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

新的采样工具，及其在量子蒙特卡罗和随机控制中的应用

• 批准号: 1217065
• 负责人: Alexandre Chorin
• 金额: $ 30万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217065&HistoricalAwards=false
• 关键词: New; Sampling; Tools; Applications; Quantum

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.

研究者和他的学生开发了多维概率密度采样的新方法。这个想法是通过隐含地制定高质量的建议密度来实现重要抽样。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态，然后通过创建一个方便的参考密度到给定概率空间的一对一映射来采样，这样模态的邻域就有很高的被采样的概率。该映射由退化代数方程的有效解算法隐式定义。两个连续的样本是独立的。该思想已被研究者成功地应用于贝叶斯估计和数据同化问题，并且研究者建议将其扩展到量子蒙特卡罗和通过路径积分公式的随机控制。对于变量离散且先前算法不适用的问题，以及概率由哈密顿量定义的问题，研究者建议首先通过在先前NSF支持下开发的快速算法估计重归一化哈密顿量序列来搜索高概率样本，然后依次对这些哈密顿量进行抽样以找到高概率样本。在第一阶段，研究人员希望将第二种想法应用于旋转玻璃和简单测量场的采样。研究者和他的学生开发了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一个物理系统的（可能不确定的）理论模型，而样本是该模型有效的事件实例；然后，可以将这些实例与（可能不确定的）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情况。这通常在气象学和经济学等领域进行。然而，在计算机资源中寻找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，但是一旦考虑到数据，大多数可能性都是极不可能的。人们想把注意力集中在可能的例子上，但通常人们事先不知道这些例子在哪里。这个困难是科学计算的主要瓶颈。研究者一直在开发采样方法，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采样来找到高概率样本；他已经成功地将这些新方法应用于海洋学和地球物理学，并建议进一步将其应用于机器人、计算化学和核物理学。