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

物理、数据科学和工程中的许多问题都涉及到所谓的智能体之间通过两两势能的相互作用。除了代理彼此独立的简单机制之外，确定代理的联合状态通常会受到随着问题维度的快速增加的复杂性的影响；这种特征称为维度诅咒。该项目的研究将通过利用凸优化方法来解决这些挑战，并将展示计算数学和数据科学的各个方面之间的协同作用。该项目将通过加速多体量子系统的计算为材料基因组计划提供新的工具，并提高从基于距离的实验测量确定蛋白质结构的能力。研究生将参与研究，并将接受跨学科培训。该项目将开发各种分而治之和多尺度技术，以显著提高交互代理算法的可扩展性。此外，还将开发张量压缩策略，以加速子问题的求解。该项目将展示该战略在几种情况下的有效性。在数据科学领域，PI将通过提出的优化方法的镜头，研究传感器网络定位问题和多边际最优传输问题。在物理和化学领域，将开发另一种范式来求解强关联系统的基态能量，如量子伊辛和哈伯德模型。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.