Inverse a sparse matrix

Question

I have a sparse singular matrix W where I want to find its inverse Q. My current method is to use $W*Q = I$ for an optimization process of approximating the convergence of cost function norm($I-W*Q$).

Within each iteration:

1. Compute the objective function value: norm($I-W*Q$)
2. Updating Q as the following: $Q = Q + stepSize*(W' * (I-W*Q))$
3. I applied a mask M that has the same size as Q so that the dot product $Q = Q.*M$ only keeps values in Q that has its corresponding position in M.
4. repeat iteration

The sparse matrix W has values of 1 for all of its elements. The above method could not make cost function approximate to a small enough value and so when I check $Q$ with $W^-1$, they look quite different.

Is there a method that I can try to get a better result?

Sparse Matrix Inverse with Topology