The are two eigen-decomposed matrices $A$ = $U_1$$V_1$$U_1$$^H$, $B$ = $U_2$$V_2$$U_2$$^H$, in which $V_1$ and $V_2$ are the eigen-matrices formed by the non-negative eigenvalues and the eigenvalues are all less than 1, $U_1$ and $U_2$ are unitary matrices formed by the eigenvectors. Is there any efficient way (including any efficient iterative solution) to calcualte the following vector?

$y$ = ($A$+$B$+$I$)$^{-1}$$x$

in which $I$ is the identity matrix, and $x$ could be an arbitrary vector.

Thanks for any discussions.