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Questions about the properties of vector spaces and linear transformations, including linear systems in general.
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How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?
You could use Woodbury matrix identity. In your case, this is
$$
(A + D)^{-1} = A^{-1} - A^{-1} (D^{-1} + A^{-1})^{-1} A^{-1}.
$$
You still need to compute $(D^{-1} + A^{-1})^{-1}$, which may not be e …