I am working with Conjugate Gradient method to solve for 𝐴𝑥=𝑏, where 𝐴 is an extremely large PSD and Singular matrix. I cannot directly access the elements of 𝐴. The only thing I can do is computing 𝐴𝑣 for any vector 𝑣, which is a rather expensive operation.

Now, I would like to solve for 𝐴𝑥=0, which is effectively finding one vector in the null space of 𝐴. I understand that it is slow to solve it with CG directly. I wonder if there are other iterative methods that would give a reasonably good solution with only the access of 𝐴𝑣 very quickly?