I have to compute the following matrix inverse
$$ \left( \begin{bmatrix}2\bf{B} & -\bf{B} & \bf{0} \\ -\bf{B} & 2\bf{B}& -\bf{B} \\\bf{0} & -\bf{B}& \bf{B} \end{bmatrix} + \begin{bmatrix}\bf{X_1} & \boldsymbol{0} &\bf{0} \\ \bf{0} & \bf{X_2}& \bf{0} \\\bf{0} & \bf{0}& \bf{X_3} \end{bmatrix}\right)^{-1} $$
where $\bf{B}$ and the matrices $\bf{X_i}$ for $i=1,2,3$ are square symmetric positive definite of dimension $N \times N$. Furthermore $\bf{B}$ is diagonal. I want to understand if the blocks of the inverse can be written as a combination of my original matrices. These blocks would be variances and covariances of random vectors, which are my objects of interest.
I think I can exploit in some way the fact that the matrix is tridiagonal, symmetric and pd but I didn't get anything useful by now.