Sobolev quasi-periodic solutions of NLS in any dimension d > 1

We prove the existence of quasi-periodic solutions of Schr\"odinger equations on any d-dimensiona torus with nonlinearities which are merely differentiable functions. Our solutions have only Sobolev regularity both in time and space. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators along scales of Sobolev spaces. We prove these linear estimates via a new multiscale inductive analysis.