Multiscale Sampling with Applications

该建议的要点是开发新的蒙特卡罗采样方法，其中要采样的密度是由其边缘的嵌套序列预先设定的。边际的概率密度将随着采样的进行而确定，使用类似于Kadanoff实空间重正化中使用的逐次连杆的展开。将探讨两种实现：一种是并行采样目标密度及其一系列边际，偶尔在几个并行计算之间进行交换，依赖于边际的较短相关时间来加速收敛；在另一种情况下，具有可用边际的最小子集到最大子集将进行一次扫描，并根据权重分配进行校正步骤；最后一个实现的时间相关时间恰好为零。目前还不清楚这两种方法中哪一种更有效，不过可以合理地假设这取决于应用程序。第一个应用将是三维安德森-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滤波和数据同化。更长远的目标是开发更有效的神经网络训练技术。在许多物理和统计问题中，有必要对具有大量变量的复杂概率分布进行抽样。目前的方法经常失败，因为它们产生的连续样本没有足够的独立性。目前的建议建议通过创建一系列先后较简单的问题来解决这个问题，以这样一种方式，对每个问题的采样使得对下一个较难的问题的采样变得更容易；该建议的核心是使该程序自洽的方法。如果成功的话，这个想法的第一个应用将是对自旋玻璃模型的分析；这是材料科学中一个非常有趣的问题，因为它很难取样，所以它是这里的方法的一个很好的试验场。下一个应用是数据同化；当一个人试图在部分理论和嘈杂的观测的基础上做出预测时，这个问题就出现了，就像人们在许多领域经常要做的那样，例如在天气预报或经济学中；在这种情况下，对大型数据数组进行采样的困难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，更长远的目标是在这些令人兴奋的领域中使用这里开发的方法。

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.