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

研究者和他的学生开发了多维概率密度抽样的新方法。我们的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。对于变量是连续的问题，研究者首先定位要采样的密度的模式，然后通过创建一个一对一的映射来采样，并从一个方便的参考密度映射到给定的概率空间，以便模式的邻域具有很高的被采样概率。这个映射是由退化代数方程的有效解算法隐式定义的。两个连续的样本是独立的。这一想法已成功地应用于调查贝叶斯估计和数据同化的问题，调查人员建议将其扩展到量子蒙特卡罗和随机控制通过路径积分公式。对于变量是离散的问题，以前的算法是不适用的，其中的概率是由一个哈密顿定义的，研究人员建议搜索高概率的样本，首先通过一个快速算法开发与以前的NSF支持估计一个序列的重正化哈密顿，然后采样这些哈密顿顺序找到高概率的样本。在第一阶段，研究者希望将第二个想法应用于自旋玻璃和简单规范场的采样，研究者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。我们可以把给定的概率分布看作是物理系统的一个（可能是不确定的）理论模型的体现，把样本看作是模型有效的事件的实例;然后我们可以把这些实例与（可能是不确定的）数据进行对比，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，在计算机资源中找到有用的样本通常是困难和昂贵的，因为可能性的数量通常是巨大的，但一旦考虑到数据，其中大多数都是极不可能的。人们希望关注可能的实例，但通常人们事先并不知道这些实例在哪里。这一困难是科学计算中的一个主要瓶颈。研究者一直在开发抽样方法，即使没有先验知识，也可以通过优化问题的适当抽象版本的抽样来找到高概率样本;他已经成功地将这些新方法用于海洋学和海洋物理学，并建议进一步发展它们用于机器人，计算化学和核物理学。

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.