Intrinsic Complexity of Random Fields and Its Connections to Random Matrices and Stochastic Differential Equations

随机场的内在复杂性及其与随机矩阵和随机微分方程的联系

基本信息

批准号：
1821010
负责人：
Hong-Kai Zhao
金额：
$ 10万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-15 至 2020-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1821010&HistoricalAwards=false
关键词：
Intrinsic Complexity Random Fields Its

项目摘要

Scientific computing together with effective use of data plays a more and more important role in many science and engineering applications. Modeling and understanding uncertainty or randomness inherited in these application is of utmost importance. Random fields are commonly used for modeling of space (or time) dependent stochastic processes in science and engineering problems, such as image and signal processing, Bayesian inference, data analysis, uncertainty quantification, and many other applications. It has the flexibility and generality to model randomness with spatial structures or vice versa. This project will characterize the intrinsic complexity of a random field.For the purpose of analysis as well as modeling and computation in practice, a separable representation or approximation of a random field in the form of separating deterministic and stochastic variables is very useful. This project will characterize the intrinsic complexity of a random field by providing accurate and computable lower bounds on the number of terms needed in a separable approximation of a random field for a given accuracy. This characterization can be related to the well-known notion of Kolmogorov n-width in information theory. It can reveal the intrinsic degrees of freedom (or richness) of a random field. It is also useful for an estimation of the intrinsic complexity of a system that is modeled upon a random field in real applications. For example, the investigator will study the intrinsic complexity of the solution space for partial differential equations that involve random material properties and develop efficient numerical methods that can explore low dimensional structures in these systems. By regarding a set of random vectors as the discrete sampling of a random field and vice versa, the investigator will also study the question of random vector embedding and explore its connections to random matrix theories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

科学计算与数据的有效利用在许多科学和工程应用中发挥着越来越重要的作用。建模和理解这些应用中继承的不确定性或随机性至关重要。随机场通常用于科学和工程问题中空间（或时间）相关随机过程的建模，例如图像和信号处理、贝叶斯推理、数据分析、不确定性量化和许多其他应用。它具有用空间结构对随机性进行建模的灵活性和通用性，反之亦然。该项目将表征随机场的内在复杂性。为了在实践中进行分析以及建模和计算，以分离确定性变量和随机变量的形式对随机场进行可分离的表示或近似是非常有用的。该项目将通过为给定精度的随机场的可分离近似所需的项数提供准确且可计算的下界，来表征随机场的内在复杂性。这种表征可以与信息论中众所周知的柯尔莫哥洛夫 n 宽度概念相关。它可以揭示随机场的内在自由度（或丰富度）。它对于估计根据实际应用中的随机场建模的系统的内在复杂性也很有用。例如，研究人员将研究涉及随机材料属性的偏微分方程解空间的内在复杂性，并开发可以探索这些系统中低维结构的有效数值方法。通过将一组随机向量视为随机场的离散采样，反之亦然，研究人员还将研究随机向量嵌入问题并探索其与随机矩阵理论的联系。该奖项反映了 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。