Ground state determination, ground state preserving fit for cluster expansion and their integration for robust CE construction

Abstract

In this thesis, we propose strategies to solve the general ground state problem for arbitrary effective cluster interactions and construct ground state preserving cluster expansions. A full mathematical definition of our problem has been formalized to illustrate its generality and clarify our discussion. We review previous methods in material science community: Monte Carlo based method, configurational polytope method, and basic ray method. Further, we investigate the connection of the ground state problem with deeper mathematical results about computational complexity and NP-hard combinatorial optimization (MAX-SAT). We have proposed a general scheme, upper bound and lower bound calculation to approach this problem. Firstly, based on the traditional configurational polytope method, we have proposed a method called cluster tree optimization method, which eliminates the necessity of introducing an exponential number of variables to counter frustration, and thus significantly improves tractability. Secondly, based on convex optimization and finite optimization without periodicity, we have introduced a beautiful MAX-MIN method to refine lower bound calculation. Finally, we present a systematic and mathematically sound method to obtain cluster expansion models that are guaranteed to preserve the ground states of the reference data.