Approximate global minimizers for pairwise interaction problems
David Shirokoff, NJIT
A wide range of material systems exhibit energy driven pattern formation governed by an underlying non-convex energy functional. Although numerically finding and verifying local minima to these functionals is relatively straight-forward, the computation and verification of global minimizers is much more difficult. Here the verification of global minimizers is often important in understanding the material phase diagram, especially at low temperatures. In this talk I will examine a general class of model functionals: those arising in non-local pairwise interaction problems. I will present a new approach for computing approximate global minimizers based on a convex relaxation of the underlying energy landscape, along with a recovery technique. The approach is sometimes exact, and also provides a numerical recovery guarantee for the approximate minimizer that is often within a few percent of the global minimum. For example, the approach sometimes predicts exact minimizers which are in the form of a lattice. Finally, the approach also generates a dual decomposition for the energy that can be used to compute the emergence of new low energy length scales seen in the collective behavior of interacting particles.