Recovering Mesh Geometry from a Stiffness Matrix

We introduce the following class of mesh recovery problems: Given a stiffness matrix A and a PDE, construct a mesh M such that the finite-element formulation of the PDE over M is A. We show, under certain assumptions, that it is possible to reconstruct the original mesh for the special case of the Laplace operator discretized on an unstructured mesh of triangular elements with linear basis functions. The reconstruction is achieved through a series of techniques from graph theory and numerical analysis, some of which are new and can find application in other scientific areas. Finally, we discuss extensions to other operators and some open questions related to this class of problems.