New Sampling Tools, with Applications to Quantum Monte Carlo and Stochastic Control

这位研究人员和他的学生开发了抽样多维概率密度的新方法。其想法是通过隐式地制定高质量的建议密度来实现重要性抽样。对于变量连续的问题，调查者首先定位要采样的密度的模式，然后通过创建一对一的采样并从方便的参考密度映射到给定的概率空间来进行采样，从而使得模式的邻域具有很高的被采样的概率。这种映射是由退化代数方程的有效求解算法隐式定义的。两个连续的样本是独立的。这一思想已被研究者成功地应用于贝叶斯估计和数据同化中的问题，并建议将其推广到量子蒙特卡罗和通过路径积分公式进行随机控制。对于变量是离散的，以前的算法不适用，并且概率由哈密顿量定义的问题，研究者提出了搜索高概率样本的方法，首先通过使用先前的NSF支持开发的快速算法估计重整化的哈密顿量序列，然后顺序采样这些哈密顿量以找到高概率样本。在第一阶段，研究人员希望将第二个想法应用于自旋玻璃和简单规范场的采样。研究人员和他的学生开发出有效的算法来寻找给定概率分布的样本。人们可以认为给定的概率分布体现了物理系统的(可能不确定的)理论模型，并且样本作为该模型对其有效的事件的实例；然后，人们可以将这些实例与(可能不确定的)数据进行比较，并找出在给定数据和模型的情况下可能发生的事情。这在气象学和经济学等领域很常见。然而，寻找有用的样本通常是困难和昂贵的计算机资源，因为可能性的数量通常是巨大的，但一旦考虑到数据，大多数被证明是极不可能的。人们想要专注于可能的情况，但通常情况下，人们不会事先知道这些情况在哪里。这一困难是科学计算中的一个主要瓶颈。研究人员一直在开发采样方法，通过在适当抽象的问题版本中优化采样，即使在没有先验知识的情况下，也可以找到高概率样本；他已经成功地将这些新方法用于海洋学和地球物理，并建议进一步开发它们，用于机器人学、计算化学和核物理。

The investigator and his students develop new methods for sampling multidimensional probability densities. The idea is to achieve importance sampling by formulating high-quality proposal densities implicitly. For problems where the variables are continuous the investigator does this by first locating the modes of the density to be sampled, and then sampling by a creating a one-to-one and onto mapping from a convenient reference density onto the given probability space so that the neighborhoods of the modes have a high probability of being sampled. This map is defined implicitly by an efficient solution algorithm for a degenerate algebraic equation. Two successive samples are independent. This idea has been successfully applied by the investigator to problems in Bayesian estimation and in data assimilation, and the investigator proposes to extend it to quantum Monte Carlo and to stochastic control via path integral formulations. For problems where the variables are discrete and the previous algorithm is not applicable and where the probabilities are defined by a Hamiltonian, the investigator proposes to search for high-probability samples by first estimating a sequence of renormalized Hamiltonians via a fast algorithm developed with previous NSF support, and then sampling these Hamiltonians sequentially to find high-probability samples. In a first stage, the investigator expects to apply this second idea to the sampling of spin glasses and simple gauge fields.The investigator and his students develop efficient algorithms for finding samples of given probability distributions. One can think of the given probability distributions as embodying a (possibly uncertain) theoretical model of a physical system, and the samples as instances of events for which the model is valid; one can then contrast these instances with (possibly uncertain) data and find out what is likely to happen given both the data and the model. This is commonly done in fields such as meteorology and economics. However, finding useful samples is typically difficult and costly in computer resources because the number of possibilities is typical colossal, but most of them turn out to be highly unlikely once data are taken into account. One wants to focus on likely instances, but in general one does not know in advance where these are. This difficulty is a major bottleneck in scientific computing. The investigator has been developing sampling methods that can find the high probability samples, even without prior knowledge, by optimizing the sampling in suitably abstract versions of the problems; he has successfully used these new methods in oceanography and geophysics, and proposes to develop them further for use in robotics, computational chemistry, and nuclear physics.