
this paper, we demonstrate that the VSIE matrix\hyp vector products needed to
calculate the electromagnetic field produced by an MRI coil inside a numerical
body model with $1$ mm$^3$ voxel resolution, could be performed in $\sim 33$
seconds in a GPU, after compressing the associated coupling matrix from $\sim
80$ TB to $\sim 43$ MB.
