# Inverse of the summation of Kronecker products with positive definite matrices

I would like to to obtain the inverse $\textbf{C}^{-1}$ of a matrix having the following form:

$$\textbf{C} = \sum_{k=1}^K A_k \otimes B_k + I,$$

where $I$ denotes the identity matrix and where $\otimes$ denotes the Kronecker product. All the $A_k$'s are $n\times n$ positive definite matrices whereas all the $B_k$'s are $m \times m$ positive semi-definite matrices.

The direct calculation is computationally expansive, since it involves the inversion of a $mn \times mn$ matrix. Does it exists a way to break up the inversion into smaller pieces, so that the computations are more efficient?

Many thanks!