Is there any property or low-complexity algorithms to calculate matrix inverse where its inverse matrix is digonal dominant?

Question

Given a symmetric positive semidefinite matrix $A \in \mathbf{R}^{n \times{} n}$ whose inverse $A^{-1}$ is diagonally dominant:

1. Is there any property for the matrix $A$?

2. Are there lower-complexity (perhaps some approximation algorithms) algorithms to calculate $A^{-1}$ of this kind of matrices?