Multiscale Sampling with Applications

多尺度采样及其应用

• 批准号: 0705910
• 负责人: Alexandre Chorin
• 金额: $ 44.39万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705910&HistoricalAwards=false
• 关键词: Multiscale; Sampling; Applications

The gist of the proposal is the development of new Monte Carlo sampling methods, where the density to be sampled is preconditioned by a nested sequence of its marginals. The probability densities of the marginals are to be determined as the sampling proceeds, using an expansion in successive linkages similar to the one used in Kadanoff's real-space renormalization. Two implementations will be explored: in one the target density and a series of its marginals will be sampled in parallel, with occasional swaps among the several parallel computations, relying on the shorter correlation times of the marginals to accelerate convergence; in the other, a single sweep from the smallest to the largest subset with available marginals will be effected, with a correction step based on an assignment of weights; this last implementation will have exactly zero temporal correlation time. At this point it is not clear which of the two may be more efficient, though it is reasonable to assume that this depends on the application. The first application will be a computer study of the three-dimensional Anderson-Edwards near-neighbors spin glass model. The second application will be to filtering and data assimilation for stochastic partial differential equations. A more distant goal is the development of more efficient training techniques for neural networks.In many problems of physics and of statistics it is necessary to sample complicated probability distributions with a very large number of variables. Current methods often fail because the successive samples they produce fail to be sufficiently independent. The present proposal suggests solving this problem by creating a sequence of successively simpler problems, in such a way that the sampling of each one makes it easier to sample the next harder one; the heart of the proposal is a methodology for making this procedure self-consistent. The first application of the idea, if it is successful, will be to the analysis of a spin glass model; this is a problem of great interest in material science, and as it is known to be very hard to sample, it is a good testing ground for the methods here. The next application will be to data assimilation; this problem arises when one tries to make predictions on the basis of a partial theory and noisy observations, as one often has to do in many fields, for example in weather forecasting or in economics; the difficulty in sampling large arrays of data is often a major roadblock in this type of situation. The spin glass model is closely related to models useful in neural networks and in neurology, and a more distant goal is to use the methods developed here in these exciting areas.

该提案的要点是发展新的蒙特卡罗采样方法，其中要采样的密度由其边缘的嵌套序列预处理。边值的概率密度将随着采样的进行而确定，使用类似于Kadanoff实空间重整化中使用的连续链接的扩展。将探讨两种实现方式：在一种实现方式中，目标密度及其一系列边缘将被并行采样，在几个并行计算之间偶尔交换，依靠边缘的较短相关时间来加速收敛;在另一种实现方式中，将实现从具有可用边缘的最小子集到最大子集的单次扫描，其中校正步骤基于权重的分配;这最后的实现将具有精确的零时间相关时间。在这一点上，还不清楚这两个可能是更有效的，虽然它是合理的假设，这取决于应用程序。第一个应用程序将是三维安德森-爱德华兹近邻自旋玻璃模型的计算机研究。第二个应用将是随机偏微分方程的过滤和数据同化。一个更遥远的目标是发展更有效的神经网络训练技术，在许多物理和统计问题中，需要对具有大量变量的复杂概率分布进行采样。目前的方法经常失败，因为它们产生的连续样本不能足够独立。本提案建议通过创建一系列依次较简单的问题来解决这个问题，以这种方式，每个问题的抽样使得下一个更难的问题更容易抽样;提案的核心是使这个程序自洽的方法。这个想法的第一个应用，如果它是成功的，将是一个自旋玻璃模型的分析;这是一个在材料科学中非常感兴趣的问题，因为它是众所周知的，很难取样，这是一个很好的测试地面的方法在这里。下一个应用将是数据同化;当人们试图根据不完整的理论和嘈杂的观测结果进行预测时，就会出现这个问题，在许多领域，例如天气预报或经济学领域，人们经常不得不这样做;在这种情况下，难以对大量数据进行抽样往往是一个主要障碍。自旋玻璃模型与神经网络和神经学中有用的模型密切相关，更远的目标是在这些令人兴奋的领域使用本文开发的方法。