I am a beginner in differential geometry and I am investigating $C^1$ isometric embedding of Riemannian manifolds to Euclidean space for computer vision.
I know that the PDE produced for a metric $g$ for this embedding is $\Sigma_{k=1}^{q} \partial_{i} u^{k} \partial_{j} u^{k}=g_{i j}$
I have no idea on how to use numerical methods for solving this PDE and the runtime analysis of such methods.