Divide-and-Conquer Approach for Strongly Interacting Systems via Convex Optimization

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Many problems in physics, data science and engineering involve interactions of so-called agents via pairwise potentials. Beyond the simple regime where the agents are independent of each other, determining the joint states of agents often suffers from complexity that increases quickly with the dimension of the problem; this feature is called the curse-of-dimensionality. Research in this project will address these challenges by exploiting a convex optimization approach and will demonstrate synergy between various aspects of computational mathematics and data science. The project will provide new tools for the Material Genome Initiative by accelerating the computation of many-body quantum system as well as improve the capability of protein structure determination from distance-based experimental measurements. Graduate students will be involved in research and will receive interdisciplinary training. The project will develop a variety of divide-and-conquer and multiscale techniques to significantly improve the scalability of algorithms for interacting agents. In addition, tensor compression strategies will be developed to accelerate the solution of subproblems. The project will demonstrate the effectiveness of the strategy for several scenarios. In the domain of data science, through the lens of the proposed optimization methods, the PI will investigate the sensor-network localization problem and multimarginal optimal transport problem. In physics and chemistry, alternative paradigms will be developed for solving for the ground state energy of strongly correlated systems such as quantum Ising and Hubbard models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.