Multi-Scaling Theory and Methods for Random Fields

随机场的多尺度理论与方法

• 批准号: 9803391
• 负责人: Edward Waymire
• 金额: $ 9万
• 依托单位: Oregon State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803391&HistoricalAwards=false
• 关键词: Multi; Scaling; Theory; Methods; Random

9803391WaymireThe main focus of this research is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena in multiplicative structures that arise naturally in a wide variety of modern applications, such as river basin hydrology, fluid turbulence, financial securities markets, spin glasses etc. The essential mathematical ingredients are a space-time random field defined by a multiplicative cascade random process and a random network represented by a random tree graph, together with a system of equations directing the evolution. Much of the data collected and reported on these structures is in the form of log-log plots of some quantity versus a length scale. This leads to the introduction and refinements of new classes of self-similar spatial/temporal models whose scaling structure is inferred from empirically observed sample realizations. Thus we seek to calculate certain large-sample (fine scale) limits of statistical estimators of exponents and limit laws governing fluctuations. In addition, self-similarities and scaling exponents are sought for the cascade and network models. Then connections between the scaling exponents of the flow processes and those of the cascade and network exponents may be investigated. The prospect of a theory which computes structure functions, (i.e., multiscaling exponents) for extreme flows from corresponding structure function calculations on the inputs and network defines the frontiers of this research.A fundamental problem from environmental science is to determine the structure of river flows (e.g. extremes) from a basin given data on the local climate (e.g. rainfall) and topography (river network structure, soil moisture). In most parts of the world the information available for the planning of dams, flood insurance, military tactics etc. is in the form of remotely sensed local climate and topography. The mathematical formulation is based on a stochastic system of conservation equations (mass, momentum) which relate the flows to the multiplicative stochastic rainfall inputs and complex network topography via scaling and multiscaling exponents which are estimated from remotely sensed data. One of the practical aspects of results of this type is to assist hydrologists and engineers in extrapolating localized observations to larger scales, and to regionalize predicted flows. However, the broad mathematical framework contributes to our understanding of diverse natural stochastic phenomena such as fluid turbulence, stochastic investment yields, renewable natural resource distributions, spin glass magnets etc., which are intrinsically multiplicative in space and/or time.

9803391WaymireThe main focus of this research is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena in multiplicative structures that arise naturally in a wide variety of modern applications, such as river basin hydrology, fluid turbulence, financial securities markets, spin glasses etc. The essential mathematical ingredients are a space-time random field defined by a multiplicative cascade random process and a random network represented通过随机树图，以及指向演化的方程式系统。 在这些结构上收集和报告的许多数据都是以一定数量与长度标度的对数模块图的形式。 这导致了新的自相似空间/时间模型的引入和改进，这些模型的缩放结构是根据经验观察到的样本实现推断出来的。 因此，我们试图计算指数指数统计估计值的某些大样本（精细）限制，并限制了控制波动的法律。 此外，要为级联模型寻求自相似性和缩放指数。 然后，可以研究流程的缩放指数与级联指数和网络指数的缩放指数之间的连接。 计算结构功能的理论的前景，即（即多刻度指数），从相应的结构函数计算上对输入的相应结构功能计算和网络定义了这项研究的前沿。环境科学的基本问题是为了确定河流的结构（例如，从当地气候（例如，雨水网络）和河流网络（例如，河流）结构（例如，河流）和河流的结构（例如，河流）和河流的结构（例如，河流）和河流的结构（河流）和河流结构（河流）和河流结构（河流）结构（河流）和河流结构。 在世界上大多数地区，可用于规划大坝，洪水保险，军事战术等的信息是遥感的当地气候和地形的形式。 数学公式基于自然保护方程式（质量，动量）的随机系统，该系统通过缩放和多刻度指数从远程感知的数据估算，将流量与乘法随机降雨输入和复杂的网络地形相关联。 这种类型结果的实际方面之一是协助水文学家和工程师将局部观测提取到更大的尺度，并将预测的流量进行区域化。 然而，广泛的数学框架有助于我们理解多种自然随机现象，例如流体湍流，随机投资产量，可再生自然资源分布，自旋玻璃磁铁等，它们在空间和/或时间上具有本质上的乘法性。