Collaborative Research: Particles and Proxies for Sampling

该项目涉及高维采样，这对许多学科都很重要，包括计算化学、材料科学、气候模型、电网、交通模型或病毒和流行病研究的分子动力学模拟。该项目将开发新的仿真算法以及对现有算法的改进。其结果将在几个方面使这些学科受益。首先，算法优化将提供新的工具，从业者可以使用它来加速他们的计算。其次，这些方法的严谨结果将为从业者提供对其预测的信心。最后，将开发开源软件。学生将参与并接受跨学科培训。该项目解决了采样和复杂能量景观（如原子系统中的势能）引起的相关问题的挑战；贝叶斯推理问题中的负对数似然；或者是机器学习问题中的损失函数。在马尔可夫链蒙特卡罗方法中，这些景观通常定义了采样某些目标分布的马尔可夫链的演变。该项目将开发马尔可夫链上遍历平均的高效计算和方法，通过减少所需的迭代次数或降低每次迭代的成本来降低遍历平均的计算成本。新的技术和分析将基于代理景观和相互作用的粒子系统。代理可以减少每次迭代的成本或导致更快的收敛，而相互作用的粒子系统可以减少代理的偏差或减少方差。研究了有限粒子数下参数选择对加权系综粒子法方差的影响；开发重量校正粒子系统，以解释代理的偏差；并分析了克服与粗糙景观相关的采样困难的方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project addresses sampling in high dimensions which is important for a variety of disciplines, including computational chemistry, materials science, and molecular dynamics simulations for climate models, power network, traffic models, or the study of viruses and pandemics. The project will develop new simulation algorithms as well as improvements of existing algorithms. The outcomes will benefit these disciplines in several ways. First, the algorithmic optimizations will provide new tools that practitioners could use to accelerate their computations. Second, rigorous results on these methods will provide practitioners with confidence in their predictions. Finally, open source software will be developed. Students will be involved and receive interdisciplinary training. The project addresses challenges in sampling and related problems arising from complex energy landscapes such as in potential energy in an atomistic system; the negative log-likelihood in a Bayesian inference problem; or the loss function in a machine learning problem. In Markov Chain Monte Carlo methods, these landscapes often define the evolution of a Markov chain that samples some target distribution. This project will develop efficient computations of ergodic averages over Markov chains and methods that reduce the computational cost of ergodic averages, by either reducing the number of required iterations or reducing the per-iterate cost. The new techniques and analyses will be based on proxy landscapes and interacting particle systems. Proxies can reduce per-iterate cost or lead to faster convergence, while interacting particle systems can reduce the bias from proxies or cut down on variance. The project includes a study of how parameter choices affect the variance of the weighted ensemble particle method at finite particle number; the development of a weight-corrected particle system to account for bias from proxies; and an analysis of methods for overcoming sampling difficulties associated with rough landscapes.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.