Inverse problems are generally known to be hard. Here is a commentary on a couple of papers published in PRL which discusses one such problem – namely, finding a potential that gives rise to a given type of lattice. One of the papers referred to in the commentary linked above has this to say:
In general, proving that a certain configuration is the ground state of a given potential is a very hard problem. In fact, the exact nature of the ground state is not rigorously known even for simple interactions such as the Lennard-Jones potential . In this Letter we have described
a direct method to design potentials for targeted self-assembly of lattices, a problem usually approached using iterative methods involving repeated relaxations of the system [2,3]. From our construction follows the somewhat counterintuitive observation that it is actually simpler to find a potential with a given configuration as a ground state than to determine the ground state(s) of a given potential.
Another thing about these papers is the use of the concept of the reciprocal space. May be I can use these papers to tell students the power of reciprocal space based techniques when I teach my mathematical methods course next semester.