Suppose we are given a set of edge lengths $\{e_j\}$ and want to recover vertex positions $\{x_i\}$ of a valid graph embedding that realizes the given edge lengths as best as possible. More precisely, we are looking for a graph embedding that minimizes the residuals $r_j:=\vert (dx)_j - e_j \vert^2$, where $(dx)_j$ is the differential of the coordinates and represents the actual edge lengths of a valid embedding.
I have read that this problem is in approximation equivalent to solving the Poisson problem $$ \Delta x = \nabla e$$
Is this equivalence correct? If so, how can one proof this statement?